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Aspects of Calabi-Yau Fourfold Compactifications Sebastian Walter Greiner

ISBN: 978-90-393-7038-4 Cover design by 123FreeVectors Printed by Gildeprint

Aspects of Calabi-Yau Fourfold Compactifications

Aspekten van Compactificaties op Calabi-Yau Variëteiten van Dimensie Vier (met een samenvatting in het Nederlands)

Proefschrift

ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magniﬁcus, prof. dr. H.R.B.M. Kummeling, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op maandag 24 september 2018 des middags te 6.00 uur

door

Sebastian Walter Greiner geboren op 9 september 1990 te Günzburg, Duitsland

Promotor: Prof. dr. S.J.G. Vandoren Copromotor: Dr. T.W. Grimm

Contents

1 Introduction 1.1 String theory as quantum gravity . . . . . . . . . . . . . 1.2 Dimensional reduction in string theory . . . . . . . . . . 1.3 F-theory on Calabi-Yau fourfolds . . . . . . . . . . . . . 1.4 Novel features of Calabi-Yau fourfolds with non-trivial cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 6 10

. . . . . . . . . odd . . . . . .

13 16

2 Calabi-Yau fourfolds with non-trivial three-form cohomology 2.1 Basic properties of Calabi-Yau fourfolds . . . . . . . . . . . . 2.2 Special ansatz for three-forms . . . . . . . . . . . . . . . . . .

19 19 22

3 Dimensional reduction of Type IIA supergravity 3.1 The eﬀective action from a Calabi-Yau fourfold reduction . . 3.2 Comments on two-dimensional N = (2, 2) supergravity . . . . 3.3 Legendre transforms from chiral and twisted-chiral scalars . .

27 27 32 36

4 Mirror symmetry at large volume and large complex structure 41 4.1 Mirror symmetry for complex and Kähler structure deformations 42 4.2 Mirror symmetry for non-trivial three-forms . . . . . . . . . . 44 5 Applications for F-theory and Type IIB orientifolds 5.1 M-theory on Calabi-Yau fourfolds . . . . . . . . . . . . . . . . 5.2 M-theory to F-theory lift . . . . . . . . . . . . . . . . . . . . . 5.3 Orientifold limit of F-theory and mirror symmetry . . . . . .

47 48 50 54

Contents 6 Geometry of toric Calabi-Yau fourfold hypersurfaces 6.1 Basic construction of toric Calabi-Yau hypersurfaces . . . . . 6.2 Cohomology of Y4 via the Gysin-sequence . . . . . . . . . . . 6.3 The geometry of toric divisors of a Calabi-Yau hypersurface . 6.4 Poincaré Residue of toric hypersurfaces . . . . . . . . . . . . . 6.5 Hodge variation in semi-ample hypersurfaces . . . . . . . . . . 6.6 Counting the Hodge numbers of a semiample Calabi-Yau fourfold hypersurface . . . . . . . . . . . . . . . . . . . . . . . . . 7 The 7.1 7.2 7.3

59 60 67 70 75 79 82

intermediate Jacobian of a Calabi-Yau fourfold hypersurface 85 Three-form periods from Riemann surfaces . . . . . . . . . . . 85 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface 92 Calabi-Yau fourfold hypersurfaces in weighted projective spaces100

8 Calabi-Yau fourfold examples 107 8.1 General aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.1.1 Weierstrass-form and non-trivial three-form cohomology 108 8.1.2 Sen’s weak coupling limit . . . . . . . . . . . . . . . . 110 8.2 Example One: An F-theory model with two-form scalars . . . 114 8.2.1 Toric data and origin of non-trivial three-forms . . . . 114 8.2.2 Picard-Fuchs equations on T 2 . . . . . . . . . . . . . . 118 8.2.3 Weak string coupling limit: a model with two-form scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.3 Example Two: An F-theory model with Wilson line scalars . 125 8.3.1 Toric data and origin of non-trivial three-forms . . . . 125 8.3.2 Comments on the weak string coupling limit . . . . . . 128 9 Conclusions and Outlook

131

Appendices 139 A Three-dimensional N = 2 supergravity on a circle . . . . . . . 139 B Twisted-chiral to chiral dualization in two dimensions . . . . . 143 Summary

147

Contents Samenvatting

151

About the Author

155

Aknowledgements

157

Bibliography

159

1 Introduction

The two cornerstones of modern fundamental physics are quantum ﬁeld theory and general relativity. Einstein’s general relativity as an extension of Newton’s theory of gravity is based on the idea that matter and energy densities curve its surrounding space-time as dictated by Einstein’s famous equations. This accounts for phenomena on astrophysical scales like the gravitational red-shift, gravitational lensing or gravitational waves as detected in 2016 by the LIGO experiment [1]. The framework of quantum ﬁeld theory is tested to be realized in nature as the Standard Model of particle physics. This theory elevates classical matter like electrons to quantum ﬁelds that interact via mediating ﬁelds, for instance photons describing light in quantum electrodynamics. Quantum electrodynamics is up to date the most well-tested theory as its predictions were observed to an astonishing accuracy beyond any other physical theory. The great success of quantum ﬁeld theory in particle physics was crowned in 2012 with the discovery of the Higgs particle at LHC [2, 3] as suggested by theorists in the 60s. This proves the realization of the Higgs mechanism in nature accounting for the generation of masses of particles. Although both theories provided great achievements for theoretical physics, unifying both into a single consistent theory of nature proves troublesome. The diﬃculties arise from the fact that general relativity loses is predictive power in the regime of high energies where a classical description of nature breaks down and we enter the realm of quantum ﬁeld theory. In nature this

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1 Introduction occurs for instance in the vicinity of black holes where the extraordinary mass density within the horizon makes a high energy description of gravity necessary. Here the size of the black hole can theoretically be smaller than its Compton wave-length requiring a full quantum treatment. In the quantum description of gravity the gravitational force is mediated by gravitons, quantized particles whose interactions with matter model the attractive forces between the latter. Due to the positive mass-dimensions of the graviton self-coupling in Einstein gravity, the graviton propagator is plagued by divergences at high energies in the standard quantum ﬁeld theory approach to gravity. Therefore a ﬁeld theoretic description of general relativity can only be an eﬀective theory at large distances which means that it only accounts for phenomena up to a certain energy scale. At this scale new physics is needed for a consistent quantum description of gravity dubbed quantum gravity. This new physics will contain additional degrees of freedom that are too massive to be observed at low energies and their interactions with the large distance spectrum is hoped to render the full theory ﬁnite. Although many alternative approaches to realize this new physics exist, the most promising candidate for a well-behaved quantum gravity is string theory [4–9].

1.1

String theory as quantum gravity

String theory is the concept of modelling the fundamental constituents of nature not via point-like, but extended objects, one-dimensional strings. The extended nature of the string allows for rich dynamics that can be split into center-of-mass movements as well as vibrational modes, similar to a violin string vibrating at a speciﬁc eigenmode. The kind of the eigenmode of the string determines its quantum numbers and hence the sort of particle it represents to an observer agnostic about the strings’ extended nature. Among this inﬁnite number of eigenmodes is also a massless symmetric two-tensor mode which behaves as a graviton in space-time. Furthermore, there are modes corresponding to matter and non-abelian gauge ﬁelds. In this sense string theory is a quantum theory of gravity, as it contains a gravitational sector that is coupled to fermions or gauge-bosons. The inﬁnite number of

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1.1 String theory as quantum gravity massive eigenmodes of the string provides the additional degrees of freedom for the new physics and resolves the high energy divergences. So far no inconsistencies were found in the string theory framework, therefore it is conjectured that string theory provides a uniﬁed framework for a quantum description of the interactions between matter and gravity. Common to all string theories is a massless real scalar ﬁeld in their spectrum called the dilaton which can be viewed as a dynamical coupling constant of the string interactions. For a weak string coupling, i.e. a small vacuum expectation value of the dilaton, we can recover a perturbative description of string theory. The interactions in perturbative quantum ﬁeld theory of point particles following their worldlines are described by one dimensional graphs called Feynman diagrams. These are replaced in perturbative string theory by two-dimensional Riemann-surfaces describing the worldsheet of a string, the two-dimensional analogue of the worldlines of point particles. In this sense perturbative string theories whose eﬀective theories might describe our world enjoy a twofold perturbative expansion in two scales, the string length and the string coupling. The perturbative expansion of scattering amplitudes of strings, perturbative in string coupling, can be viewed as summing over all possible worldsheet topologies where the contribution from each topology is suppressed by a power of string coupling corresponding to the number of holes (and handles) of the two-dimensional worldsheet. This expansion is the analogue of regular Feynman diagrams smearing out the localized interaction points of point particles to the worldsheet of interacting strings, hereby removing divergencies in the scattering amplitudes. The loops of Feynmandiagrams correspond here to the holes and handles of the worldsheet of the string. These remarkable properties put, however, strong constraints on the consistent string theories. To avoid tachyons rendering a possible vacuum unstable and to contain fermions in its spectrum, a string theory needs to be supersymmetric. A quantum ﬁeld theory endowed with supersymmetry shows a symmetry between its bosonic and fermionic degrees of freedom. The generators of these symmetries are called supercharges and map bosons to fermions and vice versa enhancing the regular symmetry group of space-time

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1 Introduction containing translations and rotations. The advantage of unbroken supersymmetry in a theory is an improved stability of the underlying quantum theory, as certain quantum corrections of fermions and bosons cancel due to the restrictions imposed by their additional symmetry. In such theories bosons and fermions combine into multiplets, irreducible representations of the enhanced Lorentz group. The number of supercharges is usually denoted by N -supersymmetry with N counting the number of irreducible representations in which the supercharges occur. In four space-time dimensions we have minimal or N = 1 supersymmetry, if there are four supercharges in the theory transforming as one real Majorana fermion or two chiral Weyl fermions. Further necessary constraints on such a superstring theory are the requirement of conformal symmetry on the world-sheet and the string propagating in a ten dimensional space-time to avoid anomalies in the two-dimensional conformal quantum ﬁeld theory. The conformal symmetry is in part responsible for the well-behaved nature of the string as it forbids the theory to have any scale allowing for the same description at all energy scales. The scale set by the string coupling only emerges by setting a vacuum expectation value for the dilaton which we choose small to ﬁnd a perturbative description of the theory. The only scale left is the string length α′ that serves as an over-all normalization of the world-sheet action. It is usually assumed to be of the same order as the Planck length. After imposing the constraints only ﬁve distinct perturbative superstring theories are left: Type IIA and Type IIB string theory, the heterotic string theory with gauge group either E8 × E8 or SO(32) and Type I string theory. As proposed by Witten in the mid 90s these ﬁve string theories are all related by dualities and can be viewed as diﬀerent limits of an eleven-dimensional theory called M-theory [10–13]. The crucial insight that led to this development was the realization that string theory contains (p + 1)-dimensional extended objects called p-branes, see [14–16] and [17] for a modern review, that are not included in the perturbative spectrum of the theory. The branes are non-perturbative as their mass scales inversely with the string coupling. For certain branes, the D-branes of Type II string theories, their ﬂuctua-

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1.1 String theory as quantum gravity tions around a background conﬁguration are, however, described by open strings ending on the D-branes that can be viewed as solitonic solutions of the underlying theory. The massless open string degrees of freedom describe a supersymmetric gauge theory, possibly non-Abelian, conﬁned to the worldvolume of the brane. As the masses of the the open strings are determined by their tension and the distance between branes, we can ﬁnd massless states at the intersection of branes. These intersections of branes can lead to chiral fermions and Yukawa couplings in their massless spectrum localized on the intersection of several branes where the open strings become massless. If we stack several branes, we obtain non-Abelian gauge theories on their common world-volume. This is a convenient way to realize the standard model on the world-volume of a D-brane which is known in the literature as brane-world scenario. At large length scales or low energies when the extended nature of the string can not be resolved by an observer, we recover point-particle physics. In this limit the tension of the strings becomes inﬁnitely high and therefore its higher excitations inﬁnitely massive. The states that survive this limit ﬁt into spectra of supergravities coupled to gauge ﬁelds, a unique ten-dimensional theory for each string theory and one unique eleven-dimensional supergravity as the low-energy limit of M-theory. Supergravity is the gauged version of supersymmetry and due to the fact that the (anti-)commutator of two supercharges is proportional to the momentum operator of the underlying space-time, gauging supersymmetry automatically implies diﬀeomorphisminvariance of the entire theory and hence it necessarily includes gravity. These supergravities are therefore eﬀective low-energy descriptions of a UVcomplete theory, provided by string theory. In this sense we recovered an eﬀective description of the low-energy physics of a consistent theory of quantum gravity. Branes appear in this context as singularities of the space-time extending the notion of a black hole to higher dimensional objects. The extended nature of a brane breaks the translation invariance of the underlying spacetime and therefore in general also some supersymmetry. Their perturbative degrees of freedom are captured by the so called Dirac-Born-Infeld (DBI)

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1 Introduction action, coupling gravity to the world volume of the brane, and topological Chern-Simons terms that model the interactions of the gauge-potentials with the background. In the weak coupling description this gives rise to gauge potentials coupled minimally to supergravity to leading order on the brane. The DBI and Chern-Simons actions of a brane are just added to the supergravity theory, in this sense we put branes into the supergravity theory by hand. The higher order interactions of the DBI theory are very hard to compute and unknown, but necessary to obtain a consistent theory. These corrections provide the interactions of the bulk or closed string degrees of freedom with the brane or open string degrees of freedom. The emerging backreaction of the branes onto the ambient geometry, however, is important to phenomenological applications as they might be responsible for conjectured no-go theorems constraining the validity of the perturbative low-energy description of string theory. A possible solution to this problem is F-theory, a non-perturbative description of type IIB string theory with seven-branes.

1.2

Dimensional reduction in string theory

Wait, we don’t live in ten space-time dimensions! So how does that ﬁt with the observation of a four dimensional space-time? One answer is compactiﬁcation, the idea that the additional dimensions predicted from string theory are curled up in tiny compact dimensions that hence avoid detection. The compact space is called the internal space, as we do not observe it, whereas the extended dimensions are called external space of the compactiﬁcation. Truncating the theory to its massless content we recover an eﬀective lowenergy description in four dimensional space-time, which is nicely described in [18]. This process to obtain a low-energy eﬀective theory from a higher dimensional theory with small extra dimensions is called dimensional reduction. We will illustrate this concept by considering a theory of a free massless scalar φ(x0 , . . . , x4 ) in ﬁve-dimensional space-time M4,1 with one compact direction Z ∂M φ ∂ M φ , M4,1 = M3,1 × S 1 . (1.1) S4,1 = M4 ,1

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1.2 Dimensional reduction in string theory The compact direction parametrizes a circle S 1 of radius R with periodic coordinate y. Expanding φ(x0 , . . . , x3 , y) into its Fourier modes φk (x0 , . . . , x3 ) as X φ(x0 , . . . , x4 ) = eiky/R φk (x0 , . . . , x3 ) , (1.2) k∈Z

we obtain an inﬁnite number of four-dimensional scalars φk (x0 , . . . , x3 ) with mass-squares k 2 . (1.3) m2k = 2πR These ﬁelds φk are often called the tower of Kaluza-Klein (KK) modes, because their masses (the levels of the tower) increase with the internal momentum k ∈ Z along the circle. In an eﬀective theory where the additional dimension avoids detection, at energies E ≪ 1/R only the lowest component of the KK-tower φ0 can be excited and we ﬁnd an eﬀective four-dimensional theory of a free scalar Z ∂µ φ0 ∂ µ φ0 . (1.4) S3,1,ef f = 2πR M3 ,1

This illustrates dimensional or Kaluza-Klein reduction that allows to derive an eﬀective theory for small additional dimensions. This eﬀective theory is a perturbation theory around a vacuum which is a solution of the full equations of motion of the higher dimensional theory. In the above example the vacuum solution is given by the space M3,1 × S 1 where the ﬁeld φ has a zero vacuum expectation value. The motivation of Kaluza [19] and Klein [20] for their original work was somewhat diﬀerent than our modern approach coming from string theory. Their goal was to unify Einstein’s general relativity with Maxwell’s theory of electro-magnetism or in today’s language a U (1)-gauge-ﬁeld theory. Starting from a ﬁve-dimensional theory of pure gravity, they compactiﬁed it on a circle and derived the eﬀective theory around a ﬂat vacuum. In this construction the ﬁve-dimensional metric g(5) splits into the four-dimensional metric g(4) , a massless four-dimensional vector A and a massless real scalar φ. (4) g A (5) g = . (1.5) AT φ

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1 Introduction Remarkably, the ﬁve-dimensional diﬀeomorphism invariance descends to a U (1) gauge-symmetry of the four-dimensional vector which can be traced back to the periodicity of the circle. The real scalar is basically the radius or size of the circle and controls the coupling of the four-dimensional gaugetheory. In broader terms the size of the circle is a modulus of the internal geometry, a ﬁeld with no potential. This expectation value of the modulus, here the radius of our background, determines the couplings in our eﬀective theory. We ﬁnd in particular that the geometry of the internal space has a crucial inﬂuence on the physics of the eﬀective theory. Although the reduction as proposed by Kaluza and Klein uniﬁes indeed gravity and electro-magnetism, it only does so on the classical level as the ﬁve-dimensional quantum theory already suﬀered from inconsistencies. The modern approach is to take as a starting point ten- or eleven-dimensional supergravity which we argued to be UV-completed by string theory or Mtheory and perform the dimensional reduction on a six- or seven-dimensional internal space. In this work we will consider only compactiﬁcations to ﬂat Minkowskispace and the background values of the ﬁeld strengths of the gauge-potentials will be set to zero. Furthermore do we only consider classical space-times as vacua where also the fermions have vanishing vacuum expectations value. For the lower dimensional theory to retain a certain amount of supersymmetry restricts the internal space to have a so called Killing spinor, i.e. the internal geometry allows for fermionic symmetries. This restricts the holonomy of the internal D-dimensional Riemannian manifold which we assume to allow for spinors. In this case is the holonomy group generated by the transformations of a spinor when parallel transported along closed curves in the manifold and is in general Spin(D). To preserve supersymmetry we need to restrict to subgroups of the holonomy group. In D = 6 dimensions Calabi-Yau manifolds serve as convenient backgrounds for string compactiﬁcations [21– 24], as they have SU (3) ⊂ Spin(6) ≃ SU (4) holonomy allowing for Killing spinors. They not only preserve supersymmetry, but are also Ricci-ﬂat and therefore solve Einstein’s equation for space-times without matter. The sixdimensional Calabi-Yau manifolds which are complex manifolds are called

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1.2 Dimensional reduction in string theory Calabi-Yau threefolds as they have three complex dimensions. The eﬀective low-energy descriptions of string theory in four ﬂat spacetime dimensions M3,1 preserving supersymmetry are therefore chosen to be perturbations around a supergravity background of the form M9,1 = M3,1 × Y3 ,

ehφi = gs ,

(1.6)

where Y3 is a Calabi-Yau threefold of a very small length scale. The vacuum expectation of the dilaton φ, the real scalar common to a all superstring theories determines the string coupling gs of the perturbation theory. It can be thought of as the radius in the compactiﬁcation of M-theory on a circle. Giving the remaining ﬁelds background values breaks supersymmetry in general spontaneously and generates a potential for the ﬁelds rendering them massive. This ﬁxes the ﬁelds at the minimum of the potential, i.e. stabilizes them at their vacuum expectation values. Non-zero vacuum expectation values of the various p-form ﬁeld strengths are called ﬂuxes. Compactiﬁcations on Calabi-Yau backgrounds usually leads to hundreds of massless ﬁelds without potential, the moduli of the eﬀective ﬁeld theory. These moduli need to be stabilized at high mass scales as they are not observed in the four dimensional space-time. A convenient scenario to do so are called ﬂux compactiﬁcations, [25–29], where the non-zero ﬁeld strength values of the background generate a potential for the moduli. So far no dynamical mechanism to determine the correct vacuum for the eﬀective low-energy theory is known and therefore we need to construct these backgrounds by hand. A convenient framework for these constructions are the Type II theories, as a background of the form (1.6) preserves eight supercharges, N = 2 supersymmetry, which is then spontaneously broken by ﬂuxes in the internal space. This approach provides a certain amount of stability to corrections. The downside is that these backgrounds do not allow for non-Abelian gauge theories or chiral matter in the four-dimensional external space. This can be cured by introducing space-time ﬁlling D-branes into the background. The corresponding D-branes need to wrap non-trivial cycles of the internal space to account for their tension and create a stable background. Since they break translational invariance of the internal space, they also break part of

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1 Introduction the supersymmetry. The tension of the D-branes and their charges sourcing the gauge ﬁelds provide a non-trivial energy-momentum density of the background and hence it can no longer be Ricci-ﬂat to solve Einstein’s equations. There are two solutions to this problem, either we consider a complex internal manifold B3 with positive curvature which is no longer Calabi-Yau or we introduce negative tension objects called orientifold planes or O-planes on the CalabiYau space Y3 . Both solutions are related by a so called orientifold-involution σ, mapping Y3 holomorphically to itself with ﬁxed point loci the O-planes. The quotient of Y3 by the action of σ is precisely the positive curvature background B3 which contains only D-branes. The four-dimensional eﬀective theories and their properties were derived in a series of papers [30–34]. Constructing these so called Type II orientifold ﬂux backgrounds is a tedious task as we need to pay attention to various consistency conditions, like tadpole constraints, anomaly cancellation and preserving a weak coupling description. An elegant geometric solution to these complications is given by F-theory.

1.3

F-theory on Calabi-Yau fourfolds

The F-theory framework as proposed by Vafa in 1996, see [35], is the best understood description of string theory as it allows to include certain nonperturbative eﬀects not accessible by other approaches. Starting point is Type IIB string theory whose massless bosonic spectrum contains a complex scalar, called the axio-dilaton τ , deﬁned as the combination τ = C0 + ie−φ ,

eφ = gs .

(1.7)

Here C0 is the zero-form gauge potential of Type IIB and φ the dilaton, whose vacuum expectation value determines the string coupling gs . Type IIB string theory and its low-energy supergravity obey a self-duality called S-duality under which the axio-dilaton transforms as aτ + b a b ∈ SL(2, R) , (1.8) , τ → c d cτ + d

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1.3 F-theory on Calabi-Yau fourfolds which is further broken to SL(2, Z) in the full string theory due to quantum eﬀects. This duality contains in particular the mapping 1 τ →− , τ

(1.9)

which allows for a strongly coupled region in the ten-dimensional space-time to have a dual weakly coupled description. The discrete duality in (1.8) is the reparametrization invariance of the complex structure modulus τ of a torus T 2 = C/Z + τ Z . (1.10) Therefore, we can interpret the ten-dimensional Type IIB string theory as a compactiﬁcation of a putative twelve-dimensional F-theory on a torus T 2 with complex structure modulus τ varying over the ten-dimensional spacetime. This geometry is called a torus ﬁbration over the space-time. The problems with this picture are that there is no interpretation for the volume of the torus, which should appear in a compactiﬁcation as a modulus in the eﬀective theory, and the fact that there is no twelve-dimensional supergravity with signature (1, 11). The more practical approach is to understand F-theory as a certain limit of M-theory. Here M-theory is compactiﬁed on a torus ﬁbration with the torus complex structure modulus τ , as for example described in [36]. We call the two cycles of the torus A-cycle and B-cycle which are both circles. M-theory on a circle can be interpreted as Type IIA String Theory with coupling the radius of the circle. As we go to strong coupling in Type IIA we decompactify the space-time and recover the eleven-dimensional M-theory description. This can also be seen as the deﬁnition of M-theory. The second duality we will exploit is the duality between Type IIA String Theory on a cirlce and Type IIB String Theory on a circle with inverse radius. This duality is known as T-duality and interchanges states with momentum along the circle direction with stringy states that wind around the circle. To obtain the desired F-theory vacuum, we ﬁrst compactify M-theory on the A-cycle of the torus leading to Type IIA String Theory and then compactify further on the B-cycle and use T-duality to obtain a Type IIB vacuum with axio-dilaton τ in nine dimensions. In the following we will restrict to

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1 Introduction a nine-dimensional space-time of the form M2,1 × B3 with B3 a three complex dimensional manifold which is Kähler. We further restrict to elliptic ﬁbrations for which τ depends on the coordinates of B3 holomorphically, preserving a certain amount of supersymmetry in the eﬀective theory. An elliptic curve is a torus with a marked point, for example the origin in (1.10). It can be shown in particular that due to the holomorphiciity of the ﬁbration the volume of the elliptic ﬁber is constant over B3 . Sending then this volume modulus of the torus to zero, the B-cycle radius becomes inﬁnite, the vacuum decompactiﬁes and we recover an eﬀectively four-dimensional Lorentz-invariant Type IIB vacuum with axio-dilaton τ . To preserve minimal supersymmetry in four dimensions the total space of the elliptic ﬁbration over B3 needs to be Calabi-Yau. The resulting space has four complex dimensions and is hence called an elliptically ﬁbered Calabi-Yau fourfold Y4 whose geometry and the eﬀective physics of its string theory compactiﬁcations will be the central topic of this thesis. They were ﬁrst studied in [37–42] and their eﬀective theories were discussed in [43–47] for Type II string theories, in [48, 49] for M-theory and in [50] for F-theory. The advantage of this construction is that the elliptic ﬁbration can degenerate, i.e. a cycle along the torus ﬁber may shrink to zero size over complex codimension one loci. At these points τ has a pole indicating the presence of a charged object which is called a space-time ﬁlling seven-brane. These are eight dimensional objects in M3,1 × B3 that wrap four-cycles in B3 and are charged magnetically under C0 . If there is more then one seven-brane at the same locus in the base B3 , for example they stack or intersect, the elliptic ﬁber degenerates further and the Calabi-Yau fourfold Y4 becomes singular. At these degeneration loci in B3 we ﬁnd non-Abelian gauge groups for stacks of seven-branes where the number of branes in the stack corresponds to the rank of the gauge group. At the intersection of two branes matter localizes and at points where three branes intersect Yukawa couplings emerge. These properties are captured by the geometry of the elliptic ﬁbration of Y4 . For further details we refer to the great review [51]. These F-theory vacua allow in general only for local weakly coupled descriptions due to the local SL(2, Z) symmetry of our theory. Vacua for which

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1.4 Novel features of Calabi-Yau fourfolds with non-trivial odd cohomology such a global weak coupling limit exists are scarce and were ﬁrst discussed by Sen [52, 53]. This scarcity renders the F-theory framework much more general than the regular weakly coupled Type IIB description. In the dual M-theory picture on the Calabi-Yau fourfold Y4 the degrees of freedom of the seven-branes are geometric moduli of Y4 , geometrizing the seven-branes and accounting for their full dynamics. This geometrization blurrs the distinction of closed (bulk) and open (brane) string degrees of freedom in the dual type IIB orientifold compactiﬁcations and allows a description of the eﬀective four-dimensional physics of Type IIB string theory at ﬁnite coupling (general τ ). From another point of view, there is no need any more for the Type IIB interpretation that distinguishes between open and closed strings as both are captured by the geometric deformations of the Calabi-Yau fourfold geometry. In general this distinction does not even exist! This motivates the study of the geometry of Calabi-Yau fourfolds and its deformations as it is essential to the computation of the eﬀective ﬁeld theory dynamics as for example described in [54,55]. In particular, it is possible to construct explicit examples of these Calabi-Yau fourfolds and to derive the couplings of the theory in terms of geometrical quantities. A simple class of these in general very complicated constructions, toric hypersurfaces, will be central to the second part of this work.

1.4

Novel features of Calabi-Yau fourfolds with non-trivial odd cohomology

In this short subsection we want to highlight the novel results that were obtained in course of our studies. As we conveyed so far, F-theory on elliptically ﬁbered Calabi-Yau fourfolds provide an interesting arena for string phenomenology, the study of the low-energy eﬀective theories of string theory. The goal of this active research ﬁeld is to derive observable consequences of string theory. As emphasized earlier, F-theory provides enough ﬂexibility to derive the details of its eﬀective theory, but also contains brane dynamics and is inherently non-perturbative. To achieve these features it uses the

13

1 Introduction machinery of algebraic geometry that provides us with powerful tools that allow calculations of spectra and couplings. As F-theory can be interpreted as a generalization of the weakly coupled Type IIB orientifold vacua, it also contains novel features and uniﬁes complicated physics of the perturbative theory. The particular feature we study in this work are non-trivial harmonic three-forms on the underlying Calabi-Yau fourfold. The corresponding massless modes have no analogue in Calabi-Yau threefolds that are characterized by their geometric moduli, the Kähler moduli determining volumina and the complex structure moduli that describe the shape of the underlying threefold. The three-form moduli of the F-theory description, however, defy a simple geometric interpretation and behave diﬀerently as they arise not from the metric of the theory, but from the three-form gauge-potential of M-theory or Type IIA string theory. Consequently, the inclusion in the topological string theory framework like in [56,41,57] is elusive, even for type II compactiﬁcations on Calabi-Yau fourfolds. This can be related to the fact that in the conformal ﬁeld theory approach, for example using orbifolds of Landau-Ginzburg vacua [58, 59], the states corresponding to the three-forms arise from twisted sectors and are not marginal deformations of the N = (2, 2) SCFT as they have Rcharges (2, 1). Consequently, the topological twists of the SCFT that lead to the familiar A-model (Kähler deformations) and to the B-model (complex structure deformations) do not apply. Therefore, their kinetic terms, usually given by the metric on the moduli space of marginal deformations of the SCFT can not be described by the Zamolodchikov metric [60, 61]. Due to this complications and the fact that the simplest examples of Calabi-Yau fourfolds have a trivial three-form cohomology, the inclusion of three-forms is hardly discussed in the literature. In course of this thesis we will ﬁnd that in the models when non-trivial three-forms are present, we can no longer discuss Kähler and complex structure deformations independently as the metric of the three-form moduli depends on both. Another approach to include the three-form moduli into the low-energy eﬀective theory is to start with a supergravity theory and perform a dimensional reduction on a general Calabi-Yau fourfold. This was done ﬁrst by

14

1.4 Novel features of Calabi-Yau fourfolds with non-trivial odd cohomology Haack and Louis [43] that compactiﬁed Type IIA supergravity on a fourfold with non-trivial non-trivial three-forms. For technical reasons it became evident that in the expansion of the three-form gauge-potential a three-form basis depending on the complex structure moduli was necessary. This was artiﬁcially solved by introducing tensors mapping a topological (constant) harmonic three-form basis to a non-constant three-form basis. Consistency required these tensors to depend in a complicated way on the complex structure moduli described by a set of diﬀerential equations. A geometric interpretation of these tensors and a solution of the diﬀerential equations was, however, not found. First progress in this direction was made ten years later in [50] as a reﬁned ansatz for the harmonic three-forms was put forward. This enabled a clearer derivation of the eﬀective theory and showed that a certain subset of the three-form moduli are dual to U (1) gauge-bosons in the weak coupling description of F-theory in four dimensions. The complex structure dependence of this novel ansatz for the harmonic three-forms was captured by a matrix valued function fAB (z) that depends on the complex structure moduli z and determines the coupling of the resulting U (1) gauge-theory. In a later publication [62], it was observed that a diﬀerent kind of three-form moduli gives rise to axions in the eﬀective theory and again fAB determined the coupling and hence the axio-decay constant relevant for inﬂationary models. Due to the crucial role of the matrix fAB as coupling in the eﬀective theory, the question we strive to answer in this thesis is What is fAB ? In course of this work we will not only give a beautiful geometric interpretation of this function fAB , but also ﬁnd the right ﬁeld basis to include the three-form moduli in the general supergravity framework and calculate their couplings. Furthermore will we extend the usual toric constructions of Calabi-Yau fourfolds to account for this novel three-form sector and discuss the geometry of elliptic ﬁbrations with non-trivial three-form cohomology. As we go through our discussion we will encounter a number of interesting and often puzzling features that deserve further research, as for instance the

15

1 Introduction shift-symmetries of the eﬀective theories investigated in [63]. In the upcoming section we will give a brief outline of this thesis.

1.5

Outline

We start this thesis by introducing the geometric properties of Calabi-Yau fourfolds and their harmonic forms in chapter 2 where we focus especially on the harmonic three-forms whose physical properties are determined by the complex structure dependence of their so called normalized period matrix. The ﬁrst part of the thesis focuses on eﬀective theories of Calabi-Yau fourfolds. In chapter 3 we perform the dimensional reduction of Type IIA supergravity on a general Calabi-Yau fourfold and ﬁnd a N = (2, 2) dilatonsupergravity in two dimensions. Here we determine the massless spectrum and the kinetic potential of the resulting supergravity. We conjecture an extension of the usual Kählerpotential of supergravity theories by a term depending on the dilaton of the theory to account for the kinetic coupling of the novel three-form scalars. Subsequently we apply mirror symmetry in chapter 4, a duality of the same supergravity theories for two diﬀerent CalabiYau fourfolds, to determine the structure of the normalized period matrix at the large complex structure point in moduli space. Following this discussion we extend our analysis to the dimensional reduction of eleven-dimensional supergravity in chapter 5. Afterwards we lift the resulting three-dimensional N = 2 supergravity to the eﬀective theory of F-theory on elliptically ﬁbered Calabi-Yau fourfolds leading to an eﬀective N = 1 supergravity description in four dimensions. We close by discussing the implications of non-trivial three-form cohomology and defer some technical details of the ﬁrst part into two appendices in A,B. In the second part of this thesis we construct explicit Calabi-Yau fourfold examples via hypersurfaces in toric varieties. We begin in chapter 6 with the general construction of toric spaces and their hypersurfaces. We focus on Calabi-Yau fourfolds realized as so called semi-ample hypersurfaces avoiding the Lefschetz-hyperplane theorem that forbids non-trivial three-form coho-

16

1.5 Outline mology. This is followed by a discussion of the origin of non-trivial harmonic forms on the hypersurface and we state well known formulas for the number of the various harmonic forms in terms of toric data. We added several clariﬁcations and extensions of the published work to include also non-toric and non-algebraic deformations of toric hypersurfaces. In chapter 7 we discuss the moduli space of the three-form scalars which is called intermediate Jacobian of a Calabi-Yau fourfold whose complex structure dependence reduces to the complex structure dependence of Riemann surfaces. The Kähler dependence is captured by a so called generalized sphere-tree that can be computed in terms of the ambient space. We determine in this situation the geometrical quantities necessary to derive the eﬀective theories discussed before. This part is completed by a lengthy discussion of two simple examples of Calabi-Yau fourfold geometries with non-trival three-form cohomology in chapter 8. We conclude the thesis in chapter 9 giving an outlook of possible future research directions. This thesis is based on two papers [64, 65] in collaboration with Thomas W. Grimm and one proceedings article [66]. • S. Greiner and T. W. Grimm, “Three-form periods on Calabi-Yau fourfolds: Toric hypersurfaces and F-theory applications”, JHEP 1705 (2017) 151, arXiv:1702.03217 [hep-th]. • S. Greiner and T. W. Grimm, “On Mirror Symmetry for Calabi-Yau Fourfolds with Three-Form Cohomology”, JHEP 1609 (2016) 073, arXiv:1512.04859 [hep-th]. • S. Greiner, “On Mirror Symmetry for Calabi-Yau Fourfolds with ThreeForm Cohomology”, PoS CORFU 2016 (2017) 102, arXiv:1704.07658 [hep-th].

17

2 Calabi-Yau fourfolds with non-trivial three-form cohomology

In this section we lay the groundwork for our dimensional reductions int the coming sections by introducing the central points of Calabi-Yau fourfold geometries. Here we begin with a precise deﬁnition of Calabi-Yau fourfolds and move then on to the geometric properties like the Kähler- and complex structure. For the mathematical details we refer to [24, 67, 68]. In this ﬁrst part we will also account for the non-trivial harmonic forms of this class of manifolds that will later determine the spectrum of our eﬀective theories. In the second part of this section we will discuss in detail the harmonic threeforms for which we will choose a special and novel ansatz that will be central to the whole body of this thesis.

2.1

Basic properties of Calabi-Yau fourfolds

We deﬁne in this thesis a Calabi-Yau fourfold Y4 as a eight-dimensional Riemannian manifold with metric g and holonomy group the full special unitary group SU (4). As a consequence, Y4 will be Kähler and hence admit a closed Kähler two-form J compatible with both the complex structure and the metric of Y4 . In addition, as was conjectured by Calabi and shown by Yau, there is a unique Ricci-ﬂat Kähler metric within the class of J ∈ H 1,1 (Y4 ). On such Calabi-Yau fourfolds exists a unique, up to rescaling by a holomorphic

19

2 Calabi-Yau fourfolds with non-trivial three-form cohomology function, nowhere-vanishing holomorphic (4, 0)-form Ω ∈ H 4,0 (Y4 ). This holomorphic (4, 0)-form Ω only depends on the complex structure, whereas the Kähler form J depends only on the Kähler moduli of the underlying space. The complex structure moduli can be interpreted as deﬁning the shape of Y4 and the Kähler moduli specify the volumes of its cycles. From Ω and J we can construct top-forms of degree eight, which have to be related. This relation is given by 1 1 ¯, J ∧J ∧J ∧J = Ω∧Ω 4! |Ω|2

1 |Ω| = V 2

Z

Y4

¯ Ω∧Ω

(2.1)

where we deﬁned indirectly the total volume of Y4 that only depends on J and hence the Kähler moduli as Z 1 J ∧J ∧J ∧J. (2.2) V= 4! Y4 Due to the SU (4) holonomy of the underlying space the existence of a complex covariantly constant spinor with deﬁnite chirality. This allows to obtain J and Ω as bilinear contractions of this spinor and hence to show their existence. The invariant spinor is obtained from the splitting of the general chiral spinor representation 8s of an eight-dimensional Riemannian manifold that admits spinors as Spin(8) → SU (4)

8s → 6 ⊕ 1 ⊕ 1 ,

(2.3)

where the two singlets combine to the complex covariantly constant spinor. Therefore compactiﬁcations of Y4 without ﬂuxes to a ﬂat space-time preserve 1/8 of the supercharges in the eﬀective theory. Another important feature of Calabi-Yau fourfolds is the very restrictive Hodge diamond displaying the dimensions of the complex cohomology groups hp,q = dim H p,q (Y4 ) ,

n M p=0

20

H n−p,p(Y4 ) = H n (Y4 , C) ,

(2.4)

2.1 Basic properties of Calabi-Yau fourfolds in the following way: h0,0 h1,0 h2,0 h3,0 h4,0

h3,1

h4,1

h4,3

h0,4

h1,4

h2,4

h2,1

0 =

h3,1

1

0 h1,1

0

h0,3

h1,3

h2,3

h3,3

0

h0,2

h1,2

h2,2

h3,2

h4,2

h0,1

h1,1

h2,1

1

h2,2

h2,1

0

0

h4,4

0

h3,1

h2,1

h1,1

0

h3,4

0

h2,1

1

0

0

0 1

Due to the severe restrictions arising from the speciﬁc holonomy group there can be no non-trivial holomorphic forms beside the trivial zero-form and Ω. Hodge-duality and Poincaré duality lead to symmeties of Hodge numbers between the horizontal and the vertical axis. The axis indicated on the left hand side of the graphic is the symmetry axis of mirror symmetry, a relation between two diﬀerent manifolds with Hodge numbers symmetric with respect to this axis. This will be discussed in great detail later in this work. Applying index theorems, one can show that also h2,2 can be calculated from h1,1 , h1,2 and h3,1 as h2,2 (Y4 ) = 2(22+2h1,1 +2h3,1 −h2,1 ) , χ(Y4 ) = 6(8+h1,1 +h3,1 −h2,1 ) (2.5) P where χ = p,q (−1)p+q hp,q is the Euler characteristic of Y4 . This makes h1,1 , h1,2 and h3,1 the independent quantities of Y4 that will specify the spectrum of the resulting eﬀective theory. The Hodge number h1,1 counts the number of independent Kähler deformations denoted by δv Σ . We have the expansion for the Kählerform J = v Σ ωΣ ,

Σ = 1, . . . , h1,1 ,

(2.6)

where ωΣ are harmonic (1, 1)-forms. In contrast h3,1 counts the number of complex structure deformations denoted by δz K and we use the complex

21

2 Calabi-Yau fourfolds with non-trivial three-form cohomology structure coordinates zK ,

K = 1, . . . , h3,1 .

(2.7)

This is very similar to the threefold case. In contrast to the threefold, however, also h2,1 counts the number of three-form moduli NA ,

A = 1, . . . , h2,1 ,

(2.8)

which have no threefold analogue. These new deformations arise from the expansion of a gauge three-form into a special ansatz for the harmonic threeforms on Y4 which will be discussed in the next section.

2.2

Special ansatz for three-forms

In this section we want to discuss the ansatz for the harmonic three-forms that will keep us busy throughout this work. In course of the upcoming dimensional reductions, it will become necessary to calculate ∗ φ(3) ∈ H 5 (Y4 , C) ,

φ(3) ∈ H 3 (Y4 , C) .

(2.9)

The problem with this ansatz is, that the Hodge-star operator ∗ of Y4 will depend on the metric g and therefore also on the Kähler- and complex structure moduli in our theory. Therefore, even if we choose φ(3) topological, ∗φ(3) will still depend on the other moduli. It can be shown that for a Calabi-Yau fourfold Y4 we can calculate the Hodge star of a three-form if it has certain Hodge type. For a Calabi-Yau fourfold the cohomology group of three-forms splits as H 3 (Y4 , C) = H 2,1 (Y4 ) ⊕ H 1,2 (Y4 ) . (2.10) The two subspaces correspond to ±iJ∧ subspaces of the Hodge-star operator ∗ on Y4 as ∗ ψ = iJ ∧ φ ψ ∈ H 2,1 (Y4 ) . (2.11) Therefore it is desirable to choose a basis of three-forms with a deﬁnite Hodge-type ψA ∈ H 2,1 (Y4 ) , 22

ψ¯A ∈ H 1,2 (Y4 ) ,

A = 1, . . . , h2,1 .

(2.12)

2.2 Special ansatz for three-forms The split of H 3 (Y4 , C) into these two subspaces is done via the induced complex structure J that varies with the complex structure moduli z K . This is a real endomorphism on H 3 (Y4 , C) that squares to the identity. H 2,1 (Y4 ) and H 1,2 (Y4 ) are the ±i eigenspaces of this endomorphism. Therefore we can choose a real basis αA , β A ∈ H 3 (Y4 , R) for a ﬁxed complex structure that satisﬁes J (αB ) = δBA β A , J (β B ) = −δBA αA . (2.13)

From this we can construct the desired basis of (2, 1)-forms as ψA = αA + iδAB β B ,

⇒

J (ψA ) = δAB β B − iαA = −iψA .

(2.14)

From the general theory of complex structure variations it is known that H 2,1 (Y4,z ) varies holomorphically with the complex structure moduli z K . Here and in the following we denote by Y4,z the Calabi-Yau fourfold with a speciﬁc complex structure z K that is a small deformation around a reference point where we deﬁned the topological basis (2.13). Therefore, we can make the ansatz ˜ AB (z)β B ∈ H 2,1 (Y4,z ) , ψA (z) = ΠA B (z)αB + Π

(2.15)

and since both matrices are invertible as seen before (they are proportional to the identity matrix at the reference point in complex structure space) we can make the ansatz ψA (z) = αA + ifAB (z)β B ∈ H 2,1 (Y4,z ) ,

(2.16)

where the real part Re fAB of the holomorphic endomorphism-valued function f (z)AB is invertible for small deformations around the reference complex structure. The matrices ΠA B and ΠAB are called period matrices and we will refer to them as three-form period matrices. The three-form periods are the column-vectors of these matrices. The quotient of these two matrices C˜ fAB = (Π−1 )A Π CB

(2.17)

is called the three-form normalized period matrix. This matrix is one of the central topics of this thesis and we will explore both its physical applications as well as calculate it for explicit examples.

23

2 Calabi-Yau fourfolds with non-trivial three-form cohomology In course of this work, it will also be important to calculate the Hodge star of ∂ ψA ∈ H 2,1 (Y4 ) ⊕ H 1,2 (Y4 ) . (2.18) ∂z K So we also want the ﬁrst derivatives of our three-form basis to have a deﬁnite Hodge-type. This can be achieved by a rescaling with the inverse of the real AB part of fAB we denote by Re f AB = (Re f )−1 1 ΨA (z, z¯) = Re f AB αB − if¯BC β C ∈ H 1,2 (Y4 ) . 2

(2.19)

This new basis is, however, not anymore (anti-)holomorphic in the complex structure moduli. The 1/2 factor in front has only cosmetically reasons. The advantage is that we can calculate ∂ ΨA = −Re f AB ∂z K Re fBC ΨC , ∂z K ∂ ¯C . ΨA = −Re f AB ∂z¯K Re fBC Ψ ∂ z¯K

(2.20) (2.21)

and hence also the derivatives of ΨA have a deﬁnite Hodge type. In course of this work, we will ﬁnd that the complex scalars NA appear in the eﬀective actions of M -theory and Type IIA-string theory. These have kinetic terms determined by a positive bilinear form Z Z ¯B . ΨA ∧ ∗ ΨB = iv Σ H(ΨA , ΨB ) = ωΣ ∧ ΨA ∧ Ψ (2.22) Y4

Y4

up to a constant factor. Here we used the Hodge-star operator ∗ on Y4 which simpliﬁes on a (1, 2)-form to ∗ΨA = −iJ ∧ ΨA and we expanded the Kähler form J as in (2.6). We can use the expansion of ΨA in (2.19) into topological three-forms to ﬁnd 1 H(ΨA , ΨB ) = − Re f BC v Σ MΣC A + ifCD MΣ DA 2 where we introduced the topological intersection numbers Z Z B B AB MΣA = ωΣ ∧ αA ∧ β , MΣ ωΣ ∧ β A ∧ β B , = Y4

24

Y4

(2.23)

(2.24)

2.2 Special ansatz for three-forms that will play an important role throughout this thesis. Note that we can choose without loss of generality a basis of topological three-forms with αA ∧ αB = 0. In contrast the intersection numbers MΣ AB do not vanish in general. The form (2.23) will be important for the derivation of eﬀective theories, for the geometric calculations on Calabi-Yau hypersurfaces in toric ambient spaces, however, evaluating H on the holomorphic (2, 1)-forms (2.15) is more convenient. This is related to (2.23) via multiplication with appropriate multiples of Re fAB and reads H(ψA , ψB ) = 2 Re fAC v Σ MΣB C + ifBD MΣ DC .

(2.25)

From the previous analysis we see that not only the normalized period matrix fAB , but also the intersection numbers MΣA B , MΣ AB need to be calculated to fully understand the metric H. As we will see in the upcoming sections these two quantities are related by mirror symmetry at certain points in moduli space. To understand this further, we will ﬁrst perform dimensional reductions of Type IIA supergravity on a Calabi-Yau fourfold Y4 in the next section.

25

3 Dimensional reduction of Type IIA supergravity

In this section we perform the dimensional reduction of Type IIA supergravity on a Calabi-Yau fourfold Y4 . Such reductions have already been performed in [43, 48, 45, 44]. Our analysis follows [43, 48, 45], but we will apply in addition the improved understanding about the three-form cohomology of section 2.2.

3.1

The effective action from a Calabi-Yau fourfold reduction

The Kaluza-Klein reduction of Type IIA supergravity can be trusted in the limit in which the typical length scale of the physical volumes of submanifolds of Y4 are suﬃciently large compared to the string scale. This limit is referred to as the large volume limit. Furthermore, these typical length scales set the Kaluza-Klein scale which we assume to be suﬃciently above the energy scale of the eﬀective action. We therefore keep only the massless Kaluza-Klein modes in the following reduction. Our starting point will be the bosonic part of the ten-dimensional Type

27

3 Dimensional reduction of Type IIA supergravity IIA action in string-frame given by 1 Z 1ˇ 1ˇ (10) −2φˇIIA ˇ ˇ ˇ ˇ ˇ ˇ R ∗1 + 2dφIIA ∧ ∗dφIIA − H3 ∧ ∗H3 SIIA = e 2 4 Z 1 ˇ 4 ∧ ˇ∗F ˇ4 + B ˇ2 ∧ Fˇ4 ∧ Fˇ4 , − Fˇ2 ∧ ˇ ∗Fˇ2 + F 4

(3.1)

ˇ 3 = dB ˇ2 is the ﬁeld strength where φˇIIA is the ten-dimensional dilaton, H ˇ2 , and Fˇp = dCˇp are the ﬁeld strengths of the of the NS-NS two-form B R-R p-forms Cˇ1 and Cˇ3 . We also have used the modiﬁed ﬁeld strength ˇ 4 = Fˇ4 − Cˇ1 ∧ H ˇ 3 . Here and in the following we will use a check to indicate F

ten-dimensional ﬁelds. The background solution around which we want to consider the eﬀective theory is taken to be of the form M1,1 ×Y4 , where M1,1 is the two-dimensional Minkowski space-time, and Y4 is a Calabi-Yau fourfold with properties introduced in chapter 2. As pointed out there such a manifold admits one complex covariantly constant spinor of deﬁnite chirality. This spinor can be used to dimensionally reduce the N = (1, 1) supersymmetry of Type IIA supergravity to obtain a two-dimensional N = (2, 2) supergravity theory. In particular, the two ten-dimensional gravitinos of opposite chirality reduce to two pairs of two-dimensional Majorana-Weyl gravitinos with opposite chirality. We will have more to say about the supersymmetry properties of the two-dimensional action in section 3.2. Furthermore, recall that Y4 admits a (8) Ricci-ﬂat metric gmn and one can thus check that a metric of the form (8) dˇ s2 = ηµν dxµ dxν + gmn dy m dy n ,

(3.2)

solves the ten-dimensional equations of motion in the absence of background ﬂuxes.2 Note that in (3.2) we denote by xµ the two-dimensional coordinates of the space-time M1,1 , whereas the eight-dimensional real coordinates on the Calabi-Yau fourfold Y4 are denoted by y m . 1 2

Note that for convenience we have set κ2 = 1. The inclusion of background fluxes complicates the reduction further. In particular, it requires to introduce a warp-factor. The M-theory reduction with warp-factor was recently performed in [69–71].

28

3.1 The effective action from a Calabi-Yau fourfold reduction The massless perturbations around this background both consist of ﬂuc(8) tuations of the internal metric gmn that preserve the Calabi-Yau condition ˇ2 , Cˇ1 , Cˇ3 and the dilaton φˇIIA . as well as the ﬂuctuations of the form ﬁelds B The metric ﬂuctuations give rise to the real Kähler structure moduli v Σ , Σ = 1, . . . , h1,1 (Y4 ) that preserve the complex structure and are given by gi¯ + δgi¯ = −iJi¯ = −iv Σ (ωΣ )i¯ ,

(3.3)

where J is the Kähler form on Y4 and ωΣ comprises a real basis of harmonic (1, 1)-forms spanning H 1,1 (Y4 ). The Kähler structure moduli appear also in the expression of the total string-frame volume V of Y4 given by Z Z 1 J ∧J ∧J ∧J. (3.4) ∗1 = V≡ 4! Y4 Y4 In addition to the Kähler structure moduli one ﬁnds a set of complex structure moduli z K , K = 1, . . . , h3,1 (Y4 ). These ﬁelds parameterize the change in the complex structure of Y4 preserving the class of its Kähler form J. Inﬁnitesimally they are given by the ﬂuctuations δz K as δg¯ı¯ = −

1 lmn (χK )lmn¯ δz K , Ω¯ı 2 3|Ω|

(3.5)

where Ω is the (4, 0)-form, the χK form a basis of harmonic (3, 1)-forms spanning H 3,1 (Y4 ), and |Ω|2 was already given in (2.1). The Kaluza-Klein ansatz for the remaining ﬁelds takes the form ˇ 2 = bΣ ω Σ , B Cˇ1 = A , ¯A Ψ ¯A , Cˇ3 = V Σ ∧ ωΣ + NA ΨA + N

(3.6)

where ΨA is a basis of harmonic (1, 2)-forms spanning H 1,2 (Y4 ) as introduced in (2.19). A discussion of the properties of ΨA was already given in section 2.2. Finally, we dimensionally reduce the Type IIA dilaton by dropping its dependence on the internal manifold Y4 . It turns out to be convenient to deﬁne a two-dimensional dilaton φIIA in terms of the ten-dimensional dilaton φˇIIA as ˇ e2φIIA . (3.7) e2φIIA ≡ V 29

3 Dimensional reduction of Type IIA supergravity In summary, we ﬁnd in the two-dimensional N = (2, 2) supergravity theory the 2h1,1 (Y4 ) + 1 real scalar ﬁelds v Σ (x), bΣ (x), φIIA (x) as well as the h3,1 (Y4 ) + h2,1 (Y4 ) complex scalar ﬁelds z K , NA . In addition there are h1,1 (Y4 ) + 1 vectors A, V Σ , which carry, however, no physical degrees of freedom in a two-dimensional theory if they are not involved in any gauging. Since the eﬀective action considered here contains no gaugings, we will drop these in the following analysis. To perform the dimensional reduction one inserts the expansions (3.3), (3.5), (3.6), and (3.7) into the Type IIA action (3.1). It reduces to the two-dimensional action Z 1 (2) (3.8) S = e−2φIIA R ∗ 1 + 2dφIIA ∧ ∗φ IIA 2 Z z L + GΣΛ dtΣ ∧ ∗dt¯Λ − e−2φIIA GKL¯ dz K ∧ ∗d¯ Z 1 Σ AB ¯B + i dΣ AB dbΣ ∧ NA D N ¯B − DNB N ¯A . − v dΣ DNA ∧ ∗D N 2 4 We note that the NS-NS part, which is summarized in the ﬁrst line of (3.1), reduces to the ﬁrst line of (3.8), while the R-R part, i.e. the second line of (3.1), reduces to the second line of (3.8). Let us introduce the various objects appearing in the action (3.8). First, we have deﬁned the complex coordinates tΣ ≡ bΣ + iv Σ ,

(3.9)

which combine the Kähler structure moduli with the B-ﬁeld moduli. Furthermore, we have introduced the metric 3 GΣΛ =

1 4V

Z

Y4

ωΣ ∧ ∗ ωΛ = −

1 1 KΣΛ − KΣ KΛ , 8V 18V

(3.10)

where V, KΣ and KΣΛ are given in terms of the quadruple intersection num3

The second equality follows from the cohomological identity ∗ ωΣ = − 21 ωΣ ∧ J ∧ J + 1 V −1 KΣ J ∧ J ∧ J. 36

30

3.1 The effective action from a Calabi-Yau fourfold reduction bers KΣΛ as Z KΣΛΓ∆ = V=

Y4

(3.11)

ωΣ ∧ ωΛ ∧ ωΓ ∧ ω∆ ,

1 KΣΛΓ∆ v Σ v Λ v Γ v ∆ , 4!

KΣ = KΣΛΓ∆ v Λ v Γ v ∆ ,

KΣΛ = KΣΛΓ∆ v Γ v ∆ .

With these deﬁnitions at hand, we can further evaluate the metric GΣΛ and show that it can be obtained from a Kähler potential as (3.12)

GΣΛ = −∂tΣ ∂t¯Λ log V .

Also the metric GKL¯ is actually a Kähler metric. It only depends on the complex structure moduli z K and takes the form R Z χcK ∧ χ ¯L ¯. Ω∧Ω (3.13) log ∂ = −∂ GKL¯ = − YR4 L K z¯ z ¯ Ω∧Ω Y Y4

4

Note that both GΣΛ and GKL¯ are actually positive deﬁnite and therefore deﬁne physical kinetic terms in (3.8). Both terms scale with the dilaton φIIA and it is easy to check that this dependence cannot be removed using a Weyl-rescaling of the two-dimensional metric. We will show in section 3.2 that this is consistent with the form of the N = (2, 2) dilaton supergravity. Let us now turn to the R-R part of the action (3.1) and discuss the couplings appearing in the second line of (3.8). First, we introduce the coupling function Z AB ¯ B = − 1 (Ref )AC MΣC B . ωΣ ∧ ΨA ∧ Ψ dΣ ≡i (3.14) 2 Y4 Here and in the following we assume the second intersection number MΣ BC in (2.24) to vanish for simplicity. This is in particular satisﬁed for the examples we will consider in chapter 6. We have used the fact that we can choose the three-form basis (αA , β A ) with αA ∧ αB = 0 to evaluate the second equality, and MΣ BC = 0 to show the third equality. One also checks the relation Z Z A B AB ¯ B = v Σ dΣ AB , ¯ =i J ∧ ΨA ∧ Ψ Ψ ∧ ∗Ψ (3.15) H ≡ Y4

Y4

31

3 Dimensional reduction of Type IIA supergravity where we have used (2.11) for the (1,2)-forms ΨA . This contraction gives precisely the positive deﬁnite metric of the complex scalars NA in (3.8). It turns out to be convenient to write 1 1 H AB = v Σ dΣ AB = − (Re f )AC v Σ MΣC B ≡ − (Re f )AC Re hC B , (3.16) 2 2 where hC B = −itΣ MΣC B with the intersection number MΣC B of (2.24). Note that H AB thus depends non-trivially on the complex structure moduli z K through the holomorphic functions fAB and on the Kähler structure moduli tΣ through the holomorphic function hC B . H AB is the metric on the threeform moduli space as deﬁned in (2.23). Second, we note that the modiﬁed derivative DNA appearing in (3.8) is a shorthand for DNA = dNA − 2 Re NC (Re f )CB ∂z K (Re fBA ) dz K .

(3.17)

¯A = DN A in the action of (3.8). Using this expresWe use the notation D N sion one easily reads oﬀ the coeﬃcient function in front of dNA ∧ ∗dz K and checks that it can be obtained by taking derivatives of a real function. In the next subsection we show that this is true for all terms in (3.8) and discuss the connection with two-dimensional supersymmetry.

3.2

Comments on two-dimensional N = (2, 2) supergravity

Having performed the dimensional reduction we next want to comment on the supersymmetry properties of the action (3.8). As pointed out already in the previous subsection the counting of covariantly constant spinors on the Calabi-Yau fourfold suggests that the two-dimensional eﬀective theory admits N = (2, 2) supersymmetry. It was pointed out in [44] that, at least in the case of h2,1 (Y4 ) = 0 one expects to be able to bring the action (3.8) into the standard form of an two-dimensional N = (2, 2) dilaton supergravity. In this work the dilaton supergravity action was constructed using superspace techniques. Earlier works in this direction include [72–74]. In the following we comment on this matching for h2,1 (Y4 ) = 0 and then discuss the general case in which h2,1 (Y4 ) > 0.

32

3.2 Comments on two-dimensional N = (2, 2) supergravity In order to display the supergravity actions we ﬁrst have to introduce two sets of multiplets containing scalars in two-dimensions: (1) a set of chiral multiplets with complex scalars φκ , (2) a set of twisted-chiral multiplets with complex scalar σ Σ . In a superspace description these multiplets obey the two inequivalent linear spinor derivative constraints leading to irreducible representations. To discuss the actions we ﬁrst focus on the case h2,1 (Y4 ) = 0 and follow the constructions of [44]. For simplicity we will not include gaugings or a scalar potential. The superspace action used in [44] is given by Z (2) (3.18) Sdil = d2 xd4 θE −1 e−2V −K . Here E −1 is the superspace measure, V is a real superﬁeld with V | = ϕ as lowest component, and K is a function of the chiral and twisted-chiral multiplets with lowest components φκ and σ Σ , respectively. To display the bosonic part of the action (3.18) we ﬁrst set e−2ϕ˜ = e−2ϕ−K ,

(3.19)

where K(φκ , φ¯κ , σ Σ , σ ¯ Σ ) is evaluated as a function of the bosonic scalars. With this deﬁnition at hand one ﬁnds the bosonic action Z (2) −2ϕ ˜ 1 (3.20) R ∗ 1 + 2dϕ˜ ∧ ∗dϕ˜ − Kφκ φ¯λ dφκ ∧ ∗dφ¯λ Sdil = e 2 σ Λ − KσΣ φ¯λ dφ¯λ ∧ dσ Σ , σ Λ − Kφκ σ¯ Λ dφκ ∧ d¯ + KσΣ σ¯ Λ dσ Σ ∧ ∗d¯ where Kφκ φ¯λ = ∂φκ ∂φ¯λ K, Kφκ σ¯ Σ = ∂φκ ∂σ¯ Σ K with a similar notation for the other coeﬃcients. It is now straightforward to compare (3.20) with the action (3.8) for the case h2,1 (Y4 ) = 0, i.e. in the absence of any complex scalars NA . One ﬁrst identiﬁes ϕ˜ = φIIA ,

φκ = z K ,

σ Σ = tΣ ,

(3.21)

and then infers that K = − log

Z

Y4

¯ + log V . Ω∧Ω

(3.22)

33

3 Dimensional reduction of Type IIA supergravity Note that we ﬁnd here a positive sign in front of the logarithm of V. This is related to the fact that there is an extra minus sign in the kinetic terms of the twisted-chiral ﬁelds σ Σ in (3.20). Clearly, the kinetic terms of the complex structure deformations z K and complexiﬁed Kähler structure deformations tΣ in the action (3.8) have both positive deﬁnite kinetic terms.4 Let us now include the complex scalars NA . It is important to note that the action (3.8) cannot be brought into the form (3.20). In fact, we see in (3.8) that the terms independent of the two-dimensional metric do not contain an φIIA -dependent pre-factor, while the terms of this type in (3.20) do admit an ϕ-dependence. ˜ Any ﬁeld redeﬁnition in (3.8) involving the dilaton seems to introduce new undesired mixed terms that cannot be matched with (3.20) either. However, we note that the action (3.8) actually can be brought to the form Z (2) −2ϕ ˜ 1 ˜ κ ¯λ dφκ ∧ ∗dφ¯λ (3.23) R ∗ 1 + 2dϕ˜ ∧ ∗dϕ˜ − K S = e φ φ 2 ˜ Σ ¯λ dφ¯κ ∧ dσ Σ , ˜ φκ σ¯ Λ dφκ ∧ d¯ ˜ σΣ σ¯ Λ dσ Σ ∧ ∗d¯ σΛ − K σΛ − K +K σ φ ˜ is now allowed to be dependent on ϕ˜ and given by where K ˜ = K + e2ϕ˜ S , K

(3.24)

Similar to K, the new function S is allowed to depend on the chiral and twisted-chiral scalars φκ , σ Σ , but is taken to be independent of ϕ. ˜ The action (3.23) trivially reduces to (3.20) for S = 0. Note that the new terms induced by S do not scale with e−2ϕ˜ . Comparison with (3.8) reveals that one can identify ϕ˜ = φIIA , φκ = (z K , NA ) , σ Σ = tΣ , (3.25) and introduce the generating functions Z ¯ + log V , Ω∧Ω K = − log

(3.26)

Y4

S = H AB Re NA Re NB , 4

H AB ≡ v Σ dΣ AB .

Our discussion differs here from the one in [44], where the sign in front of log V was claimed to be negative.

34

3.2 Comments on two-dimensional N = (2, 2) supergravity To show this, it is useful to note that dΣ AB can be evaluated as in (3.14) and depends on the complex structure moduli through the holomorphic function fAB (z) only. Let us close this subsection with two remarks. First, note that (3.23) is expected to be compatible with N = (2, 2) supersymmetry and gives an extension of the two-dimensional dilaton supergravity action (3.18). A suggestive form of the extended superspace action is Z S (2) = d2 xd4 θE −1 e−2V −K + S , (3.27)

where S is now evaluated as a function of the chiral and twisted-chiral superﬁelds. It would be interesting to check that (3.27) indeed correctly reproduces ˜ as in (3.24). the bosonic action (3.23) with K Second, the action (3.23) with the identiﬁcation (3.25) can also be straightforwardly obtained by dimensionally reducing M-theory, or rather elevendimensional supergravity, ﬁrst on Y4 and then on an extra circle of radius r. The reduction of M-theory on Y4 was carried out in [43, 48]. We give the resulting three-dimensional action in (5.5) and brieﬂy recall this reduction in section 5.1 when considering applications to F-theory. Using the standard relation of eleven-dimensional supergravity on a circle and Type IIA supergravity, one straightforwardly identiﬁes r = e−2φIIA ,

e2φIIA v Σ =

Σ vM ≡ LΣ , VM

(3.28)

Σ and V Σ where vM M are the analogs of v and V used in the M-theory reduction. Note that the scalars LΣ are the appropriate ﬁelds to appear in three-dimensional vector multiplets. Inserting the identiﬁcation (3.28) into (3.24) together with (3.25), (3.26) one ﬁnds Z ¯ + log 1 KΣΛΓ∆ LΣ LΛ LΓ L∆ ˜ M = − log Ω∧Ω K 4! Y4

+ LΣ dΣ AB Re NA Re NB ,

(3.29)

where we have dropped the logarithm containing the circle radius. Indeed ˜ M agrees precisely with the result found in [43,48,50] from the M-theory reK duction. The general discussion of the circle reduction of a three-dimensional

35

3 Dimensional reduction of Type IIA supergravity un-gauged N = 2 supergravity theory to a two-dimensional N = (2, 2) supergravity theory can be found in A.

3.3

Legendre transforms from chiral and twisted-chiral scalars

In this subsection we want to introduce an operation that allows to translate the dynamics of certain chiral multiplets to twisted-chiral multiplets and vice versa. More precisely, we will assume that some of the scalars, say the scalars λA , in the N = (2, 2) supergravity action have continuous shift symmetries, i.e. λA → λA + cA for constant cA . These scalars therefore only appear with derivatives dλA in the action. By the standard duality of massless p-forms to (D − p − 2)-forms in D dimensions, one can then replace the scalars λA by dual scalars λ′ A . Accordingly, one has to adjust the complex structure on the scalar ﬁeld space by performing a Legendre transform. In the following we will give representative examples of how this works in detail. We will see that this duality, in particular as described in the ﬁrst example, becomes crucial in the discussion of mirror symmetry of chapter 4. As a ﬁrst example, let us consider the above theory with complex scalars in chiral multiplets and complex scalars tΣ in twisted-chiral multi˜ was given in (3.24) with (3.26). plets. The kinetic potential for these ﬁelds K Two facts about this example are crucial for the following discussion. First, the ﬁelds NA admit a shift symmetry NA → NA + icA in the action, i.e. the ˜ given in (3.26) is independent of NA − N ¯A . Second, kinetic potential K the NA only appear in the term S of the kinetic potential and thus carry no dilaton pre-factor in the action. One can thus straightforwardly dualize ¯A into real scalars λ′ A . The new complex scalars N ′ A are then given NA − N by 1 ∂S + iλ′ A , (3.30) N′A = 2 ∂Re NA z K , NA

where we have included a factor of 1/2 for later convenience. Furthermore, ˜ ′ is now a function of z K , tΣ , N ′ A and given by the new kinetic potential K

36

3.3 Legendre transforms from chiral and twisted-chiral scalars the Legendre transform ˜′ = K ˜ − 2 e2ϕ˜ Re N ′ A Re NA , K

(3.31)

where Re NA has to be evaluated as a function of Re N ′ A and the other complex ﬁelds by solving (3.30) for Re NA . One now checks that the scalars N ′ A actually reside in twisted-chiral multiplets. Using the transformation (3.30) and (3.31) in the action (3.23) simply yields a dual description in which certain chiral multiplets are consistently replaced by twisted-chiral multiplets. It is simple to evaluate (3.30), (3.31) for S given in (3.26) to ﬁnd N ′ A = H AB Re NB + iλ′ A , ˜ ′ = K − e2φIIA HAB Re N ′ A Re N ′ B , K

(3.32) (3.33)

where H AB is the inverse of the matrix HAB introduced in (3.15), (3.16). It is interesting to realize that upon inserting (3.32) into (3.33) one ﬁnds that ˜ ′ evaluated as a function of NA only diﬀers by a minus sign in front of the K ˜ This simple transformation arises term linear in e2φIIA from the original K. ˜ is only quadratic in the NA . This observation will be from the fact that K crucial again in the discussion of mirror symmetry in chapter 4. As a second example, we brieﬂy want to discuss a dualization that transforms all multiplets containing scalars to become chiral. The detailed computation for a general N = (2, 2) setting can be found in B. For the example of section 3.2 we focus on the twisted-chiral multiplets with complex scalars tΣ . These admit a shift symmetry tΣ → tΣ + cΣ for constant cΣ , such that Re tΣ only appears with derivatives in the action. Accordingly, the kinetic ˜ is independent of tΣ + t¯Σ as seen in (3.24) with (3.26). Due to potential K the shift symmetry we can dualize the scalars tΣ + t¯Σ to scalars ρΣ . However, note that by using the kinetic potential (3.24), (3.26) there are couplings of tΣ in (3.23) that have a dilaton factor eϕ˜ , and others that are independent of eϕ˜ . This seemingly prevents us from performing a straightforward Legendre transform to bring the resulting action to the form (3.23) with only chiral multiplets. Remarkably, the special properties of the kinetic potential (3.24), (3.26), however, allow us to nevertheless achieve this goal as we will see in the following.

37

3 Dimensional reduction of Type IIA supergravity The action (3.23) for a setting with only chiral multiplets with complex scalars M I takes the form Z I J (2) −2ϕ ˜ 1 ¯ R ∗ 1 + 2dϕ˜ ∧ ∗dϕ˜ − KM I M¯ J dM ∧ ∗dM , (3.34) S = e 2 where KM I M¯ J = ∂M I ∂M¯ J K. In other words, the potential K is in this case actually a Kähler potential on the ﬁeld space spanned by the complex coordinates M I . For our example (3.24), (3.26) the scalars M I consist of z K , NA , and TΣ , where TΣ are the duals of the complex ﬁelds tΣ . We make the following ansatz for the dual coordinates TΣ TΣ = e−2ϕ˜

˜ ∂K ∂S ∂K + iρΣ = e−2ϕ˜ + + iρΣ , Σ Σ ∂Im t ∂Im t ∂Im tΣ

(3.35)

and the dual potential K ˜ − e2ϕ˜ Re TΣ Im tΣ . K=K

(3.36)

These expressions describe the standard Legendre transform for Im tΣ , but crucially contain dilaton factors e2ϕ˜ . This latter fact allows to factor out e−2ϕ˜ as required in (3.34), but requires to perform a two-dimensional Weyl rescaling as we will discuss below. Using (3.24) with (3.26) one straightforwardly evaluates 1 KΣ + dΣ AB Re NA Re NB + iρΣ , TΣ = e−2φIIA 3! V Z ¯ + log V . Ω∧Ω K = − log

(3.37) (3.38)

Y4

Clearly, upon using the map (3.28) this result is familiar from the study of M-theory compactiﬁcations on Calabi-Yau fourfolds [43, 48, 50]. Also note that the contribution S present in the kinetic potential (3.24) is removed by the Legendre transform in K and reappears in a more involved deﬁnition of the coordinates TΣ . At ﬁrst it appears that (3.35) induces new mixed terms involving one dϕ˜ due to the dilaton dependence in front of the derivatives of K. Interestingly, 38

3.3 Legendre transforms from chiral and twisted-chiral scalars these can be removed by a two-dimensional Weyl rescaling if K satisﬁes the conditions Σ ¯Λ (3.39) KvΣ d Im tΣ = df , KtΣ Kt t Kt¯Λ = k , for some constant k and some real ﬁeld dependent function f . In this exΣ Λ pression Kt t¯ is the inverse of KtΣ t¯Λ and KvΣ ≡ ∂Im tΣ K. In fact, one can perform the rescaling g˜µν = e2ω gµν , which transforms the Einstein-Hilbert action as Z Z −2ϕ ˜ 1 ˜ −2ϕ ˜ 1 e R˜ ∗1 = e R ∗ 1 − 2dω ∧ ∗dϕ˜ , (3.40) 2 2 while leaving all other terms invariant. Using (3.40) to absorb the mixed terms one needs to chose f k (3.41) ω = − ϕ˜ − . 2 2 The details of this computation can be found in B. Indeed, for the example (3.26) one ﬁnds f = log V and k = −4. Remarkably, the condition (3.39) essentially states that K has to satisfy a no-scale like condition. A recent discussion and further references on the subject of studying four-dimensional supergravities satisfying such conditions can be found in [75].

39

4 Mirror symmetry at large volume and large complex structure

In chapter 3 we have determined the two-dimensional action obtained from Type IIA supergravity compactiﬁed on a Calabi-Yau fourfold. We commented on its N = (2, 2) supersymmetry structure which relies on the proper identiﬁcation of chiral and twisted-chiral multiplets in two dimensions. In this section we are exploring the action of mirror symmetry. More precisely, we consider pairs of geometries Y4 and Yˆ4 that are mirror manifolds [76,41,37]. From a string theory world-sheet perspective one expects the two theories obtained from string theory on Y4 and Yˆ4 to be dual. This implies that after ﬁnding the appropriate identiﬁcation of coordinates the two-dimensional effective theories should be identical when considered at dual points in moduli space. We will make this more precise for the large volume and large complex structure point in this section. Note that in contrast to mirror symmetry for Calabi-Yau threefolds the mirror theories encountered here are both arising in Type IIA string theory.1

1

This can be seen immediately when employing the SYZ-understanding of mirror symmetry as T-duality [77]. Mirror symmetry is thereby understood as T-duality along half of the compactified dimensions, i.e. Y4 is argued to contain real four-dimensional tori along which T-duality can be performed. Clearly, this inverts an even number of dimensions for Calabi-Yau fourfolds.

41

4 Mirror symmetry at large volume and large complex structure

4.1

Mirror symmetry for complex and Kähler structure deformations

Mirror symmetry arises from the observation that the conformal ﬁeld theories associated with Y4 and Yˆ4 are equivalent. It describes the identiﬁcation of Calabi-Yau fourfolds Y4 , Yˆ4 with Hodge numbers hp,q (Y4 ) = h4−p,q (Yˆ4 ) .

(4.1)

Note that this particularly includes the non-trivial conditions h1,1 (Y4 ) = h3,1 (Yˆ4 ) ,

h3,1 (Y4 ) = h1,1 (Yˆ4 ) , h2,1 (Y4 ) = h2,1 (Yˆ4 ) ,

(4.2) (4.3)

The ﬁrst identiﬁcation (4.2) together with the observations made in chapter 3 implies that mirror symmetry exchanges Kähler structure deformations of Y4 (Yˆ4 ) with complex structure deformations of Yˆ4 (Y4 ). Accordingly one needs to exchange chiral multiplets and twisted-chiral multiplets in the eﬀective N = (2, 2) supergravity theory. The second identiﬁcation (4.3) seems to suggest that for the ﬁelds NA the mirror map is trivial. However, as we will see in section 4.2 this is not the case and one has to equally change from a chiral to a twisted-chiral description. To present a more in-depth discussion of mirror symmetry we ﬁrst need to introduce some notation. All ﬁelds and couplings obtained by compactiﬁcation on Y4 are denoted as in chapter 3. To destinguish them from the quantities obtained in the Yˆ4 reduction we will dress the latter with a hat. In particular for the ﬁelds we write Y4 : Yˆ4 :

φIIA , tΣ , z K , NA , ˆA . φˆIIA , tˆK , zˆΣ , N

(4.4)

Note that we have exchanged the indices on tˆK and zˆΣ in accordance with the fact that complex structure and Kähler structure deformations are interchanged by mirror symmetry. In other words, K = 1, . . . , h1,1 (Yˆ4 ) and A = 1, . . . , h3,1 (Yˆ4 ) is compatible with the previous notation due to (4.2).

42

4.1 Mirror symmetry for complex and Kähler structure deformations Similarly we will adjust the notation for the couplings. For example, the functions introduced in (3.26) and (2.17), (3.15) are Y4 : Yˆ4 :

fAB (z) , H AB (v, z) , ˆ AB (ˆ fˆAB (ˆ z) , H v , zˆ) .

(4.5) (4.6)

The functional form of the various couplings will in general diﬀer for Y4 and Yˆ4 . A match of the two mirror-symmetric eﬀective theories should, however, be possible when identifying the mirror map, which we denote formally by M[·]. We want to focus on the sector of the theory independent of the threeforms. Recall that in the two-dimensional eﬀective theory obtained from Y4 the kinetic terms of the complex structure moduli z K and Kähler structure moduli tΣ are obtained from the kinetic potential (3.22), (3.26) as Z 1 Σ Λ Γ ∆ ¯, Ω∧Ω (4.7) K(Y4 ) = log − log KΣΛΓ∆ Im t Im t Im t Im t 4! Y4 when used in the action (3.20). Mirror symmetry exchanges the Kähler moduli tΣ of Y4 with the complex structure moduli zˆΣ of Yˆ4 . The expression (4.7) was computed at the large volume point in Kähler moduli space, i.e. with the assumption that Im tΣ ≫ 1 in string units. Accordingly one has to evaluate K(Yˆ4 ) at the large complex structure point as Z ˆ ˆ ∧Ω ¯ = 1 KΣΛΓ∆ Im zˆΣ Im zˆΛ Im zˆΓ Im zˆ∆ , (4.8) Ω 4! Yˆ4 where now Im zˆΣ ≫ 1. Similarly, one has to proceed for the Kähler moduli part of the kinetic potential K(Yˆ4 ) and evaluate K(Y4 ) at the large complex structure point Z ¯ = 1K ˆ KLMN Im z K Im z L Im z M Im z N , Ω∧Ω (4.9) 4! Y4 ˆ KLM N are now the quadruple intersection numbers on the geometry where K Yˆ4 . Therefore, at the large volume and large complex structure point the two eﬀective theories obtained from Y4 and Yˆ4 are identiﬁed under the mirror map M tΣ = zˆΣ , M z K = tˆK , (4.10) 43

4 Mirror symmetry at large volume and large complex structure and M K(Y4 ) = −K(Yˆ4 ) ,

M φIIA = φˆIIA .

(4.11)

It is important to stress that a sign change occurs when applying the mirror map to K. This can be traced back to the fact that scalars in chiral and twisted-chiral multiplets have diﬀerent sign kinetic terms in the actions (3.20), (3.23). The quantum corrections to K were discussed using mirror symmetry in [76, 41, 37, 78] and localization in [79, 80] (using and extending the results of [81–83]).

4.2

Mirror symmetry for non-trivial three-forms

Let us next include the moduli NA arising for Calabi-Yau fourfolds Y4 with non-vanishing h2,1 (Y4 ). In chapter 3 we have seen that these complex scalars are part of chiral multiplets. Their dynamics was described by the real func˜ given in (3.24) and (3.26). For completeness tion S in the kinetic potential K we recall that S(Y4 ) = H AB Re NA Re NB ,

H AB ≡ v Σ dΣ AB ,

(4.12)

where dΣ AB is a function of the complex structure moduli of Y4 . Mirror ˆA arising in the reduction symmetry should map the ﬁelds NA to scalars N on the mirror Calabi-Yau fourfold Yˆ4 , i.e. one should have ˆ , zˆ, tˆ) , M NA = QA (N

(4.13)

where we have allowed the image of NA to be a non-trivial function that will be determined in the following. In fact, note that the map cannot be ˆA . As already pointed out in [44] the mirror duals as simple as M(NA ) = N M(NA ) need to be, in contrast to the NA , parts of twisted-chiral multiplets. To achieve this we need to use the results of section 3.3. Let us therefore consider the reduction on Yˆ4 using the same notation as in chapter 3 but with hatted symbols. The two-dimensional theory will contain ˆA that reside in chiral multiplets. We can transform a set of complex scalars N

44

4.2 Mirror symmetry for non-trivial three-forms them to scalars in twisted-chiral multiplets using (3.32) and (3.33). In other ˆ ′A deﬁned as words, we ﬁnd a dual description with scalars N ˆ′ A , ˆ′A = H ˆ AB Re N ˆB + iλ N

(4.14)

ˆ AB is a function of the mirror complex structure moduli zˆΣ and where H Kähler moduli vˆK . The dual kinetic potential takes the form ˜ ′ (Yˆ4 ) = K(Yˆ4 ) − e2φˆIIA H ˆ AB Re N ˆ ′ A Re N ˆ′B . K

(4.15)

The mirror map (4.10), (4.11) and (4.13) exchanges chiral and twisted-chiral states and therefore has to take the form ˆ ′ A (N ˆ , zˆ, tˆ) , M tΣ = zˆΣ , M z K = tˆK , (4.16) M NA = N ˜ 4 ) = −K ˜ ′ (Yˆ4 ) , (4.17) M K(Y M φIIA = φˆIIA .

ˆA , zˆΣ and tˆK by using (4.14). and is evaluated as a function of N Using these insights we are now able to infer the mirror image of the ˜ 4 ). To do that, we apply the mirror map to function HAB appearing in K(Y ˜ the kinetic potential K. Note that ˜ 4 ) = −K(Yˆ4 ) + e2φˆIIA M S(Y4 ) , M K(Y

(4.18)

where we have used (4.11). Furthermore, we insert (4.17) into (4.12) to ﬁnd X AB ˆ ′A Re N ˆ ′B . M S(Y4 ) = M H Re N (4.19) A,B

We next apply (4.17) together with (4.15) which requires X A,B

! ˆ ˆ ′ A Re N ˆ′B , ˆ ′A Re N ˆ ′B = HAB Re N M H AB Re N

(4.20)

and thus enforces ! ˆ AB . M H AB = H

(4.21)

We therefore ﬁnd that the mirror map actually identiﬁes H AB with the inˆ AB of H ˆ AB . This inversion is crucial and stems from the exchange of verse H

45

4 Mirror symmetry at large volume and large complex structure chiral an twisted-chiral multiplets under mirror symmetry. In the ﬁnal part of this section we evaluate the condition (4.21) at the large complex structure point, since H AB given in (4.12) was computed at large volume. Using the mirror map we are now able to determine the holomorphic function fAB appearing in the deﬁnition of HAB at the large complex structure point. Note that (3.16) translates on Y4 and Yˆ4 to 1 H AB = − Re f AC Re hC B , 2 ˆC B , ˆ AB = − 1 Re fˆAC Re h H 2

hC B = −itΣ MΣC B ,

(4.22)

ˆ C B = −itˆK M ˆ KC B , h

where on the mirror geometry we introduced the intersection numbers Z B ˆ MKA = ω ˆK ∧ α ˆ A ∧ βˆB . (4.23) Yˆ4

Using (4.16), (4.17), (4.21), and (4.22) in the mirror map one infers that a possible identiﬁcation is 2 ˆ KA B . Re fAB = Im z K M

(4.24)

By holomorphicity of fAB we ﬁnally conclude ˆ KA B . fAB = −iz K M

(4.25)

Having determined the function fAB at the large complex structure point we have established a complete match of the two two-dimensional eﬀective theories obtained from Y4 and Yˆ4 under the mirror map M[·]. The result (4.25) is not unexpected. In fact, from the variation of Hodge-structures one could have expected a leading linear dependence on z K . Furthermore, we will ﬁnd agreement with a dual Calabi-Yau threefold result when using the geometry Y4 as F-theory background and performing the orientifold limit. This will be the task of the next section. 2

Note that in general the basis (αA , β A ) might not directly map to (α ˆ A , βˆA ) on the ˆ mirror geometry Y4 . In this expression we have assumed that there is no non-trivial base change under mirror symmetry.

46

5

Applications for F-theory and Type IIB orientifolds

In this section we want to apply the result obtained by using mirror symmetry to compactiﬁcations of F-theory and their orientifold limit. The F-theory eﬀective action is studied via the M-theory to F-theory limit. Therefore, we will brieﬂy review in section 5.1 the dimensional reduction of M-theory on a smooth Calabi-Yau fourfold including three-form moduli. This reduction was already performed in [48], but we will use the insights we have gained in the previous sections to include the three-form moduli more conveniently. In section 5.2 we will then restrict to a certain class of elliptically ﬁbered Calabi-Yau fourfolds and perform the M-theory to F-theory limit. This allows us to identify the characteristic data determining the four-dimensional N = 1 F-theory eﬀective action in terms of the geometric quantities of the internal space [50]. We note that for certain fourfolds the holomorphic function fAB lifts to a four-dimensional gauge coupling function. Starting from these F-theroy settings we will then perform the weak string coupling limit in section 5.3. In this limit fAB can be partially computed by using mirror symmetry for Calabi-Yau threefolds and we show compatibility with the fourfold result of chapter 4.

47

5 Applications for F-theory and Type IIB orientifolds

5.1

M-theory on Calabi-Yau fourfolds

In this subsection we review the dimensional reduction of M-theory on a Calabi-Yau fourfold Y4 in the large volume limit without ﬂuxes. The ansatz here is similar to the one used for Type IIA supergravity in section 3.1. We start with eleven-dimensional supergravity as the low-energy limit of M-theory. Its bosonic two-derivative action is given by Z 1ˇ 1 1 (11) S = Rˇ ∗ 1 − Fˇ4 ∧ ˇ∗ Fˇ4 − Cˇ3 ∧ Fˇ4 ∧ Fˇ4 , (5.1) 2 4 12 with Fˇ4 = dCˇ3 the eleven-dimensional three-form ﬁeld strength. This will be dimensionally reduced on the background (3) (8) dˇ s2 = ηµν dxµ dxν + gmn dy m dy n ,

(5.2)

where η (3) is the metric of three-dimensional Minkowski space-time M2,1 and g(8) the metric of the Calabi-Yau fourfold Y4 . This is the analog to (3.2) and, as we brieﬂy discussed at the end of section 3.2, the Type IIA supergravity vacuum can be obtained by a circle-reduction of this Ansatz. To perform the dimensional reduction one inserts similar expansions of (3.3), (3.5) and (3.6) into the eleven-dimensional action (5.1). For the metric deformations consisting of Kähler and complex structure deformations, this is exactly the same as (3.3) and (3.5), hence we obtain h1,1 (Y4 ) real scalars Σ by expanding the M-theory Kähler form J as vM M Σ JM = vM ωΣ

(5.3)

and h3,1 (Y4 ) complex scalars z K in three dimensions. Since the elevendimensional three-form Cˇ3 is the common origin of the Type IIA ﬁelds ˇ2 , Cˇ3 , we expand B ¯A Ψ ¯A . Cˇ3 = V Σ ∧ ωΣ + NA ΨA + N

(5.4)

This yields h2,1 (Y4 ) three-dimensional complex scalars NA and h1,1 (Y4 ) vecΣ into three-dimensional tors V Σ . The latter combine with the real scalars vM

48

5.1 M-theory on Calabi-Yau fourfolds vector multiplets, whereas z K , NA give rise to three-dimensional chiral multiplets. Combining the expansions (3.3), (3.5) and (5.4) with the action (5.1) by using the notation of section 3.1 and section 3.2 we thus obtain the three-dimensional eﬀective action 1 Z 1 1 (3) S = R ∗ 1 − GKL¯dz K ∧ ∗d¯ z L − d log VM ∧ ∗d log VM (5.5) 2 2 Σ Λ 2 M Σ Λ − GM ΣΛ dvM ∧ ∗dvM − VM GΣΛ dV ∧ ∗dV 1 Σ AB ¯B + i dΣ AB dV Σ ∧ NA D N ¯B − N ¯B DNA . dΣ DNA ∧ ∗D N − vM 2 4 Note the GM ΣΛ takes the same functional form as (3.10), but uses the M-theory Σ. Kähler structure deformations vM The three-dimensional action given in (5.5) is an N = 2 supergravity theory. The proper scalars in the vector multiplets are denoted by LΣ and Σ Σ as LΣ = vM , as already given in (3.28). are expressed in terms of the vM VM The complex scalars in the chiral multiplets are collectively denoted by φκ = ˜M (z K , NA ). The action (5.5) can then be written using a kinetic potential K as S

(3)

=

Z

1 ˜M 1 ˜M 1 (3) Σ Σ R ∗1+ K ∧ ∗dLΛ + K ∧ ∗dV Λ Σ LΛ dL Σ Λ dV L 2 4 4 L L ˜ MΣ κ dφκ ) , ˜ Mκ ¯λ dφκ ∧ ∗dφ¯λ + dV Σ ∧ Im(K −K L φ φ φ

(5.6)

˜ M. ˜ M , and K ˜ MΣ κ = ∂LΣ ∂φκ K ˜ K ˜ Mκ ¯λ = ∂φκ ∂ ¯λ K ˜ MΣ Λ = ∂LΣ ∂LΛ K, where K φ L φ L L φ φ Comparing (5.5) with (5.6) the kinetic potential obtained for this M-theory reduction therefore reads Z M ¯ + log 1 KΣΛΓ∆ LΣ LΛ LΓ L∆ ˜ (5.7) Ω∧Ω K = − log 4! Y4 + LΣ dΣ AB Re NA Re NB , and was already given in (3.29). Recalling the discussion at the end of section 3.2 it is not hard to check that (5.5) reduces to the Type IIA result 1

The action has been Weyl-rescaled to the three-dimensional Einstein frame by introducnew old ing gµν = V −2 gµν

49

5 Applications for F-theory and Type IIB orientifolds found in section 3.1 upon a circle compactiﬁcation. The detailed circle reduction is performed for a general three-dimensional un-gauged N = 2 theory in A.

5.2

M-theory to F-theory lift

Let us now lift the result (5.6) of the M-theory reduction on a general smooth Calabi-Yau fourfold Y4 to a four-dimensional eﬀective F-theory compactiﬁcation. To do so, we need to restrict Y4 to be an elliptic ﬁbration π : Y4 → B3 over a base manifold B3 which is a three-dimensional complex Kähler manifold. This four-dimensional theory exhibits N = 1 supersymmetry. In the following we will not need to consider the full four-dimensional theory, but will rather focus on the kinetic terms of the complex scalars and vectors without including gaugings or a scalar potential. Supersymmetry ensures that these can be written in the form [84] S (4) =

1 I F ¯J R ∗ 1 − KM IM ¯ J dM ∧ ∗ dM 2 Z 1 1 − Re fΛΣ F Λ ∧ ∗ F Σ + Im fΛΣ F Λ ∧ F Σ . 2 2 Z

(5.8)

In this expression we denoted by M I the bosonic degrees of freedom in chiral F multiplets, and by F Λ the ﬁeld strengths of vectors AΛ . The metric KM IM ¯J F is Kähler and thus can be obtained from a Kähler potential K F via KM IM ¯J = F ∂M I ∂M¯ J K . The gauge-kinetic coupling function fΛΣ is holomorphic in the complex scalars M I . In order to determine the Kähler potential K F and the gauge coupling function fΛΣ via M-theory one next would have to compactify (5.8) on a circle. The resulting three-dimensional theory then has to be pushed to the Coulomb branch and all massive modes, including the excited Kaluza-Klein modes of all four-dimensional ﬁelds, have to be integrated out. The resulting three-dimensional eﬀective theory can then, after a number of dualizations, be compared with the M-theory eﬀective action (5.5). Performing all these steps is in general complicated. However, a relevant special case has been

50

5.2 M-theory to F-theory lift considered in [50] and will be the focus in the following discussion.2 Despite the fact that we could refer to [50] we will try to keep the derivation of K F and fΛΣ in this subsection self-contained. Let us therefore assume that Y4 is an elliptically ﬁbered Calabi-Yau fourfold that satisﬁes the conditions h2,1 (Y4 ) = h2,1 (B3 ) ,

h1,1 (Y4 ) = h1,1 (B3 ) + 1 .

(5.9)

It is not hard to use toric geometry to construct examples that satisfy these conditions (see, for example, refs. [85]). From the point of view of F-theory, or Type IIB string theory, the ﬁrst condition in (5.9) implies that all scalars NA in (5.5) lift to R-R vectors AA in four dimensions. In other words, one can compactify Type IIB on the base B3 and obtain vectors Al by expanding the R-R four-form as C4 = AA ∧ αA − A˜A ∧ β A + . . . .

(5.10)

The vectors A˜A are the magnetic duals of the AA and can be eliminated by using the self-duality of the ﬁeld-strength of C4 . The second condition in (5.9) implies that there are no further vectors in the four-dimensional theory, i.e. there are no massless vector degrees of freedom arising from seven-branes. The two-forms used in (5.3) and (5.4) split simply as ωΣ = (ω0 , ωσ ) , (5.11) where ω0 is the Poincaré-dual of the base divisor B3 and ωσ is the Poincarédual of the vertical divisors D σ = π −1 (Dbσ ) stemming from divisors Dbσ of B3 . Accordingly one splits the three-dimensional vector multiplets in (5.6) as LΣ = (R, Lσ ) , V Σ = (A0 , Aσ ) . (5.12) One can now evaluate the kinetic potential (5.7) for the special case (5.9). The only relevant non-vanishing quadruple intersection numbers are given 2

The geometries of the other two cases will be considered later in chapter 8. They require more involved geometries as we will see.

51

5 Applications for F-theory and Type IIB orientifolds by K0σλγ =

Z

Y4

ω0 ∧ ωσ ∧ ωλ ∧ ωγ ≡ Kσλγ ,

(5.13)

which are simply the triple intersections Kσλγ of the base B3 . Crucially, for an elliptic ﬁbration one has Kσλγδ = 0. Furthermore, note that due to (5.9) all non-trivial three-forms come from the base B3 and we can chose the basis (αA , β A ) such that Z B B ω0 ∧ αA ∧ β B = δA , MσA B = 0 , M0A = (5.14) Y4

with MΣA B introduced in (2.24). Inserting (5.13) and (5.14) into (5.7) one ﬁnds Z M ¯ + log 1 Kσβγ Lσ Lβ Lγ + log(R) ˜ (5.15) Ω∧Ω K = − log 3! Y4 1 − R Re f AB Re NA Re NB , 2 B Re f AB = − 1 R Re f AB , and where we have used that LΣ dΣ AB = − 12 LΣ MΣA 2 we have dropped terms in the logarithm that are higher order in R. In order to compare this kinetic potential with the result of the circle reduction of (5.8) we next have to dualize (Lσ , Aσ ) into three-dimensional complex scalars Tσ , and NA into three-dimensional vectors (ξ A , AA ). Due to our assumption (5.9) leading to (5.14) we can perform these dualizations ˜ M is similar to independently. The change from (Lσ , Aσ ) to Re Tσ = ∂Lσ K (3.37). It is conveniently parameterized by the base Kähler deformations vbσ and the base volume Vb deﬁned as [32, 50]

Lσ =

vbσ , Vb

Vb =

1 Kσβγ vbσ vbβ vbγ . 3!

(5.16)

The dualization of the complex scalars Nk into three-dimensional vectors is similar to the dualization yielding (3.30), (3.31) and (3.32), (3.33). First, one introduces ˜M , ξ A = ∂Re NA K

52

˜ M→F = K ˜ M − ξ A Re NA , K

(5.17)

5.2 M-theory to F-theory lift and then dualizes the ﬁeld Im NA with a shift symmetry into the vector AA . Together both Legendre transforms yield Z M→F ¯ − 2 log Vb + log R + 1 Re fAB ξ A ξ B , (5.18) ˜ Ω∧Ω K = − log 2R Y4 which has to be evaluated as a function of z K , ξ A and ˜ M + iρσ = Tσ = ∂Lσ K

1 Kσβγ vbβ vbγ + iρσ . 2!

(5.19)

The kinetic potential (5.18) is now in the correct frame to be lifted to four space-time dimensions. To derive K F , fAB one reduces (5.8) on a circle of radius r with the usual Kaluza-Klein ansatz the four-dimensional metric and vectors as (3) 2 0 0 2 0 gpq + r Ap Aq r Aq A 0 A A (4) , AA (5.20) gµν = µ = (Ap + Ap ζ , ζ ) , 2 0 2 r Ap r where we introduced the three-dimensional indices p, q = 0, 1, 2 and the Kaluza-Klein vector A0 . Note that we use for three-dimensional vectors the same symbol AA as in four dimensions. Furthermore, we introduced the new three-dimensional real scalars r, ζ A into the theory. We next deﬁne ˆ

ξ A = (R, Rζ A ) ,

R = r −2 ,

ˆ

AA = (A0 , AA ) .

(5.21)

The three-dimensional theory obtained by reducing (5.8) has thus the ﬁeld content: chiral multiplets with complex scalars M I and vector multiplets ˆ ˆ (ξ A , AA ). Its action can be written in the form (5.6) with a kinetic potential ˜ ¯ , ξ) = K F (M, M ¯ ) + log(R) − 1 Re fAB (M )ξ A ξ B , K(M, M R ˆ

(5.22)

ˆ

when replacing LΣ → ξ A , V Σ → AA , and φκ → M I . Finally, comparing (5.22) with (5.18) implies that one ﬁnds M I = {Tσ , z K } Z F ¯ − 2 log VΛ , Ω ∧ Ω) (5.23) K = − log( Y4

fAB

1 = fAB . 2

(5.24)

53

5 Applications for F-theory and Type IIB orientifolds In the next section, we want to derive the orientifold limit of this result relating the data of F-theory on Y4 to Type IIB supergravity with O7/O3planes on the closely related Calabi-Yau three-fold Y3 , a double cover of B3 .

5.3

Orientifold limit of F-theory and mirror symmetry

In this ﬁnal subsection we investigate the orientifold limit of the F-theory eﬀective action introduced above. More precisely, we assume that the Ftheory compactiﬁcation on the elliptically ﬁbered geometry Y4 admits a weak string coupling limit as introduced by Sen [52, 53]. This limit takes one to a special region in the complex structure moduli space of Y4 in which the axiodilaton τ = C0 + ie−φIIB , given by the complex structure of the two-torus ﬁber of Y4 , is almost everywhere constant along the base B3 . The locations where τ is not constant are precisely the orientifold seven-planes (O7-planes). In the weak string coupling limit the geometry Y4 can be approximated by Y4 ∼ σ = (Y3 × T 2 )/˜

(5.25)

where we introduced the involution σ ˜ = (σ, −1, −1) with σ being a holomorphic and isometric orientifold involution such that Y3 /σ = B3 . The two one-cycles of the torus are both odd under the involution, but its volume form is even. It was shown in [52, 53] that the double cover Y3 of B3 is actually a Calabi-Yau threefold. The location of the O7-planes in Y3 is simply the ﬁxed-point set of σ. More reﬁned picture of the weak coupling limit will be developed in chapter 8 that holds for the examples considered there. In the limit (5.25) we can check compatibility of the mirror symmetry results of chapter 4 with the mirror symmetry of the Calabi-Yau threefold Y3 . By using the mirror fourfold Yˆ4 of Y4 we have found that the normalized period matrix fAB is linear in the large complex structure limit of Y4 . Here we recall that the weak string coupling expression gives a compatible result. Using the mirror Yˆ3 of Y3 one shows that the period matrix fAB is linear in

54

5.3 Orientifold limit of F-theory and mirror symmetry the large complex structure limit of Y3 . This can be depicted as follows: F-theory on Y4

weak coupling

−−−−−−−−−−−−→

Type IIB orientifolds Y3 /σ l

physical mirror duality

(5.26)

Type IIA orientifolds Yˆ3 /ˆ σ Note that mirror symmetry of Y3 and Yˆ3 gives a physical map between Type IIB and Type IIA orientifolds. The mirror map between Y4 and Yˆ4 has no apparent physical meaning in F-theory. Nevertheless, using the geometry Y4 in Type IIA compactiﬁcations it can be used to calculate fAB as we explained in chapter 4. Let us now introduce the function fAB for the geometry (5.25). In the p,q orientifold setting one splits the cohomologies of Y3 as H p,q (Y3 ) = H+ (Y3 )⊕ p,q ∗ H− (Y3 ), which are the two eigenspaces of σ . We denote their dimensions as K hp,q ± (Y3 ). As reviewed, for example, in [36] the complex structure moduli z of Y4 split into three sets of ﬁelds at weak string coupling. First, there is the axio-dilaton τ , which is now a modulus of the eﬀective theory as it is constant α over the internal space. Second, there are h2,1 − complex structure moduli z of the quotient Y3 /σ. Third, the remaining number of complex structure deformations of Y4 correspond to D7-brane position moduli. The last set are open string degrees of freedom and are not captured by the geometry of Y3 . For simplicity, we will not include them in the following discussion. With this simplifying assumption one ﬁnds that the pure complex structure part of the F-theory Kähler potential (5.23) splits as Z i h Z ¯ 3 + . . . , (5.27) Ω3 ∧ Ω Ω ∧ Ω) = − log − i(τ − τ¯) − log i − log( Y4

Y3

where Ω3 is the (3, 0)-form on Y3 that varies holomorphically in the complex structure moduli z α . The dots indicate that further corrections arise that are suppressed at weak string coupling −i(τ − τ¯) ≫ 1. Taking the weak coupling limit for the Kähler potential (5.23) of the Kähler structure deformations is

55

5 Applications for F-theory and Type IIB orientifolds 1,1 (Y3 ) and identimore straightforward. The deformations are counted by h+ ﬁed with the Kähler structure deformations vbσ of the base B3 introduced in (5.16). The orientifold Kähler potential for this set of deformations is then simply the second term in (5.23) and the Kähler coordinates are given by (5.19). Turning to the gauge theory sector, we note that the number of R-R vectors A A arising from C4 as in (5.10) are counted by h2,1 + (Y3 ) in the orientifold setting. The gauge coupling function for these vectors is determined as function of the complex structure moduli z α of Y3 in [32].3 It is given by

fAB (z α ) = −iFAB |(z α ) ≡ ∂z A ∂z B F|(z κ ) ,

(5.28)

where F is the pre-potential determining the moduli-dependence of the Ω3 of the geometry Y3 . To evaluate (5.28) one ﬁrst splits the complex structure 2,1 A α moduli of Y3 into h2,1 − (Y3 ) ﬁelds z and h+ (Y3 ) ﬁelds z . The pre-potential F(z κ , z A ) of Y3 at ﬁrst depends on both sets of ﬁelds. Then one has to take derivatives of F with respect to z A and afterwards set these ﬁelds to constant background values compatible with the orientifold involution σ. This freezing of the z A is indicated by the symbol | in (5.28). Using mirror symmetry for Calabi-Yau threefolds it is well-known that the pre-potential at the large complex structure point of Y3 is a cubic function of the complex structure moduli z α and z A . Taking derivatives and evaluating the expression on the orientifold moduli space one thus ﬁnds ˆ αAB , fAB (z α ) = −iz α K

(5.29)

R ˆ αAB = ˆ ω where K ˆA ∧ ω ˆ B are the triple intersection numbers of the Y3 ˆ α ∧ ω mirror threefold Yˆ3 . This result agrees with the one for Type IIA orientifolds, which have been studied at large volume in [33]. Hence, we ﬁnd consistency with the F-theory result (4.25) obtained by using mirror symmetry for Y4 at the large complex structure point. To obtain a complete match of the results ˆ αA B of Yˆ4 is identiﬁed with the triple intersection the intersection matrix M ˆ αAB of Yˆ3 . K 3

Note that we have slightly changed the index conventions with respect to [32] in order to match the F-theory discussion.

56

5.3 Orientifold limit of F-theory and mirror symmetry To close this section we stress again that we have only discussed the matching with the orientifold limit for special geometries satisfying (5.9). Furthermore, we have not included the open string degrees of freedom on the orientifold side. Clearly, our result for fAB obtained in chapter 4 can be more generally applied. For example, a simple generalization is the inclusion of A h1,1 − (Y3 ) moduli G into the orientifold setting, which arise in the expansion of the complex two-form C2 − τ B2 . In F-theory the same degrees of freedom appear from the expansion (5.4) into non-trivial three-forms ΨA that have two legs in the base B3 and one leg in the torus ﬁber, i.e. are not present in the geometries satisfying (5.9). In the orientifold setting one ﬁnds that the ﬁelds GA correct the complex coordinates (5.19). We read oﬀ the result from [32] to ﬁnd 4 Tσ =

1 1 Kσβγ vbβ vbγ + KσAB Im GA Im GB + iρσ . 2! 2 Imτ

(5.30)

Comparing this expression with (3.37) we read oﬀ that N A = iGA ,

dσAB =

1 1 KσAB , 2 Imτ

fAB (τ ) = iτ δAB ,

(5.31)

in order to match the F-theory result as already done in [34]. Again we ﬁnd that the result is linear in one of the complex structure moduli, namely the ﬁeld τ , of the Calabi-Yau fourfold Y4 in the orientifold limit (5.25). In chapter 8 we will generalize these results further and also include Wilson line moduli in the discussion. These are proper open string degrees of freedom that can also be derived from the three-forms of the fourfold geometry. In order to gain intuition about the general situation we will study explicit examples using toric techniques that we will introduce in the next section.

4

Note that compared with [32] we have redefined ρσ to make the terms in Tσ involving the GA real.

57

6 Geometry of toric Calabi-Yau fourfold hypersurfaces

In this section we will introduce the basic notions of Calabi-Yau fourfold hypersurfaces in toric varieties as already studied in [41, 37]. A variety is the algebraic equivalent of a manifold that allows for singularities and is hence a more ﬂexible geometric object. We start out with some basic deﬁnitions in section 6.1 introducing concepts like toric varieties, their homogeneous coordinate rings and the Newton polyhedron of a divisor. In section 6.2 we explain how to use the Gysin-sequence to deduce the non-trivial cohomology groups of a semi-ample Calabi-Yau fourfold hypersurface from its toric subvarieties. The geometry of these toric subvarieties is then explained in section 6.3 and reduced to the study of toric divisors that have certain ﬁbration structures. Introducing the Poincaré residue in section 6.4 to represent non-trivial forms as rational functions of the homogeneous coordinates allows us to study the complex structure dependence of these forms in section 6.5. The ﬁbration structures combined with the Gysin-sequence arguments allow us to compute the Hodge numbers of the fourfold in section 6.6 which determines the massless spectrum of the eﬀective theories derived in previous sections. The ﬁeld of toric geometry is a broad subject and we give the condensed treatment necessary to construct Calabi-Yau fourfolds. For a more thorough introduction and conventions we refer to [86, 87] for the basics of toric geometry and for general algebraic geometry to [88]. Great

59

6 Geometry of toric Calabi-Yau fourfold hypersurfaces introductions are also given in [89, 24]

6.1

Basic construction of toric Calabi-Yau hypersurfaces

Let us start with the basic construction of hypersurfaces with trivial anticanonical class in ﬁve dimensional toric varieties, i.e toric Calabi-Yau fourfold hypersurfaces. We begin with the d-dimensional complex ambient space Ad which will be a toric variety. The special cases we will consider are d ≤ 5 as we will later also discuss the toric subvarieties Ad of A5 . This toric ambient space is deﬁned by a convex polyhedron ∆∗ ⊂ NQ in the rational extension of a lattice N ≃ Zd . The integral points of the polyhedron will be denoted νi∗ ∈ ∆∗ ∩N and the rays from the origin through νi∗ we denote by τi . A toric variety is constructed from a so called fan Σ, as for example explained in detail in [86]. A fan Σ is in our deﬁniton a set of convex rational polyhedral cones σ that are spanned by one-dimensional rays τ = Q+ ν ∗ where τ is a ray from the origin of the lattice N through a lattice point ν ∗ ∈ N ∩ ∆∗ of the polyhedron ∆∗ as σ = {r1 ν1∗ + . . . + rs νs∗ | ν ∗ ∈ N ∩ ∆∗ , ri ∈ Q+ } .

(6.1)

A cone is called simplicial if it is generated by linearly independent vectors. The simplest example is the cone spanned by the unit vectors ei in N . The dual lattice M of N is deﬁned as M = Hom(N, Z)

(6.2)

with MQ its rational extension. Identifying M ≃ Zd we have m(n) = P hm, ni = i mi ni using the standard inner product on Zd . The dual cone σ ∨ of σ is σ ∨ = {u ∈ MZ | hu, vi ≥ 0 , ∀v ∈ σ} . (6.3) To the integral points in Sσ = σ ∨ ∩M we can associate abstract characters χν satisfying χν1 · χν2 that form a commutative algebra C[Sσ ] with unit 1 = χ0 and therefore gives rise to a complex aﬃne variety Aσ = Spec(C[Sσ ]) , 60

Sσ = σ ∨ ∩ M .

(6.4)

6.1 Basic construction of toric Calabi-Yau hypersurfaces The characters χν can be interpreted as monomials of a polynomial ring. The cone σ generated by the unit vectors e∗i of Zd , the dual cone is also generated by the unit vectors e∗i of the dual and we can associate to them the coordinates Xi = χei and hence we have Aσ = Cd = Spec(C[Xi ]) .

(6.5)

Simplicial cones will in general result in aﬃne varieties with orbifold singularities along a toric subvariety. A simple example is given by the cone σ spanned by 2e1 + e2 , e2 in Q2 leading to C[Sσ ] = C[X, Y 2 X −1 ] ≃

C[X, Y, Z] , XY − Z 2

Aσ = C2 /Z2 ,

(6.6)

with the singularity at (0, 0) ∈ C2 /Z2 . which is a cone. These aﬃne toric varieties Aσ can be glued together to form a toric variety Ad and the corresponding information is contained in the fan Σ. To ensure a well-deﬁned gluing of the cones and hence the aﬃne varieites Aσ , σ ∈ Σ need to intersect each other only in faces that are cones that are also part of Σ. If we want to obtain a compact or rather complete toric variety Ad , the cones of Σ need to cover the full space NQ . We will denote the fan Σ(∆∗ ) a fan that is deﬁned by the polyhedron ∆∗ and the lattice N . Another interpretation is to view Ad as a compactiﬁcation of (C∗ )d with C∗ = C − {0} an algebraic torus giving the toric variety its name. For simplicity we will assume in the following that all cones are simplicial and hence each correspond to a simplicial subpolyhedron of a face of ∆∗ . In higher dimensions d > 3 this can not be guaranteed, but will simplify our discussion. The polyhedron ∆∗ is triangulated, i.e. it is divided into subpolyhedra with integral vertices that generate together with the origin the convex cones for the fan Σ(∆∗ ). This is called a star-triangulation of ∆∗ , since all cones have their tip at the same point in the lattice, the origin. We will work with a maximal star-triangulation of ∆∗ with all rays τ through integral points ν ∗ ∈ N ∩ ∆∗ . Such a triangulation is in general not unique and we will assume henceforth that all cones arise from such a maximal star-triangulation and are simplicial and hence Ad will only have Zn -orbifold singularities along subspaces of codimension greater or equal to one.

61

6 Geometry of toric Calabi-Yau fourfold hypersurfaces The Calabi-Yau fourfold hypersurface Y4 in A5 will be described via the language of divisors and line-bundles over a toric variety Ad . A Weil-divisor DW eil is a formal sum of codimension one irreducible subvarieties Vi X DW eil = bi Vi , codim(Vi ) = 1 , bi ∈ Z . (6.7)

A Cartier divisor D is a set of non-zero rational functions over each aﬃne coordinate patch that can be glued together to form a line-bundle O(D) over Ad . If this line-bundle is trivial, i.e. deﬁned by a global non-zero rational function on Ad , the divisor is principal. A Cartier-Divisor deﬁnes a WeilDivisor as X DCartier = ordV (D) · V , (6.8) codim(V )

where ordV (D) is the order of vanishing of the deﬁning function of D along the subvariety D. Two Weil-divisors are linear equivalent if they diﬀer by a principal divisor and their group of equivalence classes is called Chow-group Ad−1 (Ad ). The T-invariant principal divisors are given by the global rational functions χu , u ∈ M which is therefore d-dimensional. A T-invariant Cartier divisor is given by a set (Aσ , χu(σ) ) where u(σ) ∈ Sσ and Aσ cover Ad . As the toric divisors Di of codimension one are deﬁned by the rays τi = νi∗ · Q in νi∗ ∈ ∆∗ ∩ N , we have X [div(χu(σ) )] = hu(σ), νi∗ )iDi , (6.9) i

we will discuss the toric subvarieties of Ad later in more detail. On the toric varieties we consider we have the isomorphisms P ic(Ad ) ⊗ C ≃ Ad−1 (Ad ) ⊗ C ≃ H 2 (Ad , C) .

(6.10)

Therefore, each toric divisor Di , νi∗ ∈ ∆∗ ∩ N modulo rational equivalence deﬁnes an element of H 2 (Ad , C) and a line bundle O(D) ∈ P ic(Ad ). The principal divisors deﬁned by one global rational function are give rise to trivial line-bundles. The deﬁning rational function of a divisor [D] can hence be viewed as a global section of a line-bundle L∆ ∈ P ic(Ad ) over Ad . The global sections 62

6.1 Basic construction of toric Calabi-Yau hypersurfaces H 0 (Ad , L∆ ) of L∆ are deﬁned via a so called Newton-polyhedron ∆ ⊂ MQ . A basis of global sections of L∆ is given abstractly by χν ∈ H 0 (Ad , L∆ ) ,

ν ∈ ∆∩M .

(6.11)

P The relation to a divisor D∆ = i bi Di with Di the toric divisors is given by ∆ = {u ∈ MQ | hu, vi ≥ −bi , ∀v ∈ ∆∗ } . (6.12) A speciﬁc global section of the line bundle L∆ is hence deﬁned by the linear combination X p∆ (aj ) = aj χν ∈ H 0 (Ad , L∆ ) , aj ∈ C . (6.13) νj ∈∆∩M

Varying the aj preserves hence the class [D∆ ] ∈ Ad1 (Ad ) of the divisor, but changes in general its representative D. This enables us to describe a family of hypersurfaces by one divisor-class. To each of the toric divisors Di we can associate a coordinate Xi such that Di = {Xi = 0}. These coordinates form the ring of homogeneous coordinates of Ad as deﬁned in [90] we denote it by Sd = C[Xi , νi∗ ∈ ∆∗ ∩ N ] .

(6.14)

This ring has a natural grading by divisor classes α ∈ Ad−1 (Ad ). For a given monomial Y b X Xi i , deg(f ) = α , α = [ f= bi Di ] . (6.15) i

i

We can interpret the elements of Sd of degree β as the global holomorphic section vanishing over β as H 0 (Ad , OAd (D)) = Sβ ,

β ∈ [D]

(6.16)

as for example shown in [91,90]. The coordinate ring Sd and the toric variety Ad are related by the proj-construction Ad = Proj(Sd ) = Proj(C[Xi , νi∗ ∈ ∆∗ ∩ N ]) ,

(6.17)

63

6 Geometry of toric Calabi-Yau fourfold hypersurfaces relating a graded ring to a projective variety. The projective toric varieties we consider are therefore in particular Kähler. This ring is called the homogenous coordinate ring of Ad , since we can choose global homogeneous coordinates, denoted by [X1 : X2 : . . .] = [X i ] = [λℓi Xi ] ,

X

νi∗ ℓi = 0 .

(6.18)

i

These are equivalence classes under rescalings by powers of a non-zero complex number λ ∈ C∗ = C − {0} which is an algebraic torus from which the ∗ name toric variety is derived. The ℓi ∈ Zℓ(∆ )−1−d induce the grading of ∗ Sd and generate a cone in Qℓ(∆ )−1−d where ℓ(∆∗ ) − 1 denotes the number integral points in the boundary of ∆∗ . It can be shown [86, 92] that Ad−1 (Ad ) the Chow group of Aˆd is given by Ad−1 (Ad ) =

C[Di ] hP [Σ(∆∗ )]i

(6.19)

hm, νi∗ iDi : m ∈ M i

(6.20)

with the projective equivalence P [Σ(∆∗ )] = h

X

νi∗ ∈∆∗ ∩N

which is a d-dimensional sub-module of Ad−1 (Ad ). The equivalence classes are denoted by [Di ]. We note here that it can be shown that the toric divisors generate the full cohomology of a complete simplicial toric variety Ad which reads C[Di ] H ∗ (Ad , C) = (6.21) hP [Σ(∆∗d )] ⊕ SRi where the [Di ] have grading (1, 1) viewed as their dual two-forms ωi ∈ H 1,1 (Ad ). In particular, all non-trivial cohomology classes of toric varieties have Hodge-type and can be represented by the intersection of p toric divisors. The Stanley-Reissner ideal SR provides the information for the product in the cohomology corresponding to the intersection of divisor-classes. SR = {Di1 · · · Dik | {τi1 , . . . τik } 6⊂ σ , ∀σ ∈ Σ(∆∗ )} . 64

(6.22)

6.1 Basic construction of toric Calabi-Yau hypersurfaces In addition, we note that from description we can calculate h1,1 (Ad ) as h1,1 (Ad ) = ℓ(∆∗ ) − (d + 1) .

(6.23)

Here ℓ(∆∗ ) counts the number of integral points in ∆∗ ⊂ N and subtracts the dimension of P [Σ(∆∗ )] which is d and the −1 corresponds to the origin, the integral point of ∆∗ to which no divisor is associated. Let us illustrate this with a simple example, the two dimensional complex projective space P2 . The polyhedron is spanned by e1 , e2 , −e1 − e2 ∈ N ≃ Q2 and each two of these give rise to a simplicial cone, isomorphic to C2 . Another way to see this is to deﬁne S2 = C[X1 , X2 , X3 ] ,

A2 = P roj(S2 ) = P2 ,

(6.24)

where Xi all have the same degree. All the [Di ] are linearly equivalent as P [Σ(∆∗ )] = h(D1 − D3 ), (D2 − D3 )i

(6.25)

and can be represented by the hyperplane class [H] = [Di ] ∈ A1 (P2 ). The SR-ideal is simply given by H 3 ≃ D1 D2 D3 and the full cohomology is therefore H ∗ (P2 , C) = C[H]/H 3 . (6.26) This is a ﬁrst example of a weighted projective space that we will consider in more detail in section 7.3. The special divisor class that will give rise to a Calabi-Yau hypersurface is the so called anti-canonical divisor class of Ad given by X D∆ = −KAd = Di . (6.27) νi∗ ∈∆∗ ∩N

The details of this construction were established in the seminal paper of Batyrev [93]. The corresponding anti-canonical hypersurface D∆ and all members of the same class are Calabi-Yau if the associated anti-canonical line-bundle L∆ is reﬂexive or equivalently, if both polyhedra are convex and contain only one interior point. This interior point can then always be shifted to the origin of M . In the second case, we can describe ∆ as ∆ = (∆∗ )∗ = {u ∈ MQ hu, vi ≥ −1 , ∀v ∈ ∆∗ } . (6.28) 65

6 Geometry of toric Calabi-Yau fourfold hypersurfaces The resulting hypersurface will in general be singular and we denote it by sing Yd−1 . This is the vanishing set of a global section of −KAd given by p∆ (aj , Xi ) =

X

ν∈∆∩M

aj

Y

νi∗ ∈∆∗ ∩N

hνj ,νi∗ i+1

Xi

∈ Sd (−KAd ) .

(6.29)

We have chosen implicitly a maximal star-triangulation of ∆∗ by associating to every integral point νi∗ ∈ ∆∗ ∩ N a homogeneous coordinate Xi . Due to the Bertini-theorem, that basically states that we can always change the aj inﬁnitesimally varying the hypersurface in its divisor-class, we can assing sume that all singularities of Yd−1 arise from the ambient space Ad . The n-dimensional toric subvarieties An of Ad along which we have singularities sing are then intersected by Yd−1 also in hypersurfaces of codimension one. We can resolve these singularities by adding new homogeneous coordinates corresponding to integral points of N that will also change ∆∗ , its triangulation and therefore also the fan of Ad . Adding a new ray corresponding to such an additional integral point in N that is not contained in the boundary of ∆∗ will also aﬀect the number of integral points in ∆. Hence this will change the number of possible deformations of the hypersurface. If the chosen integral point is in the boundary of ∆∗ the resolution is called crepant, preserving the anti-canonical class −KAd . We will henceforth assume that we can resolve the singularities of Ad that will be inherited by the hypersurface via crepant resolutions. 1 In addition, we are going to assume that we can always choose a transverse and quasi-smooth hypersurface in the anti-canonical hypersurface class, intersecting all toric subvarieties of the ambient space in smooth varieties of codimension one. The resolved smooth Calabi-Yau hypersurface will be denoted by Yd−1 and the fully resolved ambient space Aˆd . With the most important basics and notations introduced, we will discuss the precise origin and representations of the non-trivial cohomology classes of the Calabi-Yau hypersurface Y4 in A5 in the upcoming section. 1

For d > 4 this is in general not possible, but it simplifies our discussion.

66

6.2 Cohomology of Y4 via the Gysin-sequence

6.2

Cohomology of Y4 via the Gysin-sequence

In this subsection, we want to describe the origin of the non-trivial cohomology classes that will give rise to the massless ﬁelds in the spectrum of our eﬀective ﬁeld theories. The key-point here is that we need to circumvent the Lefschetz-hyperplane theorem for quasi-smooth Fano-hypersurfaces to obtain non-trivial three-form cohomology on Y4 . This is done via quasi-Fano or semiample hypersurfaces as discussed by Mavlyutov in [94]. The standard way to calculate the cohomology groups of the hypersurface Y4 is to apply the Lefschetz-hyperplane theorem as for example stated in [91]. This theorem states that for a quasi-smooth hypersurface D of a (d + 1)-dimensional complete simplicial toric variety Ad+1 deﬁned by an ample divisor we have the isomorphism ≃

ι∗ : H j (Ad+1 , C) −−−−→ H j (D, C) ,

j ≤d−1

(6.30)

This is the induced map of the inclusion ι : D ֒→ Ad+1 . Furthermore, we have the inclusion ι∗ : H d (Ad+1 , C) ֒→ H d (D, C) .

(6.31)

This implies basically that all non-trivial cohomology of degree less than d is induced from the ambient toric variety Ad+1 . Combing this with (6.21) which implies H i,j (Ad+1 ) 6= 0 ⇒ i = j , (6.32) we see that this imposes strong restrictions on the cohomology of the hypersurface. In particular, it follows that a quasi-smooth four-dimensional Calabi-Yau hypersurface in a toric ambient space with ample anti-canonical line-bundle can not have non-trivial three-forms. A variety with ample anticanonical line-bundle is called a Fano variety. A line-bundle over a variety is ample, iﬀ for every point we can ﬁnd a global section that doesn’t vanish over that point. An example for this is the sextic, the anti-canonical hypersurface in P5 which does not support odd cohomology, as studied in [95]. Therefore, we want to consider ambient spaces more general than regular projective spaces.

67

6 Geometry of toric Calabi-Yau fourfold hypersurfaces As was shown in [96] the non-trivial cohomology classes of degree less than d in a semiample hypersurface D∆ in a complete simplicial toric variety Ad+1 arise from the toric divisors Di of Ad+1 Di = {Xi = 0} ,

Di′ = Di ∩ D∆ ,

νi∗ ∈ ∆∗ ∩ N .

(6.33)

In toric geometry, see [86] section 3.4. it can be shown that for an ample Cartier divisor over a complete toric variety with polyhedron ∆∗ we have a one-to-one correspondence between vertices of ∆ and maximal-dimensional cones in ∆∗ . Due to the fact that crepant resolutions of a Calabi-Yau hypersurface subdivide the maximal cones of ∆∗ , but leave the dual polyhedron ∆ invariant, a crepant resolution renders the resulting ambient space non-Fano. The exceptional divisors of the toric resolutions are the Di that will induce non-trivial cohomology on the hypersurfaces D∆ . These divisors carry themselves non-trivial cohomology that lifts to the full Calabi-Yau hypersurface. Therefore, we call a hypersurface semiample if it is obtained from crepant resolutions of an ample hypersurface in a possibly singular Fano toric ambient space. This can be viewed as starting from a polyhedron ∆∗ with a triangulation that only contains rays through the vertices deﬁning the singular space Ad+1 . Then we add subsequently the rays through all integral points of ∆∗ such that we resolve all singularities on the hypersurface which leads to the new toric ambient space Ad+1 that is simplicial and complete. In terms of n-dimensional polyhedra ∆∗n , we hence call a hypersurface ksemiample, k ≤ n, if the Newton-polyhedron ∆k of the hypersurface class has dimension k. A semiample hypersurface of An is therefore n-semiample. The precise composition of the cohomology groups of a Calabi-Yau fourfold Y4 realized as a semiample divisor in the toric simplicial complete ambient space A5 is given by a number of exact sequences as found in equation (7) of [96]. The ﬁrst one, inducing non-trivial two-forms is given by M νi∗

⊕i ιi,∗

H 0,0 (Di′ ) −−−−−−−→ H 1,1 (Y4 ) ,

ιi : Di′ ֒→ Y4 .

(6.34)

L 0,0 (D ′ ) basically provides a two-form for every toric We note that i νi∗ H divisor class of Y4 . It is, however, possible to obtain h0,0 (Di′ ) > 1 after

68

6.2 Cohomology of Y4 via the Gysin-sequence intersecting with the hypersurface, but we always have h0,0 (Di ) = 1 for the purely toric setting. We will see this in more detail in the next section, when we discuss the cohomology of toric divisors of a Calabi-Yau hypersurface. The map ιi,∗ is the Gysin-map induced by the inclusion of the toric divisor Di′ into Y4 , which we will discuss shortly. It can be viewed as dual to taking the non-trivial cycles of Di′ as cycles of Y4 . The next sequence clariﬁes the origin of non-trivial three-forms in Y4 and reads M ⊕i ιi,∗ (6.35) H 1,0 (Di′ ) −−−−−−−→ H 2,1 (Y4 ) . νi∗

The requirement that a toric divisor D ′ hosts non-trivial one-forms will be a severe restriction, as we will discuss in the next section. The next sequence we want to mention will allow us to count the (3, 1)forms of Y4 , which correspond to the complex structure deformations that preserve the Calabi-Yau structure. This is, however, a true exact sequence that does not collapse to one isomorphism

0 −→

M νi∗

⊕i ιi,∗

3,1 H 2,0 (Di′ ) −−−−→ H 3,1 (Y4 ) −→ GrW (Y4 ∩ T) −→ 0 . (6.36) 4 H

3,1 (Y ∩ T) can be interpreted as the bulk We will later see, that GrW 4 4 H complex structure deformations corresponding to the deformations of the deﬁning section p∆ of the hypersurface. Therefore, these are called algebraic complex structure deformations. The torus T is the open torus (C∗ )5 ⊂ Aˆ5 of which A5 is a compactiﬁcation. The union of toric divisors is the complement of T in Aˆ5 . The other deformations arise from holomorphic (2, 0)-forms on toric divisors and are therefore called non-algebraic divisors, since they leave p∆ invariant.

The Gysin map ι∗ of a inclusion ι : N → M of an n-dimensional submanifold of a compact manifold M of dimension m is best described with the

69

6 Geometry of toric Calabi-Yau fourfold hypersurfaces following diagram ι

Hn−p (N, C)

PD

PD H p (N, C)

Hn−p (M, C)

H m−n+p (M, C)

ι∗

(6.37)

Here PD is Poincaré duality and from this we see that the Gysin-map is the dual map of the inclusion of cycles of submanifolds. This is a purely topological construction of topological manifolds and hence also applies to our setting of varieties. In general there is no reason why this map should be surjective or injective. In our case the toric divisors have the real dimension n = 2d − 2 and m = 2d. This implies for example that for p = 1 we have H5 (Di′ , C)

ιi

PD

PD H 1 (Di′ , C)

H5 (Y4 , C)

ιi,∗

H 3 (Y4 , C)

(6.38)

Here we note that due to the fact that all maps are topologically and independent of the metric or the complex structure on Y4 . Therefore, the Gysin-map is compatible with the splitting into Hodge-type and we obtain the exact sequences we stated before. In the two upcoming sections, we will discuss the geometry of toric divisors further and also see, how the algebraic deformations of a hypersurface can be desribed in detail via chiral rings and the Poincaré residue.

6.3

The geometry of toric divisors of a Calabi-Yau hypersurface

So far we have established that we can obtain the non-trivial forms of various types on a semiample hypersurface D∆ in a simplicial toric ambient space

70

6.3 The geometry of toric divisors of a Calabi-Yau hypersurface Aˆ5 from toric divisors. These toric divisors correspond to the rays through integral points ν ∗ in the boundary of the polyhedron ∆∗ deﬁning Aˆ5 . The corresponding rays can be classiﬁed by the codimension codim(θ ∗ ) of the face θ ∗ ⊂ ∆∗ such that ν ∗ ∈ int(θ ∗ )∩N . This was already suggested in [37] where this idea was used to determine the Hodge-numbers of Calabi-Yau fourfolds. In our case, since we are interested in the precise moduli dependence of the harmonic forms, we will be more explicit. First, we note that since the toric divisor Di′ = Di ∩ Y4 of the semiample hypersurface Y4 in the simplicial toric space Aˆ5 is again a semi-ample hypersurface in a toric variety Di , we review ﬁrst the construction of the n-dimensional toric subvarieities An of A5 . The subvariety An corresponds to a (4 − n)-dimensional face θ ∗ of ∆∗ ⊂ NQ . We construct An starting from the (5−n)-dimensional cone σ ⊂ NQ over the face θ ∗ with apex at the origin. This cone enables us to deﬁne new n-dimensional lattices Nn , Mn via Nn = N (σ) = N/Nσ , Mn = M (σ) = M ∩ σ ⊥ ,

Nσ = N ∩ Q · σ ⊂ N

(6.39)

σ = θ ∗ · Q+ .

Here we denoted by Q · σ ⊂ NQ the (5 − n)-dimensional vector space spanned by the elements of σ over Q. The elements in MQ that pair to zero with σ ⊂ NQ are denoted by σ ⊥ . σ ⊥ = {m ∈ M | hm, ni = 0 , ∀n ∈ σ} .

(6.40)

This is a n-dimensional vector space in MQ . The fan of the toric subvariety An,θ∗ is given by the set Star(σ) of all cones over faces of ∆∗ that share a face with θ ∗ projected to N (σ). The image of these adjacent faces under the projection form again a star subdivison of a polytope ∆∗n in N (σ) and all of σ · Q gets projected to the origin of the quotient lattice N (σ). For details we refer to [86]. Consequently, we can associate a homogeneous coordinate ring to An,θ∗ as follows Sn,θ∗ = C[Xi , νi∗ ∈ ∆∗n ] ⊂ C[Xi , ν ∗ ∈ ∆∗ ]/hXi , νi∗ ∈ θ ∗ i =

S5 /hXi , νi∗

(6.41)

∗

∈ θ i.

71

6 Geometry of toric Calabi-Yau fourfold hypersurfaces These rings inherit the grading structure of the homogeneous coordinate ring S5 of A5 , since the other toric divisors either intersect An,θ∗ transversely or not at all. If the divisor Di intersects An,θ∗ the coordinate Xi is also in Sn,θ∗ , if it does not intersect we can set Xi = 1 in Sn,θ∗ . If νi∗ ∈ θ ∗ , the coordinate Xi gets projected out, since it is equivalent to zero in Sn,θ∗ . Let us now discuss, how the hypersurface intersects the toric subvariety An,θ∗ . We have seen that the polynomial p∆ deﬁning the hypersurface is built from global sections of the anti-canonical bundle of the ambient space P −KAˆ5 . Therefore, p∆ ∈ S(−KAˆ5 ) and its monomials have degree [ i Di ] where we sum over all toric divisors Di of Aˆ5 . Restricting now to An,θ∗ where the divisors Di vanish for which νi∗ ∈ θ ∗ ∩ N , we obtain from the projection S5 (−KAˆ5 ) → Sn,θ∗ (−KAˆ5 ) ,

p∆ 7→ pθ = p∆ |An,θ∗ ,

(6.42)

the hypersurface equation pθ = 0 on An,θ∗ . Homogeneous coordinates in p∆ corresponding to divisors not intersecting An,θ∗ will be set to one. The monomials of pθ correspond to the global sections of −KAˆ5 that do not vanish over An,θ∗ determined by the integral points of the dual face θ of θ ∗ given by θ = {v ∈ ∆ | hv, wi = −1 , ∀w ∈ θ ∗ } .

(6.43)

As was discussed in [92], the toric divisors Di′ ∈ Di ∩Y4 are so called dim(θ)semiample hypersurfaces of the toric varieties Di where νi∗ ∈ int(θ ∗ ) ∩ N . In the following we will denote pairs of faces as (θα∗ , θα ) ,

dim(θα∗ ) = n ,

α = 1, . . . , kn .

(6.44)

To each of such pairs we can associate divisors Dlα with Dlα :

νl∗α ∈ int(θα∗ ) ∩ N ,

lα = 1, . . . , ℓ′ (θα∗ ) ,

(6.45)

where ℓ′ (θα∗ ) counts the number of interior points of θα∗ . Let us ﬁrst consider the toric divisors Dlα of Aˆ5 . The cones corresponding to this subvariety are the rays (6.46) τlα = ν ∗ · Q+ , Dlα = V (τlα ) = A4,νl∗ . α

72

6.3 The geometry of toric divisors of a Calabi-Yau hypersurface Here we introduced the alternative expression Dlα = V (τlα ) to make contact with the literature. The divisors Dlα admit a ﬁbration structure, which can be seen as follows. For νl∗α an inner point of a n-dimensional face θα∗ , the ray τlα is contained in the cone σα over θα∗ . This we project to N (τlα ) to obtain the polyhedron ∆∗4 ⊂ N (τlα ) of the toric variety A4,νl∗ = Dlα . α Due to the convexity of ∆∗ and νl∗α int(θα∗ ) ∩ N , we ﬁnd that θα∗ maps to a subpolyhedron of ∆∗4 containing the origin and deﬁning the subvariety Dlα . This subpolyhedron of dimension (4 − n) has an induced triangulation from ∆∗ and deﬁnes a simplicial irreducible (connected) and complete (compact) toric variety Elα . As described in the literature, for example in [86], this implies that Dlα is a ﬁbration over Aθα∗ ,n = V (σα ) = Aα with ﬁber given by Elα . This can be summarized in the following ﬁbration diagram: Elα

ilα

Dlα = V (τlα ) πlα Aα = V (σα )

(6.47)

Here πlα denotes the projection of Dlα to the base Aα and ilα the inclusion of the ﬁber. The semiample hypersurface Dl′α = Dlα ∩ Y4 inherits this ﬁbration structure, since the deﬁning polynomial pθα = pα is obtained from p∆ by setting all homogeneous coordinates corresponding to integral points in θα∗ to zero. From this we can deduce that the hypersurface equation is independent of the homogeneous coordinates of Elα and therefore, we ﬁnd a similar ﬁbration structure ilα Dl′α = V ′ (τlα ) Elα πlα Rα = V ′ (σα )

(6.48)

Here we denoted by Rα = V (σα ) the dim(θ)-semiample hypersurface Aα ∩ 73

6 Geometry of toric Calabi-Yau fourfold hypersurfaces Y4sing deﬁned by the polynomial pα . We used here Y4sing because without adding the rays through the integral inner points of θα∗ the hypersurface is general and contains singularities along Rα . Only after resolving the singularities via the introduction of the exceptional divisors Elα is Aˆ5 and therefore Y4 resolved. After the resolution, Rα is not a subvariety of Y4 , because the ﬁbration Dl′α may not have a section. Having established the ﬁbration structure of the toric divisors Dl′α , we can calculate its cohomology. The ﬁbration structure enables us to use the Leray-Hirsch theorem, [97, 67], since due to the toric ﬁbration structure of Dlα , one can show that the ﬁber does not degenerate and is locally trivial. The inclusion ilα : Elα → Dl′α satisﬁes furthermore the necessary condition that the elements i∗lα (cj ) for cj ∈ H ∗ (Dl′α , C) generate H ∗ (Elα , C). Therefore, we ﬁnd the induced isomorphism of C-modules ≃

H ∗ (Rα , C) ⊗C H ∗ (Elα , C) −−−−→ H ∗ (Dl′α , C)

(6.49)

given by the map bi ⊗C i∗lα (cj ) 7→ πl∗α (bi ) ∧ cj .

(6.50)

Note that this is an isomorphism of C-modules and not of rings. Alle morphisms appearing in this construction are independent of the Hodge structure and therefore the Hodge numbers arise from products of Hodge numbers of the base space Rα and Elα . Due to the fact that Elα is toric and irreducible, i.e. connected, its Hodge numbers are restricted to hp,q (Elα ) = 0 , p 6= q ,

h0,0 (Elα ) = hn,n (Elα ) = 1 ,

n > 1,

(6.51)

For the regular n-semiample hypersurface Rα of dimension n − 1, implying that it is ample, we ﬁnd that h0,0 (Rα ) = 1 , This will by ℓ′ (θα ) Let us dim(θ ∗ ).

74

hn−1,0 (Rα ) = ℓ′ (θα ) ,

n > 1.

(6.52)

be discussed in more detail in the upcoming section. We denoted the number of interior integral points of θα in NQ . now go through the various cases of face dimensions (4 − n) = For n = 0, we ﬁnd that A0 is a set of points in Aˆ5 and hence the

6.4 Poincaré Residue of toric hypersurfaces corresponding singularities in A5 will be avoided by a general hypersurface. In the case n = 1, we ﬁnd that Aα ≃ P1 and the hypersurface will intersect these one-dimensional subvarieties in a number of points counted with multiplicity. The number of points ptα is the degree of pα and is equal to ℓ′ (θα )+1. Resolving the singularities along Rα introduces complex three-dimensional toric varieties Elα that are irreducible. We have that dim(θα ) = 1 ,

H 0,0 (ptα ) = ℓ′ (θα ) + 1 .

(6.53)

The next case is n = 2, here we ﬁnd that the toric divisors Dlα are ﬁbrations over Riemann surfaces Rα with ﬁbers two-dimensional toric varieties Elα . The Hodge numbers in this case satisfy dim(θα ) = 2 ,

H 1,0 (Rα ) = ℓ′ (θα ) .

(6.54)

In the case n = 3 we have that Rα are surfaces Rα = Sα that are ample hypersurfaces of toric varieties and hence satisfy dim(θα ) = 3 ,

H 2,0 (Sα ) = ℓ′ (θα ) ,

H 1,0 (Sα ) = 0 .

(6.55)

as we will see in the next subsection. The hypersurfaces Sα are ample divisors of A3 and hence for them the Lefschetz-hyperplane theorem holds and h1,0 (S) = 0. Contrary, if h1,0 (S) 6= 0, the hypersurface needs to satisfy h2,0 (S) = 0 and we are in the n = 2 case. We note in particular, that the divisors giving rise to non-algebraic deformations and three-forms can not intersect. In this section we have reduced the complex structure dependence of semiample divisors to their ample bases, which are ample hypersurfaces in toric varieties. In the next section, we will give an explicit description of the non-trivial holomorphic forms on these hypersurfaces.

6.4

Poincaré Residue of toric hypersurfaces

Let us now discuss the holomorphic (n − 1)-forms on semi-ample hypersurfaces in projective and simplicial toric varieties An , following [98] for ample 75

6 Geometry of toric Calabi-Yau fourfold hypersurfaces hypersurfaces. This was generalized to semiample hypersurfaces in [96, 92]. n−1 of the The insight here is that we express the holomorphic (n−1)-forms ΩR n hypersurface R as global rational holomorphic forms ΩAn (R) of the ambient space An with poles along R. This will enable us to express the algebraic deformations of the polynomial pθ to explicit forms on the hypersurface, the periods. A global rational and holomorphic n-form on An with poles of ﬁrst order along the hypersurface R that is a restriction of the anti-canonical hypersurface in A5 is given by g dωAn g ∈ Sn (−KA5 |An + KAn )} pθ ≃ Sn (−KA5 |An + KAn ) .

H 0 (An , ΩnAn (R)) = {

(6.56)

We introduced the following notation for this: the Cartier divisors class of the restriction of the anti-canonical divisor of A5 to An is denoted by −KA5 |An ∈ An−1 (An ). In An we have the hypersurface R ⊂ An deﬁned as the vanishing locus of pθ ∈ Sn (−KA5 |An ). Similarly, we introduced the anti-canonical divisor −KAn of the subvariety An of A5 . Furthermore, we introduced the holomorphic volume-form dωAn of An that has degree [−KAn ] as we will show shortly. Finally, we denoted by Sn (−KA5 |An + KAn ) the elements of the homogeneous coordinate ring Sn of An with degree [−KA5 + KAn ] as we introduced in (6.41). For the above expression we introduced the holomorphic volume-form dωAn of An . To deﬁne this, we introduce a ﬁxed integral basis {m1 , . . . , mn } of Mn and deﬁne for each index set I = {i1 , . . . , in } of n-integral points νi∗1 , . . . , νi∗n in ∆∗n ∩ Nn the determinant det(νI∗ ) = det(hmi , νj∗ i1≤i,j≤n ) ,

(6.57)

which can be thought of as the volume of the simplex spanned by the integral points νi∗ and the origin. From this we can construct the holomorphic volumeform as Y X (6.58) Xi dXi1 ∧ . . . ∧ dXin . dωAn = det(νI∗ ) |I|=n

76

i∈I /

6.4 Poincaré Residue of toric hypersurfaces Here the sum runs over all index sets I with n elements {i1 , . . . , in }. If we assign the one-forms dXi the same degree as their coordinate counterparts Xi , we see that the degree of dωAn is given by X degSn (dωAn ) = [ Di ] = −KAn . (6.59) νi∗ ∈∆∗n ∩Nn

Note that the holomorphic volume-form is only unique up to a multiple of a constant. Having introduced the notation, we can now establish the Poincaré residue construction to ﬁnd representations for the holomorphic (n−1)-forms of the Cartier divisor R with degree [−KA5 |An ] deﬁned by pθ = 0 in the toric ambient space An . We deﬁne the Poincaré residue as the map H 0 (An , ΩnAn (R)) g dωAn pθ

n−1 ) H 0 (R, ΩR Z g dωAn . pθ Γ

→ 7→

(6.60)

We introduced here a small one-dimensional curve Γ ∈ H1 (An −R, R) around R in the complement of R in An . This integral expression deﬁnes a holomorphic (n − 1)-form, as applied to a (n − 1)-cycles α ∈ Hn−1 (R), we ﬁber Γ over α and integrate the rational (meromorphic) form in H 0 (An , ΩnAn (R)) over the resulting n-cycles to obtain a complex number. This map is well-deﬁned, but not injective, since we can partially integrate Z i g ∂X i pθ dωAn = 0, (6.61) pθ Γ which constitutes the kernel of the residue map which can also be denoted by the residue symbol ResR ( · ). Therefore, modding out the partial derivatives of pθ of Sn and deﬁne the Jacobian or chiral ring Rθ =

Sn . h∂X i pθ i

(6.62)

Here we denoted by h∂X i pθ i the ideal spanned by the partial derivatives of the deﬁning polynomial pθ . The ring Rθ inherits the grading of Sn . This renders the map Rθ (−KA5 |An + KAn )

֒→

n−1 ) H 0 (R, ΩR

(6.63)

77

6 Geometry of toric Calabi-Yau fourfold hypersurfaces deﬁned by the Poincaré residue injective, as was shown in [92]. In the case that the Calabi-Yau hypersurface Y4sing has only singularities of codimension two arising from the ambient space A5 , as is the case for our semiample situation of the previous section, we see that for n = 1 the residue just gives a constants over disjoint points, whose location is deﬁned by pθ . In this case we ﬁnd ℓ(θ) + 1 distinct points over which we have a three-dimensional exceptional divisor. the points are the zeroes of pθ . The ℓ′ (θ) count the moduli, the diﬀerences between the zeroes in P1 = A1 . n = 1 : R = {pt | pθ (pt) = 0} ,

h0,0 (R) = ℓ′ (θ) + 1 .

(6.64)

In the case n = 2, we ﬁnd the holomorphic one-forms of Riemann-surfaces n=2 :

Rθ (−KA5 |A2 + KA2 ) ≃ H 0 (R, Ω1R ) = H 1,0 (R) .

(6.65)

Similarly, we constructed all the holomorphic two-forms of a surface S as n=3 :

Rθ (−KA5 |A3 + KA3 ) ≃ H 0 (S, Ω2S ) = H 2,0 (S) .

(6.66)

The trivial case is now n = 5 which is the full Calabi-Yau fourfold Y4 in the space Aˆ5 and since we are only considering complex structure dependent quantities, it doesn’t matter if we resolve via blow-ups of the ambient space. Therefore, we ﬁnd that n=4 :

R∆ (0) ≃ H 0 (Y4 , Ω4Y4 ) = H 4,0 (Y4 ) .

(6.67)

It is easy to see that R∆ (0) = S5 (0) = h1i .

(6.68)

Note that p∆ can not be linear in any homogeneous coordinate Xi , since the Q generic monomial in p∆ has degree ν ∗ ∈∆∩N Xi . Therefore, this element has i to correspond to the holomorphic four-form Ω on Y4 and can be represented as Z d ωA5 Ω∼ ∈ H 4,0 (Y4 ) . (6.69) p ∆ Γ up to a function holomorphic in the complex structure moduli aj .

78

6.5 Hodge variation in semi-ample hypersurfaces Let us now determine the number of holomorphic (n − 1)-forms of the hypersurfaces R, i.e. hn−1,0 (R), in terms of the toric data. Given a divisor D∆ with Newton-Polyhedron ∆ over a toric variety A with polyhedron ∆∗ , the degree [D∆ ] submodule S(D∆ ) of its homogeneous coordinate ring S is given by M Y hν ,ν ∗ i S(D∆ ) = C· Xi j i . (6.70) νi ∈∆∩M

νi∗ ∈∆∗ ∩N

It can be shown, that the dimension of the quotient R∆ (D∆ ) with D∆ represented by a transverse polynomial p∆ as a C-module is the given by the number of interior integral points of ∆ M Y hν ,ν ∗ i R∆ (D∆ ) = C· Xi j i . (6.71) νj ∈int(∆)∩M

νi∗ ∈∆∗ ∩N

Therfore, we ﬁnd that hn−1,0 (Rn−1 ) = ℓ′ (θ) ,

dim(θ) = n .

(6.72)

From this we have now seen, how to represent the holomorphic forms on the various toric hypersurfaces we encountered in the Gysin-sequences of the previous sections. In the next section, we will use these representations to gain further insights into the behaviour of these holomorphic forms under complex structure variation.

6.5

Hodge variation in semi-ample hypersurfaces

In this section, we want to investigate the complex structure variations of the semiample hypersurface Y4 in the simplicial toric variety A5 . Due to the fact that these are independent of the Kähler moduli and hence the volumina of the blow-up divisors resolving singularities, it does not matter if we blow-up the singularities for the complex structure variations. The variations around a point a ∈ Mc in complex structure moduli space Mc of the Calabi-Yau fourfold Y4,a can be parametrized by H 1 (Y4,a , T Y4 ) ≃ H 3,1 (Y4,a ) ,

(6.73)

79

6 Geometry of toric Calabi-Yau fourfold hypersurfaces where the isomorphism is given by contraction with the no-where vanishing holomorphic four-form Ω(a). We will drop the a-dependence in the notation for Y4,a and just write Y4 in the following. From the Gysin-sequence (6.36) we can deduce the splitting (we are dealing with free groups) M 3,1 H 3,1 (Y4 ) ≃ GrW (Y4 ∩ T) ⊕ H 2,0 (Di′ ) . (6.74) 4 H νi∗

The ﬁrst part corresponds to the deformations of the polynomial p∆ via monomials pν with ν ∈ ∆ ∩ M , therefore we will refer to it as algebraic deformations of the complex structure. It can be shown, [92, 94] that these are given by 3,1 (Y4 ∩ T) ≃ R∆ (−KA5 ) . H 3,1 (Y4 )alg ≃ GrW 4 H

(6.75)

They are represented by the Poincaré residue construction as H 0 (A5 , Ω5 (−2KA5 )) → H 3,1 (Y4 ) ⊕ H 4,0 (Y4 ) Z pν pν dω → 7 A5 2 2 dωA5 . p∆ Γ p∆

(6.76)

H 0 (A5 , Ω5 (−2KA5 )) = S5 (−KA5 ) ,

(6.77)

Note that and the kernel is given by h∂i p∆ , νi∗ ∈ ∆∗ i. It was shown by Batyrev in [99] that the dimension of the space of these algebraic complex structure deformations is X h3,1 ℓ′ (θ) . (6.78) alg = ℓ(∆) − 6 − dim(θ)=4

Due to the residue representation we can see that Z Z pν ∂ 1 ∂ Ω= dωA5 = − 2 dωA5 . ∂aν ∂aν Γ p∆ Γ p∆

(6.79)

The remaining complex structure deformations arise from divisors that are blow-ups of singular surfaces Sα M H 3,1 (Y4 )non−alg ≃ H 2,0 (Sα ) ≃ Rα (−KA5 |A3,α + KA3,α ) . (6.80) dim(θα )=3

80

6.5 Hodge variation in semi-ample hypersurfaces The number of these deformations is ℓ′ (θα ), the number of holomorphic (2, 0)forms on the toric divisors Dlα times the number of necessary blow-ups ℓ(θα∗ ): X h3,1 = ℓ′ (θα )ℓ′ (θα∗ ) . (6.81) non−alg dim(θα )=3

Note that we have in general the map p H 0 (Ad , Ωd (pD) ≃ Sd (pβ + KAd ) → Falg =

q dωAd 7→ pp∆

Z

Γ

p M

d−1−k,k Halg (D)

(6.82)

k=0

q dωA5 , pp∆

for a quasi-smooth semiample divisor D∆ with Newton-polyhedron ∆ the zero set of p∆ ∈ Sd [D∆ ] with degree [D∆ ] ∈ Ad−1 (Ad ) and Γ the usual one-dimensional curve in the complement of D∆ in Ad . This generates the algebraic part of the (d − 1)-dimensional cohomology group of D∆ , called horizontal cohomology of D∆ . 2 This divisor D∆ needs not to be Calabi-Yau and hence we can also use this construction for bases B3 of elliptically ﬁbered Calabi-Yau fourfolds that are toric hypersurfaces with h2,1 (B3 ) > 0 as considered in section 5.2. A simple example is given by the cubic hypersurface B3 = P4 [3] in P4 , for which p∆ is just a general degree three homogeneous polynomial in the coordinates [X1 , . . . , X5 ]. In this case ∆ is not reﬂexive and its vertices are not integral, but the same construction applies. The (2, 1)-forms can be represented as Z Xi 2,1 (B3 ) , (6.83) γi = 2 dωP4 ∈ H Γ p∆ In particular, we ﬁnd the non-trivial Hodge numbers h0,0 = h1,1 = h2,2 = h3,3 = 1 (generated by the hyperplane class of P4 ) and h2,1 = 5. These threeforms do not correspond to the complex structure deformations, as there is no holomorphic no-where vanishing three-form. Note that we have h3,0 = 0 2

3,1 Note that in GrW (Y4 ∩ T) the GrW 4 H 4 is a weight grading up to order four which corresponds to the pole order of the rational forms necessary to produce all four-forms that arise from the residue construction.

81

6 Geometry of toric Calabi-Yau fourfold hypersurfaces which can be seen from the positive degree of dωP4 /p∆ . The geometry of this hypersurface was discussed in detail by Clemens and Griﬃths in [100]. In the following we will focus on the three-forms of Calabi-Yau hypersurfaces for which a similar construction can be considered. First, however, we will use the obtained representations to count the Hodge numbers of a Calabi-Yau fourfold hypersurface.

6.6

Counting the Hodge numbers of a semiample Calabi-Yau fourfold hypersurface

The number of components of an n-semiample toric divisors D ′ is given by h0,0 (D ′ ) = 1 ,

n = 2, 3

h0,0 (D ′ ) = ℓ′ (θ) + 1 ,

n = 1.

(6.84)

For each face θ ∗ we have ℓ′ (θ ∗ ) toric divisors, where ℓ′ (θ ∗ ) counts the number of interior integral points of the (4 − n)-dimensional face θ ∗ of ∆∗ , n < 4 corresponding to the blow-up divisors necessary to resolve the ambient space A5 . Combining now the insights from the Gysin-sequence of the previous section and the Poincaré representation of the holomorphic forms, we can deduce the Hodge numbers of smooth Calabi-Yau fourfold Y4 . This was already discussed in [37]. We ﬁnd for the number of (1, 1)-forms that we have three parts. The ﬁrst arises from the ambient space Ad+1 as seen in (6.21) with h1,1 (Ad+1 ) = ℓ(∆∗ ) − 6, to obtain the cohomology of the hypersurface, however, we need to substract the blow-up divisors over points that do not lie on the general hypersurface Y4 , which are the zero-semiample divisors. To account for nontoric divisors arising from one-semiample divisors we need to add the ℓ∗ (θ) components over ℓ∗ (θ ∗ ) points: h1,1 (Y4 ) = ℓ(∆∗ ) − 6 −

X

dim(θ)=0

ℓ′ (θ ∗ ) +

X

ℓ′ (θ)ℓ′ (θ ∗ ) ,

(6.85)

dim(θ)=1

where ℓ(∆∗ ) denotes all integral points of ∆∗ . This turns out to be exactly dual to the formula for h3,1 (Y4 ) which combines the algebraic and non-

82

6.6 Counting the Hodge numbers of a semiample Calabi-Yau fourfold hypersurface algebraic complex structure deformations as X h3,1 (Y4 ) = ℓ(∆) − 6 − ℓ′ (θ) + dim(θ)=4

X

ℓ′ (θ)ℓ′ (θ ∗ ) ,

(6.86)

dim(θ)=3

counting the number of elements of chiral rings with a certain degree as described in section 6.5. The duality between the two Hodge numbers of h1,1 and h3,1 is a ﬁrst manifestation of mirror symmetry on Calabi-Yau fourfolds, as described in [76, 41] and we have already seen in chapter 4. The mirror symmetry of toric hypersurfaces can be simply expressed as the exchange of the dual polyhedra ∆ and ∆∗ , as ﬁrst realized by Batyrev [93]. Due to the fact that the non-trivial three-forms arise solely from toric divisors as discussed around (6.48), the Hodge number h2,1 per two-dimensional face θα is the product of one-forms on the the Riemann surface Rα given by ℓ′ (θα ) and the number of blow-up divisors necessary to resolve the singularity along Rα given by ℓ′ (θ ∗ ). The result is X h2,1 (Y4 ) = ℓ′ (θ)ℓ′ (θ ∗ ) , (6.87) dim(θ)=2

which is invariant under the exchange of ∆ and ∆∗ as predicted by mirror symmetry. The remaining non-trivial Hodge number h2,2 can be computed from the other three via index theorems as already seen in (2.5). These split into four orthogonal parts.3 The ﬁrst two parts a horizontal, they arise from the complex structure variations of algebraic and non-algebraic (3, 1)-forms and are also primitive, i.e. their product with the Kählerform J vanishes. The second part is the vertical cohomology that arises form wedge products of (1, 1)-forms corresponding to intersections of toric and non-toric divisors. In the next section we will focus more on the origin and complex structure dependence of the non-trivial three-forms of toric Calabi-Yau fourfold hypersurfaces in toric varieties which was not studied before. 3

As we do not prove these statements here, this should be interpreted as a conjecture. It is the Calabi-Yau fourfold version of the cohomology splitting shown in [92] for general semi-ample hypersurfaces in toric varieties.

83

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface

Using the arguments presented in chapter 6, we deduced that the threeform cohomology of a semiample Calabi-Yau fourfold hypersurface of a toric variety is induced by the holomorphic one-forms of a Riemann surface. The relevant notions of the Riemann surface will be introduced in section 7.1 and then lifted to the full fourfold geometry in section 7.2. In course of this we will identify the quantities of section 2.2 of the toric hypersurface setting and introduce a class of simple example geometries, the weighted projective spaces in section 7.3.

7.1

Three-form periods from Riemann surfaces

We want to specify now to the toric divisors of the Calabi-Yau fourfold hypersurface Yˆ4 providing non-trivial three-forms. The ﬁbration structure of these divisors with base a Riemann surface will enable us to apply the wellestablished theory of periods of Riemann surfaces to ﬁnd a description of the periods of the three-forms on a Calabi-Yau fourfold realized as a toric hypersurface. To do so, we ﬁrst introduce the general theory of Riemann surfaces necessary to understand the period construction, as is by now textbook material [88]. Then we will use the representation of holomorphic one-forms on ample toric hypersurfaces as used in [98, 91, 92] to ﬁnd the periods of these

85

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface Riemann surfaces. These periods satisfy a set of second order diﬀerential equations, the Picard-Fuchs equations, as was derived similarly in [101]. In the speical case of two-semiample toric divisors Dl′α with two-dimensional Newton-polyhedron θα , the ﬁbration structure described in (6.48) reads Elα

ilα

Dl′α ⊂ Dlα πlα Rα ⊂ A2,α

(7.1)

where Rα is a (compact) Riemann surface embedded in a two-dimensional simplicial complete toric ambient space A2,α by the polynomial pα . The ﬁber Elα is two-dimensional and toric and the Hodge types of its cohomology is independent of the complex structure of Yˆ4 . In particular, we see that the full complex structure dependence of the non-trivial three-forms is captured by the holomorphic one-forms of the Riemann surfaces Rα . Therefore we will now discuss the general theory of periods on Riemann surfaces R. A Riemann surface R is a compact Kähler manifold of complex dimension one and we are interested in its non-trivial cohomology, the non-trivial one-forms and their complex structure dependence. To do this, we introduce appropriate bases for the one-forms, ﬁrst a topological and then a holomorphic basis. The Riemann surface R we consider has genus g = h1,0 (R) and is equipped ˆ a ∈ H1 (R, Z) with indices a = with a basis of integral one-cycles Aˆa , B 1, . . . , g and dual one-forms α ˆ a , βˆa ∈ H 1 (R, Z). We can choose this basis to satisfy canonically Z Z Z βˆa ∧ βˆb = 0 . (7.2) α ˆa ∧ α ˆb = 0 , α ˆ a ∧ βˆb = δab , R

R

R

For a n-dimensional Kähler manifold we can choose the holomorphic n-forms to vary holomorphically with the complex structure, [67]. In particular, we can choose a basis of holomorphic one-forms on R denoted by γa ∈ H 0 (R, Ω1R ) depending holomorphically on the the complex structure of R. The so called

86

7.1 Three-form periods from Riemann surfaces period matrices are obtained by integrating these one-forms over the basis of one-cycles as Z Z b ˆ ˆ γa . (7.3) γa , (Πa )b = (Πa ) = ˆb B

ˆb A

These g × g matrices are holomorphic in the complex structure. The period ˆ b and Π ˆ b are the column vectors of these matrices. These vectors vectors Π are obtained by integrating the full basis of holomorphic one-forms over one ﬁxed one-cycle. These 2g vecotrs are linearly indpendent over R and therefore generate a lattice M ˆ aZ ⊕ Π ˆ aZ . ˆ= Π (7.4) Λ a

Cg

H 1,0 (R).

in ≃ The period matrices allow to expand the holomorphic one-forms into the topological basis as ˆ a )b α ˆ a )b βˆb , γa = (Π ˆ b + (Π

(7.5)

and deﬁne the projection of the integral cohomology, the lattice H 1 (R, Z), to the eigenspace of complex structure H 1,0 (R) ⊂ H 1 (R, C). The object we want now to study is the Jacobian variety J 1 (R) of the Riemann surface R given by H 1,0 (R) ˆ. ≃ Cg /Λ (7.6) J 1 (R) = 1 H (R, Z) We can normalize the basis γa , since one of the two period matrices will ˆ a )b . This enables us to introduce the normalized be invertible, we choose (Π basis γ˜a ∈ H 1,0 (R) of holomorphic one-forms on R: Z −1 b ˆ γ˜a = δab , (7.7) γ˜a = (Π )a γb , ˆb A

which now depends meromorphically on the complex structure captured by the normalized period matrix c

ˆ −1 )a (Π ˆ c )b ifˆab = (Π

⇒

γ˜a = α ˆ a + ifˆab βˆb .

(7.8)

In the following we will assume our basis of holomorphic one-forms to always be of this form and hence we drop the tilde and write for the normalized basis γa ∈ H 1,0 (R). 87

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface It can now be shown that this holomorphic (in general meromorphic) normalized period matrix fˆab of the Riemann surface R satisﬁes the relations fˆab = fˆba ,

Re fˆab > 0 ,

(7.9)

the period matrix is symmetric and its real part is positive deﬁnite. Therefore, we can ﬁnd a positive deﬁnite quadratic form on H 1,0 (R) that is given in the normalized basis as Z γa ∧ γ¯b = 2 Re fˆab . (7.10) −i R

This normalized period matrix fˆab will be the physical quantity we are interested in. Especially its complex structure dependence will be of interest and we will study this by giving an explicit representation of the holomorphic one-forms for Riemann surfaces as hypersurfaces R in simplicial complete toric varieties A2 . Here we specialize to the more general construction as discussed in section 6.4 of mapping rational two-forms on A2 with poles along the hypersurface R of order r denoted by Ω2A2 (rR) to the middle cohomology of R via the Poincaré residue. The general description of these rational forms that arise from the restriction of the anti-canonical hypersurface on a ambient space A5 to A2 reads H 0 (A2 , Ω2A2 (rR)) = {

g dωA2 : g ∈ S2 (−rKA5 |A2 + KA2 )} prθ

(7.11)

≃ S2 (−rKA5 |A2 + KA2 ) . We denoted here by −KA5 |A2 ∈ A1 (A2 ) the divisor class of the restriction of the anti-canonical divisor class −KA5 of A5 to A2 . The representative of this divisor is given by R which is the zero set of pθ = p∆ |A2 ∈ S2 (−KA5 |A2 ). Similarly we denoted by −KA2 the anti-canonical divisor class of A2 and as we already seen in (6.58) the holomorphic volume form dωA2 of A2 has the same degree as the anti-canonical divisor class −KA2 . Using now the theory outlined in section 6.4 we can express the (anti-

88

7.1 Three-form periods from Riemann surfaces )holomorphic one-forms via the residue construction as Rθ (−rKA5 |A2 + KA2 )

→

q

7→

H 2−r,r−1 (R) , Z q r dωA2 . Γ pθ

r = 1, 2 (7.12)

Here Γ is a small one-dimensional curve winding around the Riemann surface R in A2 . Moving on to the study of the complex structure dependence of these forms, we see that due to the fact that R is an ample hypersurface its holomorphic one-forms are entirely determined by the polynomial pθ which is a restriction of p∆ to A2 . This implies that the complex structure of H 1 (R, C) can only depend on algebraic deformations of the Calabi-Yau fourfold and only the very few surviving the projection of the full polynomial p∆ to pθ . Recall that the family of Calabi-Yau hypersurfaces in the toric simplicial complete ambient space A5 was given as the zero set of Y X hν ,ν ∗ i+1 Xi j i (7.13) p∆ = aj νj ∈∆∩M

νi∗ ∈∆∗ ∩N

∈ S5 (−KA5 ) ≃ H 0 (A5 , OA5 (−KA5 )) . The restriction to A2 is simply given by X Y p∆ |A2 = pθ = aj νj ∈θ∩M2

hνj ,νi∗ i+1

Xi

(7.14)

νi∗ ∈θ ∗ ∩N2

∈ S2 (−KA5 |A2 ) ≃ H 0 (A2 , OA2 (−KA5 |A2 )) . Therefore, the complex structure of R is ﬁxed by the vector of the prefactors a = (aj ) of the monomials deformations pj corresponding to integral points νj of ∆. Therefore, we can denote the Riemann surface with complex structure at a point a in complex structure moduli space by Ra , this is the induced complex structure of the full Calabi-Yau fourfold Y4,a . From the previous analysis, we deduce that a one-form γ ∈ H 1 (Ra , C) on Ra will only depend on complex structure moduli corresponding to integral points in the interior of θ: ∂ γ(a) = 0 , ∀ γ ∈ H 1 (Ra , C) , νj ∈ / int(θ) . (7.15) ∂aj

89

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface In the following we will therefore consider only deformations of pθ by monomials pb corresponding to the integral points νb ∈ int(θ) and similar the corresponding complex structure coordinates ab . As we have seen the explicit representations of (1, 0)-forms γˆb (a) ∈ H 1,0 (Ra ) depend holomorphic on the coordinates ab . This makes the normalized period matrix fˆab (a) a holomorphic function of the complex structure moduli ab . Note that we will ﬁnd fˆab (a) to be a rational function of the moduli a, but we will only consider the normalized period matrix in a region of moduli space where it is holomorphic. In more mathematical language we can interpret the residue representation of the one-forms as local trivializations of the Hodge bundles with base the complex structure moduli space Mc of Y4 and ﬁbers given by H 1,0 (Ra ) and H 1 (Ra , C), respectively. These are holomorphic bundles, for details we refer to [67]. This will enable us to derive the complex structure dependence of the holomorphic (1, 0)-forms γb (a) =

Z

Γ

p′b dωA2 ∈ H 1,0 (Ra ) , pθ

νb ∈ int(θ) ∩ M .

(7.16)

Q Here we denoted by p′b = pb / ν ∗ ∈θ∗ Xi ∈ S2 (−KA5 |A2 + KA2 ). This is still i a monomial, since the corresponding integral point νb ∈ int(θ) ∩ M . We have p′b =

Y

νi∗ ∈θ ∗ ∩N2

hνi∗ ,νb i

Xi

∈ S2 (−KA5 |A2 + KA2 ) ,

νb ∈ int(θ) ∩ M .

(7.17)

We can now see easily what happens if we vary the complex structure by taking derivative with respect to the moduli ab . The ﬁrst derivative is given by Z ′ pb pc ∂ ∂ 1 γb (a) = γc (a) = − (7.18) 2 dωA2 ∈ H (Ra , C) . ∂ac ∂ab Γ pθ The ring structure of the chiral ring Rθ = S2 /h∂i pθ i determines the Hodge type of this form. Note that although S2 does not depend on a, Rθ does 90

7.1 Three-form periods from Riemann surfaces depend on a through pθ . We have Z ′ pb pc ∂ 1,0 (Ra ) , p′b pc ∈ h∂i pθ i , γb (a) = − 2 dωA2 ∈ H ∂ac p Γ Z ′θ pb pc ∂ 1 p′b pc 6∈ h∂i pθ i . γb (a) = − 2 dωA2 ∈ H (Ra , C) , ∂ac Γ pθ

(7.19) (7.20)

It can be shown in general that the one-forms γb and its ﬁrst derivatives are suﬃcient to generate all of H 1 (Ra , C) as a C-module. This implies in particular, that we can express the second derivatives of the γb in terms of lower derivatives. This can be seen from Z ′ pb pc pd ∂ ∂ γb (a) = 2 dωA2 ∈ H 1 (Ra , C) (7.21) ∂ad ∂ac p3θ Γ ⇒

p′b pc pd ∈ h∂i pθ i .

Expressing the monomials of the second derivatives p′b pc pd in terms of lower degree monomials and partial derivatives of pθ introduced coeﬃcients depending rationally on the complex structure moduli ab . These coeﬃcients are structure constants of the chiral ring Rθ . Therefore, we can write ef ∂ f ∂ ∂ γb (a) = c(1) (a)cdb + c(0) (a)cdb γf (a) . ∂ac ∂ad ∂ae ef

(7.22) f

Here we denoted the structure constants of Rθ as c(1) (a)cdb and c(0) (a)cdb , respectively. These are rational functions in the complex structure moduli ab determined by modding out the partial derivatives of pθ from the homogeneous coordinate ring S2 of the ambient space A2 . From the previous considerations, it is easy to see that these structure constant are symmetric in their lower and upper indices. The relations between the second derivative and the lower derivatives are Picard-Fuchs equations which are central in the derivation of the complex structure dependence of the holomorphic one-forms γb and the normalized period matrix fˆab . Due to the fact the Picard-Fuchs equations are determined by the structure constants of the chiral ring Rθ which is a quotient of the full Jacobian R∆ of the Calabi-Yau fourfold Y4 , the ﬂat complex structure coordinates z K (a)

91

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface for which the structure constants of R∆ trivialize, as described in [89, 41], are also ﬂat coordinates for the Hodge bundles of R trivializing the structure constants in (7.22), and we ﬁnd ∂ ∂ γb (a(z)) = 0 . ∂z L ∂z K

(7.23)

This implies that in the ﬂat coordinates γa (z) depend at most linearly on the complex structure moduli z K . Integrating these over a basis of topological ˆ b )a and one-cycles as introduced in (7.2) we obtain the period matrices (Π ˆ b )a also depending at most linearly on the ﬂat coordinates z K . Therefore (Π we ﬁnd that the normalized period matrix fˆab (z) can be expanded around the large complex structure point z ≫ 1 as ˆ Kab + Cˆab + O(z −1 ) . fˆab (z) = z K M

(7.24)

ˆ Kab , Cˆab ∈ C can be determined from boundary conditions The constants M as found in [65] for the large complex structure point. There we found that ˆ Kab correspond to certain intersection numbers of the mirthe numbers M ror Calabi-Yau fourfold. The coeﬃcients of (7.24) are likely to be further restricted by shift-symmetries of the intermediate Jacobian (7.6). In the next section we will lift these holomorphic one-forms to the full Calabi-Yau fourfold realized as a semiample hypersurface in a complete simplicial toric variety using the Gysin-map.

7.2

The intermediate Jacobian of a Calabi-Yau fourfold hypersurface

In this section we will lift the previously established theory for Riemann surfaces and their intermediate Jacobian to the three-form cohomology of the full Calabi-Yau fourfold realized as a semiample hypersurface in a simplicial complete toric variety. This will enable us to deﬁne the intermediate Jacobian of the Calabi-Yau fourfold hypersurface and express its metric in terms of the normalized period matrices of Riemann surfaces and certain intersection numbers of divisors inducing the three-form cohomology via the Gysin-map.

92

7.2 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface In the following we consider a smooth and semiample Calabi-Yau fourfold hypersurface Y4 embedded in a complete and simplicial toric variety A5 . As we have already seen, in this case the non-trivial three-form cohomology of Y4 is induced by the Gysin-map, (6.38). This is a topological mapping induced by the inclusion ιi of toric divisors Di′ into Y4 and hence respects the complex structure induced on the cohomology. We have the isomorphism M H 2,1 (Y4 ) ≃ H 1,0 (Di′ ) . (7.25) νi∗ ∈∆∗ ∩N

In section 6.3 we have seen that a toric divisor Di′ of Y4 can only have nontrivial one-forms if it is two-semiample, i.e. the integral point νl∗α of ∆∗ deﬁning the toric divisor Dl′α∗ is contained in a two-dimensional face θα∗ of ∆∗ , νl∗α ∈ int(θα∗ ). These divisors are ﬁbration over Riemann surfaces Rα deﬁned by the dual face θα of θα∗ . The ﬁbers Elα of these divisors are connected toric surfaces and can be assumed to not depend on the complex structure of Y4 , since they are toric. Therefore, we argued that M M (7.26) H 1,0 (Rα ) ⊗C H 0,0 (Elα ) . H 2,1 (Y4 ) ≃ ∗ )=2 ν ∗ ∈int(θ ∗ ) dim(θα α lα

Since we are now dealing with several Riemann surfaces Rα with genus gα we will denote their respective holomorphic one-forms for which we constructed representatives in the previous section by γaα ∈ H 1,0 (Rα ) ,

aα = 1, . . . , gα = ℓ′ (θα ) .

(7.27)

These lift to holomorphic (2, 1)-forms ψA of Y4 by ﬁrst pulling them back via πα∗ to the full toric divisor Dl′α with the projection πlα : Dα′ → Rα and then pushing forward via the Gysin-map ιlα ∗ induced by the inclusion ιl∗ : Dl′α → Y4 . Explicitly this is given by ψA = ιlα ∗ (πl∗α γaα ) ,

(7.28)

A = (α, lα , aα ) = (1, 1, 1), . . . , (k2 , ℓ′ (θα∗ ), ℓ′ (θα )) , where we made use of a multi-index A keeping track of the base Riemann surface Rα the toric ﬁber Elα and the holomorphic one-form γaα ∈ H 1,0 (Rα ). 93

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface Recall that k2 is the number of two-dimensional faces θα∗ of ∆∗ and hence counts the Riemann surfaces Rα . The projection πlα as well as the inclusion ιlα are both topological maps independent of the complex structure or the metric of Y4 . Therefore, also a basis of topological integral one-forms (ˆ αaα , βˆaα ) as introduced in (7.2) for each Rα will be pushed forward to a basis of topological three-forms (αA , β A ) of H 3 (Y4 , Z), via the same construction ˆ aα ) , β A = ιlα ∗ (πl∗α βˆaα ) ∈ H 3 (Y4 , Z) . αA = ιlα ∗ (πl∗α α

(7.29)

These also satisfy the canonical properties of (7.2) which will simplify the upcoming discussion. Note that we do only consider here integral cohomology without torsion. This implies that the intermediate Jacobian J 3 (Y4 ) also splits as a topological space using (7.26) into a direct product of intermediate Jacobian of Riemann surfaces as k2 Y ′ ∗ H 2,1 (Y4 ) (J 1 (Rα ))ℓ (θα ) . ≃ J (Y4 ) = 3 H (Y4 , Z) 3

(7.30)

α=1

This is the intermediate Jacobian we already introduced in section 2.2. In the upcoming discussion we will derive the normalized period matrix fAB (z) of Y4 , which can be seen to be a block-diagonal matrix with the blocks being the normalized period matrices of the Riemann surfaces Rα . Due to the direct sum in (7.26) these blocks are independent. At special points in complex structure, however, the lattice in H 2,1 (Y4 ) induced by H 3 (Y4 , Z) may degenerate. This will require an extension of the diagonal ansatz we consider here and we hope to come back to this situation in the future. In this work we will restrict ourselves to the non-degenerate case. As we have already seen in section 2.2, we can ﬁnd a positive quadratic form Q on the intermediate Jacobian J 3 (Y4 ) given by Z

ψA ∧ ∗ψ¯B Z Σ = −iv ωΣ ∧ ψA ∧ ψ¯B ,

Q(ψA , ψB ) =

(7.31)

Y4

Y4

94

ψA , ψB ∈ H 2,1 (Y4 ) .

7.2 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface Here we inserted the expansion of the Kählerform J = v Σ ωΣ into real harmonic two-forms ωΣ ∈ H 1,1 (Y4 ) as already introduced in (2.6). The twoforms ωΣ can be chosen to be Poincaré dual to a set of homologically inde′ of Y . As we have seen around (6.53) we can assume pendent divisors DΣ 4 these divisors to be induced by toric divisors of the ambient space and for simplicity we will assume that all the toric divisors of Y4 are connected. Therefore, we can also obtain the non-trivial two-forms as a push-forward of ′ via the the generator 1Σ of the constant functions on the toric divisors DΣ Gysin map as ωΣ = ιΣ∗ (1Σ ) ∈ H 1,1 (Y4 ) ,

′ ). 1Σ ∈ H 0,0 (DΣ

(7.32)

′ has several components, H 0,0 (D ′ ) In the case where the toric divisor DΣ Σ ′ . The generalization to has several generators, one for each component of DΣ non-connected toric divisors is straightforward, but will just clutter up the notation. Let us now evaluate Q for the constructed (2, 1)-forms of (7.28). The key insight that will allow us to reduce to Q to quantities accessible to calculation is to analyze the intersection structure implied by the integral in (7.31). Since all three of the appearing forms ωΣ , ψA , ψB arise from toric ′ , D ′ , D ′ it is natural to consider the intersection of these divisors divisors DΣ lβ lα which will result in a curve C in Y4 ′ C = DΣ ∩ Dl′α ∩ Dl′β ⊂ Y4 .

(7.33)

Evaluating the integral of (7.31) will hence only have a non-vanishing result if the homology class of C is the same as Rα and Rβ , respectively. This can be deduced from the fact that the three-forms ψA are induced by one-forms γaα that have support on Rα . Another way to see this is that C is again a hypersurface in the toric ambient space DΣ ∩ Dlα ∩ Dlβ . If the one-forms on this C lift to Y4 the hypersurface has to be two-semiample which requires C to be homologous to one of the Riemann surfaces Rα . Note that this also implies ′ to be two-semiample which is a severe restriction on the number of Kähler DΣ moduli Q can depend on. Consequently, all three divisors are ﬁbrations over the same Rα , but with possibly diﬀerent ﬁbers Elα corresponding to integral

95

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface points of int(θα∗ ). Therefore the intersections have the form ′ ˆ l n m · Rα . =M Dl′α ∩ Dn′ α ∩ Dm α α α α

(7.34)

ˆ l n m account for possible multi-components The intersection numbers M α α α of the intersection. These intersection numbers can be computed in the so S called generalized Hirzebruch-Jung Sphere-Tree, lα Elα , which is the union over all ﬁbers Elα at a point of Rα . Two components of this sphere-tree only intersect in codimension one subvarieties. The intersection numbers ˆ l n m are given by M α α α ˆl n m . Elα ∩ Enα ∩ Emβ = M α α α

(7.35)

We depicted the intersection structure in ﬁgure 7.1. In practice we can calulate these intersection numbers from the intersection numbers of Y4 by taking an intersection of four toric divisors with three of them as above ﬁbrations over Rα and the fourth one being transverse to Rα intersecting it in a single point (or several points, but then we have to take multiplicities into account). The easiest choice for the fourth divisor is the dual of Rα , we can call it Dα , which satisﬁes Dα · Rα = 1. Therefore, we ﬁnd that ′ ˆ l n m = D ′ ∩ Dn′ ∩ Dm ∩ Dα , M lα α α α α α

Dα · Rα = 1 .

(7.36)

ˆ l n m by This implies that we can compute the intersection numbers M α α α standard techniques, for example by calculating the intersection numbers of ﬁve divisors of the ambient space A5 and then choosing the ﬁfth divisor the anti-canonical divisor class of the Calabi-Yau fourfold Y4 . From this analysis we see that there is a convenient expansion of the Kählerform suited to calculate Q. Σ

J = v ωΣ =

n2 X X

α

v ωlα + . . . ,

ωlα = PD[Dlα ] .

(7.37)

α=1 lα

Here we only displayed the expansion in two-forms that will contribute to Q and used Poincaré duality to relate the divisor classes [Dlα ] and two-forms ωlα ∈ H 1,1 (Y4 ) in the natural way. Combining now the insights on the 96

7.2 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface

Enα

Elα

Emα

γaα

A2,α Rα

Figure 7.1: Intersection structure of the divisors Dl′α that are fibration over Rα with fiber Elα and holomorphic one-forms γaα .

97

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface intersection structure, we can express the positive bilinear form Q in a part depending on the intersection pattern independent of the complex structure and a second part only depending on the positive bilinear form on a Riemann surface Z lα ˆ Q(ψA , ψB ) = −iδαβ v Mlα mα nβ γaα ∧ γ¯bβ , (7.38) Rα

where we used the multi-indices A = (α, mα , aα ) and B = (β, nβ , bβ ). Geometrically this can be interpreted in the following picture. For ﬁxed Riemann surface Rα , the El -ﬁbers form a generalized Hirzebruch-Jung sphere-tree, which is usually used for the resolution of codimension two orbifold singularities, where a chain of P1 ’s (the spheres of the tree) are ﬁbered over the singularity locus. The intersection matrix of these spheres is then related to the symmetry group of the orbifold. We use here the resolution of codimension three singularities requiring complex two-dimensional resolution ﬁbers El that intersect in a more complicated pattern which is captured by the fully symmetric three-tensor Mlmn . The intersection number Mlmn are independent of the hypersurface and can also be computed directly in the ambient space geometry A5 , but depends well on the triangulation of ∆∗ as well as the Kähler moduli. The codimension three-singularity we also assume to be an orbifold singularity and it would be interesting to ﬁnd a group theoretic interpretation of the intersection matrices Mlmn . Let us now connect the result for Q in (7.38) to the general formula of section 2.2 given by (2.25). Therefore, we need to identify the intersection numbers (2.24) with (7.36) as M ˆ l m n δabα for α = β and Σ = lα α α α α MΣA B = (7.39) 0 otherwise , with mulit-indices A = (α, mα , aα ) and B = (β, nβ , bβ ). The second intersection number MΣ AB vanishes, MΣ AB = 0 ,

(7.40)

since we showed that we can choose a canonically normalized basis of topological three-forms as in (7.29) induced by the canonical basis of one-cycles

98

7.2 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface (7.2) on the Riemann surfaces Rα . This implies that the bilinear form Q will only depend on the real part of the normalized period matrix fAB as in (7.38) in contrast to the general form (2.25). As already mentioned, the normalized period matrix of (2.17) has on its (α) diagonal the normalized period matrices fˆaα bα of the Riemann surfaces Rα . This can be read oﬀ by inserting (7.10) into (7.38) ˆ l m n Re fˆ(α) , Q(ψA , ψB ) = 2 δαβ v lα M α α β aα bα and comparing to (2.25). The precise identiﬁcation is given by fˆ(α) δmα nα for α = β aα bα fAB = 0 otherwise ,

(7.41)

(7.42)

where we used again the multi-indices A = (α, mα , aα ) and B = (β, nβ , bβ ). Combining the identiﬁcations (7.39), (7.40) and (7.42) with the results on Picard-Fuchs equations of (7.22) we found the quantities of section 2.2 relevant for the three-form moduli dynamics on Calabi-Yau fourfold hypersurfaces in toric varieties. The normalized period matrix fAB shows a block structure, one block for each Riemann surface that serves as a basis of several toric divisors giving rise to the non-trivial three-forms on the hypersurface. The relations of the divisors forming a generalized sphere tree ﬁbered over a ﬁxed Riemann surface are encoded by the topological intersection number MΣA B that obtain a similar block-structure. Some of the important physical quantities relevant for eﬀective theories can be computed with this information as was done in [62, 102]. In contrast, some applications require a more general class of geometries. The non-Abelian structures discussed in [63, 103, 104] are not covered by the block-diagonal structures appearing in the hypersurface scenario and are likely to make an extension of our considerations to complete intersections necessary. This is an interesting topic that will provide exciting opportunities for the next generation of students. In the next section we will illustrate the previously encountered structures on the simplest non-trivial examples of Calabi-Yau fourfold hypersurfaces in simplicial complete toric ambient geometries. We will consider Fermat hypersurfaces in weighted projective spaces.

99

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface

7.3

Calabi-Yau fourfold hypersurfaces in weighted projective spaces

We end our discussion of the construction of three-form periods on CalabiYau fourfold hypersurfaces by giving simple examples. A particular wellsuited class of geometries are Fermat hypersurfaces in weighted projective spaces providing both simplicity as well as non-trivial features illustrated in the previous sections. These are examples of hypersurfaces in toric varieties and hence we can apply what we learned before directly in this context. The discussion here lays the groundwork for our examples we will discuss in the following section that also ﬁt into this scheme. The focus of our discussion will be on the calculation of the normalized period matrix fAB of (7.42) that was shown to only depend on Riemann surfaces that serve as bases for ﬁbrations of blow-up divisors resolving singularities along the Riemann surfaces. Therefore it is not necessary to blow-up the singularities and explicitly resolve them for the calculation of fAB , which is very practical to simplify our analysis. For the derivation of the intersection numbers MΣA B as in (7.39) the structure of these blow-ups is crucial. We close this section by specifying to geometries whose three-forms are all induced by a single toric divisor and hence by only one Riemann surface. A weighted projective space AD = PD (w1 , . . . , wD+1 ) is a simplicial complete toric space whose geometry is determined by its weights wi ∈ N. These spaces are in general singular and require blow-ups to resolve these singularities. For our ambient space A5 we will consider a weighted projective space with one weight w6 = 1 whose deﬁning polyhedron ∆∗ ⊂ Q5 = NQ can be represented by the simplex with the six vertices νi∗ = ei ∈ Z5 ,

ν6∗ = (−w1 , −w2 , −w3 , −w4 , −w5 ) ∈ Z5 . (7.43) 5 These vertices are integral, i.e. elements of the lattice N = Z with generators the unit vectors ei . Choosing w6 = 1 allows us to relate the toric divisor classes Di corresponding to the vertices to each other, as all of these divisors

100

i = 1, . . . , 5 ,

7.3 Calabi-Yau fourfold hypersurfaces in weighted projective spaces are rational equivalent to multiples of the divisor D6 = H as (7.44)

[Di ] = wi [H] .

The divisor class [H] can be interpreted as a generalization of the hyperplane class of classical projective spaces. The homogeneous coordinate ring of A5 (7.45)

S5 = C[X1 , . . . , X6 ]

has therefore the usual grading of a monomial by a positive number Y k X Xi i = wi ki ∈ Z≥0 . (7.46) degS5 i

i

In particular, we can choose globally quasi-homogeneous coordinates, denoted by [X1 : . . . : X6 ] = [λw1 X1 : . . . : λw6 X6 ] (7.47) that are invariant under a rescaling by a non-zero factor λ ∈ C − {0}. The polynomial p∆ ∈ S5 (−KA5 ) deﬁning the anti-canonical hypersurface sing Y4 ⊂ A5 has therefore degree d determined by X X X wi [H] , d = − KA5 = [Di ] = wi . (7.48) i

i

i

A polynomial p is called of Fermat type, if it is the sum of monomials containing only one variable Xi raised to certain power ki . Schematically this reads X k Xi i , ⇒ ki = d/wi ∈ N , pF ermat = (7.49) i

where we indicated already the condition on the degree d of our anti-canonical hypersurface p∆ . Choosing p∆ a deformation of a Fermat type polyhedron, we can represent the dual polyhedron ∆ as a simplex in Q5 = MQ with integral vertices νi ∈ Z5 = M corresponding to the monomials of (7.49). They are given by νi = −

X j

ej +

d ei ∈ Z5 , wi

i = 1, . . . , 5 ,

ν6 = −

X j

ej ∈ Z5 .

(7.50)

101

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface We denoted again by ei , i = 1, . . . , 5 the unit vectors generating the lattice Z5 ⊂ Q5 . Note that the assumptions that we can ﬁnd a anti-canonical hypersurface that allows for a Fermat representative of its class is a very restrictive assumption that will, however, simplify our calculations considerably. In general ∆ will not be a simplex and have more vertices. We will in the following consider small deformations of Fermat hypersurfaces that will make them non-degenerate. Pure Fermat hypersurfaces have a high degree of symmetry and hence will introduce orbifold singularities that do not stem from the ambient toric space. Small deformations by monomials in p∆ will resolve these singularities. Therefore, we consider p∆ of the form X k X p∆ = Xi i + aν pν (7.51) i

ν∈θ,codim(θ)>1

where we have chosen a set of inequivalent deformations by restricting to monomials that correspond to integral points ν not contained in vertices or edges of ∆. These monomials can be reabsorbed by linear coordinate redeﬁnitions and hence are equivalent to the remaining monomials up to derivative ∂i p∆ . We will denote a degree d hypersurface in the weighted projective space by PD (w1 , . . . , wD+1 )[d]. In particular we will consider X Y4sing = PD (w1 , . . . , w5 , w6 = 1)[d] , wi | d , d = wi . (7.52) i

For the resolved smooth fourfold Y4 to have non-trivial three-form cohomology, we need Y4sing and hence also A5 to have codimension three orbifold singularities. A toric surface A2 of C3 /Zn singularities in the ambient space A5 exists if and only if exactly three of the six weights wi have a common divisor n. Up to renaming the coordinates we can assume that n | w3 , w4 , w5 ,

n 6 | w1 , w2 , w6 .

(7.53)

Therefore A2 is the subspace of A5 given by X1 = X2 = X6 = 0 which is the intersection of three toric divisors A2 = D1 ∩ D2 ∩ D6 ⊂ A5 . 102

(7.54)

7.3 Calabi-Yau fourfold hypersurfaces in weighted projective spaces The hypersurface Y4sing will intersect A2 in general transversely and hence inherit the C3 /Zn singularities along a Riemann surface R. To resolve the hypersurface Y4sing we will need to blow-up the ambient space several times in general. This will introduce the toric divisors that give rise to the three-forms of Y4 , their complex structure dependence is, however, completely captured by R ⊂ Y4sing which justiﬁes to work with the singular geometry in order to obtain the normalized period matrix fAB . In the special situation of Fermat hypersurfaces of degree d in the weighted projective space A5 , it is easy to see that n divides d, n | d for n the order of the cyclic orbifold group Zn . From (7.54) it is easy to see that A2 is also a weighted projective space A2 = P2 (w3 , w4 , w5 ) ,

S2 = C[X3 , X4 , X5 ] .

(7.55)

From the homogeneous coordinate ring S2 and the fact that X3 , X4 , X5 have a common divisor n, it is easy to see that we can represent A2 also as A2 ≃ P2 (w3 /n, w4 /n, w5 /n) ,

(7.56)

since they have the same homogeneous coordinate ring S2 = C[X3 , X4 , X5 ]. Restricting p∆ to A2 produces pθ where θ is the face with vertices ν3 , ν4 , ν5 . The restricted polynomial pθ has the same degree d and therefore, we can represent the Riemann surface R of singulalarites in Y4sing by R = P2 (w3 /n, w4 /n, w5 /n)[d/n] .

(7.57)

This is still a Fermat hypersurface, since we obtain it from (7.51) by setting X1 = X2 = X6 = 0 and this can be written as pθ = X3k3 + X4k4 + X5k5 + X3 X4 X5

(7.58) X

deg(p′b )=w1 +w2 +w6

ab p′b (X3 , X4 , X5 ) ,

where we made contact with the notation introduced in section 7.1 using the monomials p′a ∈ S2 (−KA5 |A2 + A2 ) = S2 (w1 + w2 + w3 ) ,

νa ∈ int(θ) ∩ M ,

(7.59)

103

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface whose equivalence classes in Rθ = S2 /h∂i pθ i form a basis of the monomials of degree w1 + w2 + w6 up to linear transformations. These monomials correspond to the integral interior points νa ∈ int(θ) ∩ M . Recall that the number of these interior points ℓ′ (θ) is the genus g of the Riemann surface R. With construction of the holomorphic one-forms of R outlined in section 7.1 in mind, we will now proceed with the holomorphic volume form of A2 deﬁned in (6.58). This simpliﬁes in the situation of A2 = P2 (w3 , w4 , w5 ) drastically to dωA2 = w3 X3 dX4 ∧dX5 −w4 X4 dX3 ∧dX5 +w5 X5 dX3 ∧dX4 ∈ Ω2A2 , (7.60) which has degree w3 + w4 + w5 or in terms of divisors classes −KA2 = (w3 + w4 + w5 )[H]. This holomorphic volume-form enables us to deﬁne the global meromorphic two-forms of A2 with ﬁrst order poles along R as p′a dωA2 ∈ H 0 (A2 , Ω2A2 (R)) . pθ

(7.61)

These forms have degree zero, therefore they are independent of the quasiprojective equivalences of the weighted projective space and hence they are globally well-deﬁned on A2 . The ﬁrst order pole facilitates the residue construction which enables us to associate to the monomials p′a the holomorphic (1, 0)-forms Z ′ pa dωA2 ∈ H 1,0 (R) , νa ∈ int(θ) ∩ M , (7.62) γa = Γ pθ where Γ is as usual a small one-dimensional curve winding around R in A2 . To derive the Picard-Fuchs equations of (7.22), we need to apply the relations in the Jacobian ring Rθ of the Riemann surface to relate the second derivatives of γa with the respect to the complex structure moduli ab to its lower derivatives. This is, however, connected with a tremendous amount of work (at least O(g2 )) for which an adapted algorithm needs to be found and implemented into a computer program. We will outline the simplest case g = 1 in the upcoming section.

104

7.3 Calabi-Yau fourfold hypersurfaces in weighted projective spaces

E

π7

α ˆa

A2 R

Figure 7.2: Fibration structure of D7′ . The Riemann surface R is a hypersurface of the toric space A2 over which the toric surface E is fibered.

For general orbifold singularities along a curve R in a toric Calabi-Yau fourfold hypersurface Y4 , the toric blow-up divisors will intersect in complicated patterns, we need to understand, how to calculate the intersection numbers MΣA B of (7.39). The number of three-tensors grows with O(ℓ′ (θ ∗ )3 ), where ℓ′ (θ ∗ ) is the number of toric divisors necessary to resolve the orbifold singularity along R. To illustrate the concept, let us consider the simplest codimension-three orbifold singularity C3 /Z3 leading to n = 3. This simple singularity can be resolved by a single toric blow-up, with coordinate X7 and divisor D7′ = {X7 = 0}. This will be a ﬁbration over a Riemann surface R as before, but with only a single exceptional ﬁber E. The divisor D7′ corresponds to the

105

7 The intermediate Jacobian of a Calabi-Yau fourfold hypersurface integral interior point ν7∗ of the face θ ∗ spanned by ν1∗ , ν2∗ , ν6∗ given by 1 ν7∗ = (ν1∗ + ν2∗ + ν6∗ ) . 3

(7.63)

Following the construction of section 6.3 one can show that the exceptional ﬁber E of D7′ is E = P2 (w1 , w2 , w6 ) . (7.64) The ﬁber E is for general w1 , w2 not smooth, but resolving these singularities will not lead to new three-forms, as only H 0,0 (E) contributes in the formula (7.26), and this is unaﬀected by blow-ups. In the special situation of a single divisor D7′ inducing the non-trivial threeforms ψA on the smooth fourfold Y4 the discussion of section 7.2 simpliﬁes a lot. First, since we have only one divisor D7′ with ﬁber E the intersection numbers MΣA B of (7.39) reduce to a single number M . This number M is for a smooth ambient space a positive integer, depending on normalization this can be set to one M = 1. The holomorphic three-forms ψA correspond in this scenario exactly to the holomorphic one-forms γa of the Riemann surface R which is the base of the ﬁbration of D7′ . The multi-index A = (α, lα , aα ) = (1, 7, 1), . . . , (1, 7, g) of ψA can be replaced here by the index a = 1, . . . , g labeling the holomorphic one-forms γa ∈ H 1,0 (R). The formula (7.28) relating one-forms of R with three-forms of Y4 reads here ψA = ι7∗ (π7∗ γa ) ∈ H 2,1 (Y4 ) ,

a = 1, . . . , g .

(7.65)

For the positive bilinear form Q of (7.41) we calculate Q(ψA , ψB ) = 2 v 7 M · Re fˆab ,

(7.66)

where fˆab is the normalized period matrix (7.10) of R and v 7 the volume modulus of D7′ in the expansion of the Kähler form J as we have seen in (7.37). We conclude this section by a depiction of the encountered ﬁbration structure of D7′ as seen in Figure 7.2. This should serve the reader as a guideline for the next section, where we will discuss explicit examples of Calabi-Yau fourfold hypersurfaces in weighted projected spaces of Fermat type.

106

8 Calabi-Yau fourfold examples

In the this section we will apply the previously introduced concepts to construct two simple examples of Calabi-Yau fourfold with non-trivial fourfold cohomology. These two geometries are elliptically ﬁbered and hence allow to serve as a background for an eﬀective description of F-theory in four dimensions. We will highlight the implications of non-trivial three-form cohomology in these F-theory geometries and focus especially on the weak coupling limit following Sen. Tracing the three-form moduli and their couplings through the F-theory and weak coupling limits we strengthen the case for the necessity to extend the weakly coupled Type IIB orientifold framework to strongly coupled regions in complex structure space provided by F-theory compactiﬁcations.

8.1

General aspects

In order to describe the F-theory examples in the upcoming sections, we ﬁrst need to introduce some general aspects of F-theory on elliptically ﬁbered Calabi-Yau fourfolds realized as hypersurfaces in weighted projective spaces as we discussed in section 7.3. In the four-dimensional eﬀective theory we will ﬁnd that the three-form modulie NA yield complex scalar ﬁelds for general hypersurfaces in toric varieties. In a general F-theory compactiﬁcations these can have two possible origins. First, the NS-NS and R-R two-forms can have non-trivial zero-modes and for a second, the seven-branes may have

107

8 Calabi-Yau fourfold examples continuous Wilson-line moduli. These two types of moduli are in general indistinguishable on a Calabi-Yau fourfold as they both arise in the same way as we have seen before. They can only be separated after performing the F-theory limit and the weak coupling limit which we will describe in detail. After introducing the general aspects we will discuss simple examples that exhibit both types of moduli.

8.1.1

Weierstrass-form and non-trivial three-form cohomology

We start our discussion be specifying to a possibly singular Calabi-yau hypersurface Y4sing in a weighted projective space Y4sing ⊂ A5 = P5 (w1 , . . . , w5 , w6 = 1) ,

(8.1)

as already discussed in section 7.3. Furthermore, we will assume that we can ﬁnd a resolution by toric blow-ups of the ambient space A5 to a toric ambient space Aˆ5 with at most point-singularities that will have a anticanonical hypersurface Y4 that is in general smooth and has a non-trivial three-form cohomology. As already discussed in [93] such a resolution is not always possible for Calabi-Yau fourfolds, which is in contrast to the threefold situation where resolutions always exist. We made the complex structure dependence of the three-forms explicit in previous sections and we can derive this dependence from the induced complex structure variations of holomorphic one-forms on Riemann surfaces along which we have orbifold singularities in Y4sing . To discuss the three-form cohomology it will only be necessary to be able to resolve the singularities along these Riemann surfaces. In order to have a valid F-theory background we will restrict our considerations in the following to elliptically ﬁbered Calabi-Yau fourfolds with a section. To do so, we use a so called Weierstrass-models with the elliptic ﬁber a hypersurface in the weighted projective space Af iber = P2 (2, 3, 1). The full ﬁve-dimensional space A5 will be a ﬁbration with ﬁber Af iber over a toric basis B3 that will be a (blow-up of a) weighted projective space. B3sing = P3 (w1 , w2 , w3 , w6 = 1) .

108

(8.2)

8.1 General aspects Allowing B3 to be a blow-up of a weighted projective space enables us to consider for example generalized Hirzebruch surfaces that are P1 -ﬁbrations over two-dimensional toric varieties. Due to the basis B3 of the ﬁbration being toric it cannot carry a non-trivial three-form cohomology itself. Furthermore do we not restrict the polyhedron ∆∗base of the toric base to be convex and hence it is in general non-Fano (the anti-canonical bundle is not semiample in previous terminology). This requires often toric resolutions of A5 corresponding to adding integral vertices in the exterior or interior of ∆∗ . In contrast to the crepant resolutions we discussed before such a resolution will alter the geometry of the Calabi-Yau fourfold and hence also change the properties of the eﬀective ﬁeld theory. The weights of the full toric ambient space A5 in which our elliptically ﬁbered Calabi-Yau fourfold Y4sing will be embedded is completely determined by the geometry of the base B3sing A5 = P5 (w1 , w2 , w3 , w4 = 2w, w5 = 3w, w6 = 1) ,

(8.3)

w = w1 + w2 + w3 + w6 .

Here w is the degree of the anti-canonical divisor class of the base in its homogeneous coordinate ring S3 (−KB sing ) = S3 (w). As is common in the 3 literature, we will denote the homogeneous coordinates of A5 with weight 2w and 3w by X4 = x and X5 = y, respectively. Imposing the CalabiYau condition on the hypersurface requires Y4sing to be a hypersurface with deﬁning polynomial p∆ of degree d = 6w that has so called Tate form y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 ,

(8.4)

with aj global sections of various multiples of the anti-canonical bundle −KB3 of the base B3 : aj ∈ H0 (B3 , −jKB3 ) ≃ S3 (−jKB3 ) , (8.5) where S3 is the homogeneous coordinate ring of the base B3 . On the singular base B3sing , a weighted projective space, being a global section of −jKB sing 3 is equivalent to being a quasi-homogeneous polynomial in the homogeneous base coordinates of degree deg(aj ) = w · j. After performing toric blow-ups 109

8 Calabi-Yau fourfold examples the grading of S3 becomes more complicated and we need to use the grading by divisor classes. Performing blow-ups of the base that do not preserve the anti-canonical divisor class will hence also change the global sections aj and hence the structure of the full elliptically ﬁbered Calabi-Yau fourfold Y4 .

8.1.2

Sen’s weak coupling limit

To make contact with the regular type IIB orientifold picture as for example discussed in [34], we will introduce next the weak string coupling limit by Sen [105, 53] for which also more reﬁned versions exist [106]. Going to weak string coupling amounts to moving to a special point in complex structure moduli space of the fourfold geometry as we will now describe. Redeﬁning the variables of (8.4), we can always obtain the standard Weierstrass form given by y 2 = x3 + f x + g . (8.6) To analyze the underlying geometry we make this redeﬁnition more explicit. First, we see that we can write the global sections f, g in this form as 1 2 (b − 24ǫb4 ) ∈ S3 (−4KB3 ) , 48 2 1 g=− (−b32 + 36ǫb2 b4 − 216ǫ2 b6 ) ∈ S3 (−6KB3 ) , 864

f =−

(8.7) (8.8)

where also bi ∈ S3 (−iKB3 ) are global sections of −iKB3 that are related to the aj of the Tate form (8.4) as b2 = a21 + 4a2 ,

b4 = a1 a3 + 2a4 ,

b6 = a23 + 4a6 .

(8.9)

We also introduced in (8.7) the parameter ǫ that can be thought of as the coordinate of the complex structure moduli space measuring the distance to the weak coupling region. Sending ǫ to zero will provide us with the weak coupling description of the considered system whose details we consider next. Starting from an F-theory compactiﬁcation on the smooth elliptically ﬁbered Y4 we can ﬁnd a weak string coupling conﬁguration whose seven-branes are purely D7-branes or O7-planes. This conﬁguration can be found by sending ǫ → 0 as we will now argue. The complex structure τ of the elliptic 110

8.1 General aspects ﬁber is a function that depends on the base coordinates and is related to the Weierstrass-form (8.6) as j(τ ) =

4(24f )3 ∈ S3 (0) , ∆

∆ = 27g 2 + 4f 3 ∈ S3 (−6KB3 ) .

(8.10)

Here j is Klein’s j-function and ∆ is the discriminant whose zero-set is the degeneration locus of the elliptic ﬁber. Combining (8.10) with (8.7) we can expand ∆ and j to leading order in small ǫ as ∆=

1 2 2 ǫ b (b2 b6 − b24 ) , 64 2

j(τ ) = −

32b42 . (b2 b6 − b24 )ǫ2

(8.11)

Using the expansion j(τ ) = exp(−2πiτ ) + . . . for j around its single pole, we see that in the limit ǫ → 0 we have Im τ ∝ −log ǫ everywhere except at the vanishing locus of b2 . In Type IIB supergravity the axio-dilaton is given by τ = C0 + ie−φ with ehφi = gs the string coupling. This τ is geometrized in F-theory by the complex structure τ of the elliptic ﬁber and therefore we conclude that the limit ǫ → 0 can be interpreted as the weak string coupling limit gs → 0. As mentioned before, the non-perturbative seven-branes allow in this limit a global description by a conﬁguration of D7-branes and O7-planes. The factorisation of the discriminant in (8.11) allows to identify the locations of these seven-branes as O7 :

b2 = 0 ,

D7 :

b2 b6 − b24 = 0 .

(8.12)

The Calabi-Yau threefold Y3 of the corresponding weakly coupled Type IIB orientifold set-up can be constructed as a double-cover of the toric base B3 with branching locus the O7-planes. This can be done in the simplest fashion by representing Y3 as a hypersurface in the anti-canonical line bundle −KB3 of the base B3 with ﬁber cooridinate ξ and general deﬁning equation Y3 :

Q = ξ 2 − b2 = 0 .

(8.13)

In this description we can easily identify the holomorphic orientifold involution σ : Y3 → Y3 with σ 2 = id (id being the identity map) whose ﬁxed point 111

8 Calabi-Yau fourfold examples loci are the O7-planes. By construction σ acts only on the coordinate ξ as σ : ξ → −ξ which implies that σ = b2 = 0 is indeed the ﬁxed-point set given by (8.12). These involutions were in detail discussed in [107]. In the situation where the base B3 is a blow-up of a weighted projective space B3sing = P3 (w1 , w2 , w3 , w6 = 1) the double-cover Y3 can be embedded in a corresponding toric ambient space Aˆ4 which is a blow-up of A4 = P4 (w1 , w2 , w3 , w6 = 1, w) ,

w = w1 + w2 + w3 + 1 ,

(8.14)

as an anti-canonical hypersurface with Y3sing of degree 2w. After toric resolutions the ambient space Aˆ4 will be a P1 -ﬁbration over B3 that can be interpreted as a compactiﬁcation of the anti-canonical line-bundle −KB3 of B3 . This hypersurface description enables us in particular to apply the previously developed techniques also for the Calabi-Yau threefold Y3 . Let us now also discuss the fate of the toric divisors inducing the non-trivial three-form cohomology of Y4 under the weak coupling limit. The geometric interpretation works best in the Poincáre dual picture of non-trivial ﬁvecycles of the divisors that are non-trivial ﬁve-cycles on the full fourfold Y4 . Since the bases B3 of the elliptic ﬁbrations we consider are all toric, the non-trivial three-forms of Y4 need to have at least one-leg in the ﬁber and therefore the dual ﬁve-cycles need to be circle ﬁbrations over divisors in the base with its circle a cycle of the elliptic ﬁber. Expanding the M-theory three-form potential C3 as in [36] leads in this case to C3 = B2 ∧ α + C2 ∧ β + . . .

(8.15)

where α and β are a real basis of one-forms on the elliptic ﬁber dual to its two one-cycles A, B as we used for the general Riemann surfaces in (7.2). Having a toric divisor Dl′α of section 6.3 that is an elliptic ﬁbration and induces nontrivial three-forms immediately implies that it has the form Dl′α = Rα × Elα where Rα is the elliptic ﬁber over a toric divisor Elα of the toric B3 . Since Rα is a ﬁbration over Eα but also vice versa, the divisor Dl′α needs to be a direct product. From the direct product structure, we can deduce that Rα the normalized period matrix fˆ(α) = iτ which simpliﬁes to a single number needs to be constant over Elα . This single number is by construction the

112

8.1 General aspects axio-dilaton. In the weak coupling limit the three-form moduli N A can be identiﬁed with the so called odd moduli N A = Glα or two-form scalars that arise from the expansion of the complexiﬁed two-form 1,1 (Y3 ) . G = Glα ωlα = B2 + iτ C2 ∈ H−

(8.16)

1,1 (Y3 ) the subspace of two-forms ωlα of H 1,1 (Y3 ) that Here we denoted by H− are odd under the orientifold involution σ as σ ∗ ωlα = −ωlα . Geometrically this means that there are two four-cycles in Y3 that get interchanged under σ and map under projection to the base B3 to the same four-cycle Elα . For ﬁnite τ in (8.16) the Elα we have constructed will not intersect the O7planes and hence their double-cover in Y3 are two disjoint copies of Elα ⊂ B3 1,1 (Y3 ) are the Poincaré dual denoted by El±α ⊂ Y3 and the two-forms ωlα ∈ H− two-forms to their diﬀerence 1,1 (Y3 ) . ωlα = P D[El+α − El−α ] ∈ H−

(8.17)

For further details on the physics of odd moduli we refer to [34]. This is just a diﬀerent point of view on the situation we already encountered in section 5.3. We can, however, also ﬁnd a second kind of ﬁve-cycle. Starting with a four-chain in the base we can have a circle-ﬁbration that degenerates at the boundaries of this four-chain which are three-cycles. An easy example of this phenomenon is to represent a two-sphere as a circle ﬁbation over an interval, the one-chain, which degenerates at the boundaries of the interval. Taking the circle of the ﬁbration as a cycle of the elliptic ﬁber, we see that at the loci in the base where this cycle and hence the elliptic ﬁber degenerates the ﬁve-cycle induces three-cycles. The degeneration loci are associated to seven-branes by constructios and hence this kind of ﬁve-cycle induces possibly non-trivial three-cycles on seven-branes. On these seven-branes, which are geometrically divisors in the base, we can dualize these three-cycles to one-forms. Taking into account the monodromy properties of the resulting cohomology classes we hence deduce that the three-forms on the elliptically Calabi-Yau fourfold hypersurface split in the weak coupling limit into two

113

8 Calabi-Yau fourfold examples classes H 2,1 (Y4 )

−→

H 1,1 (Y3 ) , −

H 1,0 (S) , −

(8.18)

where S denotes a divisor in Y3 wrapped by a D7-brane. Both one-cycles of the elliptic ﬁbration are odd under the orientifold involution, as explained in [36], therefore also their linear combinations are odd and hence also the ﬁve-cycles with one leg in the ﬁber. There is a priori no reason why these arguments should not work for general elliptically ﬁbered Calabi-Yau fourfolds Y4 and therefore we conjecture that (8.18) holds in general for the three-form splitting in the weak-coupling limit. In the two subsequent sections we will give examples for the two-form scalar moduli in section 8.2 and an example of the second kind will be presented in section 8.3.

8.2

Example One: An F-theory model with two-form scalars

In this subsection we present the ﬁrst example of an elliptic Calabi-Yau fourfold geometry with non-trival three-form cohomology. In this simple geometry there will only be one non-trival (2, 1)-form whose complex structure dependence is therefore induced by a genus one Riemann-surface, a two-torus. In the speciﬁc case we discussed before this two-torus can be identiﬁed with the elliptic ﬁber over a divisor in the base over which the elliptic ﬁbration factors as a direct product as discussed at the end of the previous section. Due to the simple geometric structure of this example, we will be able to derive the Picard-Fuchs type equations for the periods explicitly and disucss the weak coupling limit in detail.

8.2.1

Toric data and origin of non-trivial three-forms

A list of elliptically ﬁbered Calabi-Yau fourfolds in weighted projective spaces was already presented in [37]. Amongst them was also a particular simple example constructed from the weighted projective space A5 = P5 (1, 1, 1, 3, 12, 18) . 114

(8.19)

8.2 Example One: An F-theory model with two-form scalars This space has two kinds of singularities that will be inherited by the anticanonical hypersurface Y4sing , that can be resolved by moving to a toric space Aˆ5 with at most point singularities. The singularity can easily be seen from the weights of A5 , since two of them have a common divisor 2, leading to C4 /Z2 singularities along a curve in A5 and three of the weights have a common divisor three leading to C3 /Z3 along a toric surface A2 in A5 . In the singular ambient space A5 the anti-canonical hypersurface is given by a polynomial p∆ that has a quasi-homogeneous degree of 36. We can introduce complex homogeneous coordinates on A5 denoted by [u : w : x : y] with the abbreviation u = (u1 , u2 , u3 ) and identiﬁcations with the usual homogenous coordinates Xi given in the table below. We can always bring the most general hypersurface equation of the type of p∆ into the form psing = y 2 + x3 + a ˆ1 xy + a ˆ 2 x2 + a ˆ3 y + a ˆ4 x + a ˆ6 , ∆

(8.20)

highlighting the elliptic ﬁbration structure and the coeﬃcient functions a ˆn that only depend on the coordinates u = (u1 , u2 , u3 ) and w which can be further split into 2n X (8.21) w2n−m cn,m (u) , a ˆn = m=0

where cn,m (u) are general homogeneous polynomials of degree 3m in u. In this description we can already ﬁnd the Riemann surface R of C3 /Z3 singularities in Y4sing that will give rise to the non-trivial three-form cohomology after resolving the singularities. It is simply given by restricting (8.20) to A2 ⊂ A5 given by u1 = u2 = u3 = 0 with equation R :

psing = y 2 +x3 + cˆ1 xy + cˆ2 w4 x2 + cˆ3 w6 y + cˆ4 w8 x+ cˆ6 w12 = 0 . (8.22) θ

Here we denoted by cˆn = cn,0 the constant non-zero coeﬃcients that remain after restricting (8.21) to A2 . The Z2 -and Z3 -singularities of the toric space A5 can be resolved by moving to the toric space Aˆ5 whose fan is uniquely determined by the cones generated from the rays through the integral points νi∗ of the following polyhedron ∆∗

115

8 Calabi-Yau fourfold examples

Toric data of Aˆ5

Example One: ν1∗

coords

ℓ1

ℓ2

ℓ3

= (

1

0

0

0

0) X1 = u1

0

1

0

ν2∗ = (

0

1

0

0

0) X2 = u2

0

1

0

ν3∗ = (

0

0

1

0

0)

X3 = w

0

0

1

ν4∗ = (

0

0

0

1

0)

X4 = x

2

0

0

ν5∗ = (

0

0

0

0

1)

X5 = y

3

0

0

ν6∗ = ( −1 −1 −3 −12 −18) X6 = u3

0

1

0

ν7∗ = (

0

−4

−6)

X7 = v

0

−3

1

ν8∗ = (

0 −1

0

0

−2

−3)

X8 = z

1

0

−2

0

.

(8.23)

In this table we also noted the three projective relations ℓi between the homogeneous coordinates of A5 . These are chosen to highlight the ﬁbration structure of the blown-up ambient space Aˆ5 and not necessarlily comprise the minimal set of generators, as is usual for the Mori-cone, the cone of all equivalence relations. This new ambient space Aˆ5 has only point singularities and therefore a general anti-canonical hypersurface in Aˆ5 will be smooth. This was already shown in [93]. In the resolved ambient space Aˆ5 we can construct a smooth Calabi-Yau hypersurface Y4 as the zero-locus of a polynomial p∆ ∈ S5 (−KA5 ). A general polynomial of this kind can always be brought into Tate form via coordinate transformations p∆ = y 2 + x3 + a1 xyz + a2 x2 z 2 + a3 yz 3 + a4 xz 4 + a6 z 6 ,

(8.24)

psing = p∆ |v=1,z=1 , ∆ where we used the homogeneous coordinates of (8.23). The coeﬃcient functions a ˆi depend on the remaining variables. The ambient space Aˆ5 can be seen to have a ﬁbration structure with ﬁber given by P2 (2, 3, 1) and coordinates [x : y : z] and coordinates for the base B3 are [u1 : u2 : u3 : v : w]. Due to the form of (8.24) we ﬁnd an elliptic curve embedded in P2 (2, 3, 1) over

116

8.2 Example One: An F-theory model with two-form scalars every point in B3 and hence we constructed an elliptically ﬁbered Calabi-Yau fourfold Y4 . The coeﬃcient function of (8.24) are explicitly given by an =

2n X

cn,m (u)w2n−m v m ,

m=0

a ˆn = an |v=1

(8.25)

where the coeﬃcient functions cn,m (u) are the same homogeneous polynomials of degree 3m in the variables u = (u1 , u2 , u3 ) as in (8.21). The toric base B3 is itself a ﬁbration over P2 with coordinates [u] and ﬁber a P1 with coordinates [v : w]. Therefore we can also interpret Y4 as a ﬁbration over P2 with ﬁber an elliptically ﬁbered K3 surface. The Hodge-numbers of Y4 can be calculated as shown in section 6.6 and are given by h1,1 (Y4 ) = 3,

h2,1 (Y4 ) = 1,

h3,1 (Y4 ) = 4358 .

(8.26)

The Calabi-Yau fourfold Y4 has therefore a single non-trivial (2, 1)-form whose origin we will describe in more detail in the following. As we have seen in chapter 7 we need an integral point in the interior of a two-dimensional face θ ∗ of ∆∗ to obtain a toric divisor with holomorphic one-forms. From the toric data in (8.23) we infer that the only integral point satisfying this condition is 1 ν7∗ = (ν1∗ + ν2∗ + ν6∗ ) , 3

D7 = {v = 0} ,

(8.27)

and the corresponding toric divisor of Aˆ5 is D7 . The induced toric divisor D7′ on Y4 is hence a hypersurface in D7 with equation p∆ |v=0 = pθ = 0, where θ is the dual face of θ ∗ . Using the scaling relation ℓ3 to set w = 1 we ﬁnd the equation for D7′ to be D7′ :

pθ = y 2 + x3 + cˆ1 xyz + cˆ2 x2 z 2 + cˆ3 yz 3 + cˆ4 xz 4 + cˆ6 z 6 = 0 , (8.28)

which is of the same form the hypersurface equation of R, (8.22), along which we had Z3 -singularities. Similarly, we have the same constant coeﬃcients cˆn = an (u, v = 0, w = 1) = cn,0 that are constant on all of Y4 , but depend

117

8 Calabi-Yau fourfold examples on the complex structure moduli. This illustrates the fact that in order to determine the complex structure dependence of the non-trival three-forms it does not matter if we deal with the full divisor D7′ or only its base R. The equation (8.28) is in Tate form1 and hence the Riemann surface is a torus and in particular the elliptic ﬁber over the base divisor v = 0. Due to the fact that the coeﬃcients cˆn of this equation do not depend on the base coordinates, it is easy to see that D7′ = R × E ,

R ≃ T2 ,

E ≃ P2 ,

(8.29)

as already advocated in subsection 8.1.2. The divisor v = 0 in the base B3 can easily be seen to be P2 with coordinates [u1 : u2 : u3 ]. The single holomorphic (2, 1)-form is therefore induced by the single holomorphic oneform on R for which we will determine its Picard-Fuchs equation in the upcoming section.

8.2.2

Picard-Fuchs equations on T 2

In this section we encounter the simplest example of theory we developed in section 7.1 and section 7.3 to calculate the Picard-Fuchs equations from which we can learn about the behavior of the normalized period matrix fAB . This section serves as an illustration of the general toric conecpts we introduced before. The form of pθ in (8.28) already implies that the toric ambient space A2 = P2 (2, 3, 1) of the Riemann surface R has coordinates [x : y : z] with weights 2, 3, 1, respectively. Therefore, its homogeneous coordinate ring S2 and the Jacobian ring Rθ are given by S2 = C[x, y, z] ,

Rθ =

C[x, y, z] , h∂x pθ , ∂y pθ , ∂z pθ i

(8.30)

and deg(pθ ) = 6. From the Poincaré residue construction, we ﬁnd therefore that H 1,0 (R) ≃ Rθ (0) , H 0,1 (R) = Rθ (6) , (8.31) 1

We will later see that it can also always be brought into Weierstrass form y 2 + x3 + f x + g = 0.

118

8.2 Example One: An F-theory model with two-form scalars and it can be shown that both are one-dimensional. The holomorphic oneforms H 1,0 (R) can be generated by Z 1 dωA2 ∈ H 1,0 (R) , (8.32) γ= Γ pθ with Γ a small one-dimensional curve in A2 − R winding around R and dωA2 the holomorphic volume-form of A2 as found in (7.60) for weighted projective spaces which can be represented by dωA2 = zdx ∧ dy − 2xdy ∧ dz + 3ydx ∧ dz .

(8.33)

We choose for pθ the representation in Weierstrass form (8.6), but other representations are possible using reparametrizations. This is equivalent to choosing an element of S2 (6) to represent the generator of Rθ (6). We use the Weierstrass form, since it allows for a comparison with the weak coupling description of the next section pθ = y 2 + x3 + z 6 + a xz 4 ,

(8.34)

with a = f the complex structure modulus. The function g is here just a constant that we can choose to be g = 1. The derivatives of the holomorphic one-form with respect to the complex structure modulus a are Z xz 4 1,0 ((R)a ) , (8.35) ∂a γ = − 2 dωA2 ∈ H p Γ θ Z 2 4 x z 2 ∂a γ = 2 dωA2 ∈ H 1 (R, C) . (8.36) p θ Γ Using the equivalence relations in Rθ we can derive the identity 3 (27 + 4a3 )x2 z 8 = 9z 8 ∂x pθ + (− az 7 + a2 z 5 x)∂z pθ , 2

(8.37)

which we can use to ﬁnd the Picard-Fuchs equation of γ at the vacuum conﬁguration with a = 0 7 (27 + 4a3 )γ ′′ + 12a2 γ ′ + aγ = 0 . 4

(8.38)

119

8 Calabi-Yau fourfold examples This is a well known diﬀerential equation appearing in the literature for example in [101] and combining these results with the boundary conditions derived in chapter 4 allows to ﬁnd a solution around a = 0. Due to the fact that the Weierstrass-form is so well studied, as for example reviewed in [36] we know already that we can ﬁnd the normalized period matrix fˆ(a) of the corresponding elliptic curve satisﬁes 24(4a)3 , ∆ = 27 + 4a3 . (8.39) ∆ Close to the three distinct zeroes of the discriminant ∆ given by ai = 3/41/3 ξ i , with ξ 3 = 1 the roots of a3 = 1, we ﬁnd j(ifˆ(a)) =

1 log(a − ai ) (8.40) 2πi up to SL(2, Z)-transformations. The boundary conditions derived in [65] are here trivially satisﬁed, fˆ = iτ , since the genus of the Riemann surface is one, and hence the coeﬃcient of the linear term is the triple intersection number of the one blow-up divisor in the mirror geometry. Due to the fact that the mirror is also smooth, this number is one. Another way to interpret this result stems from Seiberg-Witten theory, like reviewed in [108]. There the exact coupling of an SU (2) gauge theory was calculated using an elliptic curve and we ﬁnd here the same result as a coupling of scalars. The three singularities ai can be used as points around which we can expand the period-matrix f and these three coordinate patches couple the full moduli space of the gauge theory. However, two of these ai describe in SW language points of gauge enhancement. In contrast to this, we expand around the large complex structure point of the Calabi-Yau fourfold Y4 after transforming to the proper complex structure coordinates z K . In the SW theory this corresponds to the solution at inﬁnity in moduli space, i.e. deep in the Coulomb branch of the gauge theory. We have found that pθ is the equation for the elliptic ﬁber R over the divisor v = 0 in the base. This implies in particular, that pθ deﬁnes the complex structure τ |v=0 of the elliptic ﬁber R over this divisor. This is deﬁned such that up to SL(2, Z)-transformations we have a holomorphic one-form ifˆ(a) ∼

γ=α ˆ + τ βˆ ∈ H 1,0 (R) , 120

(8.41)

8.2 Example One: An F-theory model with two-form scalars for α ˆ , βˆ a canonical basis of H 1 (R, Z) as introduced in Equation 7.2. This τ is the axio-dilaton of Type IIB string theory varying over the base B3 . The important observation here is that τ |v=0 is constant along the divisor v = 0 in B3 , i.e. does not depend on the base coordinates, but does vary non-trivially with the complex structure moduli. To see this, we evaluate j(τ ) v=0 =

4(24f )3 = C(ˆ cn ) . 27g2 + 4f 3 v=0

(8.42)

In order to do that we determine f |v=0 , g|v=0 using (8.7), (8.9) with the an |v=0 determined from pθ given in (8.34). The result is a non-trivial function of the coeﬃcients cˆn of pθ , these are constants on Y4 , but do depend on the complex structure moduli z K of Y4 . Note that there are 4358 such complex structure moduli and we will not attempt to ﬁnd the precise map to the ﬁve coeﬃcients cˆn . Putting everything together, we can thus use τ |v=0 as normalized period matrix of the curve R that induces the non-trivial threeforms in the fourfold Y4 . Therefore, we have just shown that fˆ(z) = iτ |v=0 (ˆ cn ) ,

(8.43)

on the full complex structure moduli space of the Calabi-Yau fourfold.

8.2.3

Weak string coupling limit: a model with two-form scalars

We next examine the weak string coupling limit of the geometry introduced in subsection 8.2.1. Using Sen’s general procedure described in subsection 8.1.2 we add an additional coordinate ξ to the homogeneous coordinate ring of the base B3 . The scaling weight of ξ is the degree of the monomials associated to the anti-canonical bundle −KB3 , i.e. ξ has the degree of half the anticanonical class in the homogeneous coordinate ring of Aˆ4 . Therefore, we ﬁnd Y3 ⊂ A4 as the Calabi-Yau hypersurface obtained as the blow-up of the singular hypersurface Y3sing = P4 (1, 1, 1, 3, 6)[12]. Recalling that B3 is a P1 -ﬁbration over P2 , the double-cover Y3 turns out to be the double-cover of P1 ﬁbered over P2 . The double-cover of the P1 -ﬁber is a two-torus, or rather an elliptic curve, P2 (1, 1, 2)[4].

121

8 Calabi-Yau fourfold examples To make this more explicit we again use a toric description. The fan of the ambient space for the three-fold is given by the cones generated by the rays through the points Example 1: Toric data of A4

coords

ℓ1

ℓ2

ν3∗ = (

0

0

1

0)

z3 = w

1

0

ν4∗

= (

0

0

0

1)

z4 = ξ

2

0

ν6∗ = (

0

−2)

z6 = v

1

ν1∗ = (

0 −1

−3

1

0

0

0) z1 = u1

0

1

ν2∗ = (

0

1

0

0) z2 = u2

0

1

= ( −1 −1 −3

−6) z5 = u3

0

1

ν5∗

(8.44)

The hypersurface equation is then denoted by Q = 0 and from subsection 8.1.2 we can deduce that it has the form Q = ξ 2 − b2 (u, v, w)

(8.45)

in the fully blown-up ambient space with b2 = a21 + 4a2

(8.46)

speciﬁed by the Weierstrass-form of the corresponding fourfold in (8.25). One computes the Hodge-numbers to be h1,1 (Y3 ) = 3,

h2,1 (Y3 ) = 165 .

(8.47)

This example was already discussed in the context of mirror symmetry in [89]. The resulting threefold is an elliptic ﬁbration over P2 with two sections. It should be stressed that despite the fact that h1,1 (Y3 ) = 3 the toric ambient space only admits two non-trivial divisor classes. In fact, we will discuss in the following that this can be traced back to the fact that the divisor v = 0 yields two disjoint P2 when intersected with the hypersurface constraint. These are the two sections, i.e. two copies of the base. This is also noted in [107], where a classiﬁcation of orientifold involutions suitable for Type IIB orientifold compactiﬁcations is presented.

122

8.2 Example One: An F-theory model with two-form scalars To make this more precise, let us analyze the singularities of Y3sing = P4 (1, 1, 1, 3, 6)[12] and their resolutions via blow-ups further. The ambient space A4 = P4 (1, 1, 1, 3, 6) has C3 /Z3 -singularities along a curve P1 given by [0 : 0 : 0 : w : ξ]. The hypersurface intersects this curve in two points, which are identiﬁed as double cover of the point of the not yet blown up base B3sing = P3 (1, 1, 1, 3), where we ﬁnd C3 /Z3 -singularities. Blowing up this curve of singularities in the ambient space by adding ν6∗ leads to an exceptional divisor v = 0, which is a P2 ﬁbration over two points of the hypersurface. On the hypersurface Y3 we ﬁnd that the ambient space divisor v = 0 splits into two parts D6′ = {v = 0, Q(1) = 0} ∼ P2 ⊔ P2

(8.48)

c = b2 |v=0 = c21,0 + 4c2,0 .

(8.49)

√ with coordinates [u1 , u2 , u3 , v = 0, w, ± cw2 ]. Note that c is a constant, but depends on complex structure moduli. It is given by

The modulus c measures the separation between the two P2 in which D6 splits when intersecting the threefold hypersurface. For cˆ2,0 = 0 we ﬁnd that c is a perfect square. We next investigate the action of the orientifold involution σ : ξ → −ξ. From the coordinate description of D6′ we ﬁnd that the two disjoint P2 are interchanged by the involution σ. Therefore, we introduce the two non-toric ′ ′ that are the two disjoint P2 such that and D6,2 holomorphic divisors D6,1 ′ + D′ ∗ ′ ′ D6′ = D6,1 6,2 and σ (D6,1 ) = D6,2 . It is now straightforward to deﬁne an eigenbasis for the involution σ as K1+ = D4′ ,

K2+ = D6′ ,

′ ′ . − D6,2 K − = D6,1

(8.50)

Therefore, we conclude that h1,1 + (Y3 ) = 2,

h1,1 − (Y3 ) = 1 ,

(8.51)

which shows that there is one negative two-from which yields zero-modes for the R-R and NS-NS two-forms of Type IIB supergravity. Furthermore, we

123

8 Calabi-Yau fourfold examples can evaluate the intersection ring to be IY3 = 18(D6′ )3 + 144(D4′ )3 = 18(D6′ )3 − 6D1′ (D6′ )2 + 2(D1′ )2 D6′

(8.52)

′ ∩ D′ Note that D6,1 6,2 = ∅. Due to the symmetry between the components ′ of D6 and D4′ being exactly the ﬁxed point of this symmetry, we ﬁnd that ′ )3 = the intersections of K − appearing linearly vanish. We learn that (D6,1 ′ )3 = 9, (D ′ )2 D ′ = (D ′ )2 D ′ = 0 and D ′ (D ′ )2 = D ′ (D ′ )2 = 0. (D6,2 4 6,2 4 6,1 4 6,1 4 6,1

From this analysis we see that all toric divisors are invariant under the involution σ. Therefore, we can choose the divisor basis of the base B3 = ˆ 3 (1, 1, 1, 3) obtained from Aˆ4 = P ˆ 4 (1, 1, 1, 3, 6) by setting ξ = 0 . This P corresponds on the lattice level to projecting to Z3 , i.e. dropping the fourth coordinate of every vertex. Toric data of B3

coords

ℓ1

ℓ2

ν3∗ = (

0

0

1)

z3 = w

1

0

ν6∗ = (

0

0 −1)

z6 = v

1

ν1∗

−3

= (

1

0

0) z1 = u1

0

1

ν2∗ = (

0

1

0) z2 = u2

0

1

ν5∗ = ( −1 −1 −3) z5 = u3

0

1

(8.53)

As a consequence, we can use D6 and D1 as a basis for the divisors on B3 . For Y3 we can choose the corresponding basis via D4′ = 2D6′ + 6D5′ and ﬁnd 1 1 IB3 = 9D63 − 3D1 D62 + D12 D6 = (18D63 − 6D1 D62 + 2D12 D6 ) ∼ IY3 . (8.54) 2 2 ′ ′ This ﬁts the fact that Y3 double-covers B3 and D6,1 and D6,2 project down 2 to the same P in B3 . Let us now discuss what happens to the normalized period matrix fˆ = iτ |v=0 that we have derived in subsection 8.2.2, in the weak coupling limit of complex structure space. In this orientifold limit the ﬁeld τ0 = C0 + ie−φ is actually constant everywhere on Y3 /σ and becomes an independent modulus. The identiﬁcation fˆ = iτ0 then precisely yields the known moduli N = c−τ0 b

124

8.3 Example Two: An F-theory model with Wilson line scalars of the orientifold setting, where c, b are the zero-modes of the R-R and NS-NS two-forms along K− introduced in (8.50). c2 6= 0 for We close by pointing out that it is important to have c = cˆ21 + 4ˆ this weak coupling analysis to apply. Indeed, if we go to the limit c → 0 we ﬁnd a spliting of the O7-plane located at b2 = 0 into v = 0 and b′2 = 0. Not only would we ﬁnd intersecting O7-planes, but also the simple identiﬁcation fˆ = iτ0 would no longer hold.

8.3

Example Two: An F-theory model with Wilson line scalars

In this subsection we construct a second example geometry that we argue to admit Wilson line moduli when used as an F-theory background. In this example the three-forms of the Calabi-Yau fourfold stem from a genus seven Riemann surface. It turns out that this example features also other interesting properties, such as a non-Higgsable gauge group and terminal singularities corresponding to O3-planes.

8.3.1

Toric data and origin of non-trivial three-forms

Our starting point is the anti-canonical hypersurface in the weighted projective space A5 = P5 (1, 1, 3, 3, 16, 24) of degree d = 48. This space is highly singular, but admits an elliptic ﬁbration necessary to serve as an F-theory background. It is easy to see that we have a curve R along which we ﬁnd C3 /Z3 -singularities. In contrast to the ﬁrst example this curve R is not the elliptic ﬁber. It rather arises as a multi-branched cover over a P1 of the singular base B3sing . We can resolve part of the singularities of the ambient-space A5 by moving to a toric space Aˆ5 whose fan is obtained by the maximal subdivision of the 125

8 Calabi-Yau fourfold examples polyhedron ∆∗ of A5 : Example 2: Toric data of Aˆ5

coords

F

P2

B

E

ν1∗ = (

1

0

0

0

0)

z1 = w

0

0

1

1

ν2∗ = (

0

1

0

0

0) z2 = u1

0

1

0

0

ν3∗ = (

0

0

1

0

0) z3 = u2

0

1

0

0

ν4∗

= (

0

0

0

1

0)

z4 = x

2

0

0

1

ν5∗ = (

0

0

0

0

1)

z5 = y

3

0

0

0

ν6∗ = ( −1 −3 −3 −16 −24)

z6 = v

0

0

1

1

ν7∗ = ( ν8∗

= (

ν9∗ = (

0 −1 −1

−5

−8)

z7 = e

0

0

0

−3

0 −1 −1

−6

−9) z8 = u3

0

1

−3

0

0

−2

−3)

1

−3

1

0

0

0

z9 = z

(8.55)

Note already at this point, that the new ambient space Aˆ5 still contains singularities of the form C4 /Z2 :

(v, w, u3 , y)

→

(−v, −w, −u3 , −y)

(8.56)

and hence the hypersurface inherits singular points that do not allow for any crepant resolution as pointed out in [109]. This can be related to the presence of O3-planes. 2 A number of intriguing features of this model arises due to the geometry of the base B3 . It arises as a non-crepant blow-up of the weighted projective space B3sing = P3 (1, 1, 3, 3) with toric data given by Toric data of B3

P1

P2

ν1∗ = (

1

0

0)

z1 = v˜

1

0

ν2∗ = (

0

1

0) z2 = u ˜1

0

1

ν3∗ = (

0

0

1) z3 = u ˜2

0

0

1

1

ν4∗ = ( −1 −3 −3) ν5∗ = ( 2

coords

z5 = w ˜

0 −1 −1) z6 = u ˜3 −3

.

(8.57)

1

Various aspects of O3-planes have been discussed recently for example in [110, 111]

126

8.3 Example Two: An F-theory model with Wilson line scalars It can be interpreted as a generalization of a Hirzebruch surface, i.e. a P2 ﬁbration over P1 . We note in particular, that the point ν5∗ does lie in the interior of the convex hull of the remaining points and correspondingly the new polyhedron is no longer convex. The consequence is that the anti-canonical bundle −KB3 of the base has only global sections that vanish over the locus {˜ u3 = 0} ≃ P1 × P1 , i.e. −KB3 is not ample. In the F-theory picture this will lead to a non-Higgsable cluster as described in [112, 113], i.e. to the generic existence of a non-Abelian gauge group in this setting. The base B3 has been analyzed recently in detail in [114]. The ambient space Aˆ5 has the ﬁbration structure given by the projection π : Aˆ5 → B3 , which reads in homogeneous coordinates π : [v : w : u1 : u2 : u3 : x : y : z : e] 7→ [˜ v=v:w ˜=w:u ˜ 1 = u1 : u ˜ 2 = u2 : u ˜3 = eu3 ] .

(8.58)

Due to the non-Higgsable gauge group, Y4 can only be written in Tate form after blowing down the exceptional divisor e = 0, i.e. setting e = 1: p∆ = y 2 + ex3 + a ˆ1 xy + a ˆ 2 x2 + a ˆ3 y + a ˆ4 x + a ˆ6 = 0 ,

(8.59)

−1 −i these a ˆn have . Due to the properties of KB with a ˆi global sections of KB 3 3 common factors of u3 e = u ˜3 independently of the point in complex structure space. This shows that the non-Higgsable cluster with the enhanced gauge group is located on the divisor u ˜3 = 0 in the base. The singularity type can be easily read of by translating (8.59) into Weierstrass form using (8.7), (8.9). We then obtain a singularity of orders (2, 2, 4) = (f, g, ∆), where ∆ is the discriminant as above. This leads to a type IV singularity and the exact gauge group, which is either Sp(1) or SU (3), can be derived from monodromy considerations as we recall below. The generic anti-canonical hypersurface Y4 of the ambient space Aˆ5 has Hodge numbers

h1,1 (Y4 ) = 4,

h2,1 (Y4 ) = 7,

h3,1 (Y4 ) = 3443,

h2,2 (Y4 ) = 13818 . (8.60)

This implies that Y4 indeed has seven (2, 1)-forms and we claim that these arise from a single Riemann surface of genus g = 7.

127

8 Calabi-Yau fourfold examples There is only one two-dimensional face θ ∗ of the polyhedron spanned by that contains an interior integral point. This interior point is ν7∗ and we add this point to resolve the C3 /Z3 -singularity along the surface A2 = P2 (1, 1, 8) given as the subspace of A5 with w = v = x = 0. The anti-canonical hypersurface Y4 intersects A2 in a Riemann surface R given by R = P2 (1, 1, 8)[16], g = 7 . (8.61)

ν1∗ , ν4∗ , ν6∗

This can also be seen from the dual face θ whose inner points correspond to the monomials (8.62) p′a = ua1 u26−a ∈ Rθ (6), a = 0, . . . , 6 where we already divided out the common factor u1 u2 y as described in section 7.1. The exceptional divisor resolving this singularity is a ﬁbration over R with ﬁber E = P2 (1, 1, 16). Expanding the Weiserstrass form (8.6) of Y4 around the singular divisor De = {e = 0}, we ﬁnd g = g2 e2 + O(e3 ) ,

g2 = g2 (u1 , u2 )

(8.63)

and this g2 is precisely the degree 16 polynomial in u1 , u2 deﬁning the Riemann surface R by R : pθ = y 2 − g2 = 0 . (8.64) The resulting gauge group over D3 = {˜ u3 = 0} in B3 is Sp(1) for general 2 g2 and if g2 = γ , i.e. for g2 a perfect square, we have an enhancement to SU (3).

8.3.2

Comments on the weak string coupling limit

So what happens to this curve in the weak coupling limit? For a IV singularity, there should be no straightforward perturbative limit in which τ can be made constant and Im τ can be made very large over the base. The general hypersurface equation derived from the naive Sen limit is Q = ξ 2 − b2 = ξ 2 − u ˜3 · b′2 = 0 , 128

(8.65)

8.3 Example Two: An F-theory model with Wilson line scalars implying that the O7-plane splits in two intersecting branches, u ˜3 = 0 and b′2 = 0. At the intersection of the O7-planes perturbative string theory breaks down and hence there is no weak coupling description. However, we can still try to learn some of the aspects of the D7-branes in this setting. In fact, in the following we want to connect the curve (8.64) and Wilson line moduli located on D7-branes. As explained in [115] the number of Wilson line moduli arising from a D7-brane image-D7-brane on a divisor S ∪ σ(S) of the threefold Y3 is given by Number of Wilson line moduli on S :

h1,0 − (S ∪ σ(S)) .

(8.66)

These are the (1, 0)-forms on the union of S and its image that get projected out when considering the orientifold quotient. Therefore, we suggest that the Wilson lines arise in S ∪ σ(S) as arcs in S that connect two components of S ∩ σ(S). These arcs close to one-cycles in S ∪ σ(S), but get projected out when we take the quotient Y3 /σ = B3 . Note here that S ∩ σ(S) is equal to O7 ∩ S. In our situation Y3 is still a ﬁbration over P1 with coordinates [v : w] and hence this will also hold for S ∩ σ(S), i.e. we suggest that S ∩ σ(S) is a covering space of the base P1 given by S ∩ σ(S) = {ξ = 0, u ˜3 = 0, g2 = 0} ⊂ Y3 ,

(8.67)

where ξ = u ˜3 = 0 is the location of one branch of the O7-plane in Y3 . We also note that the divisor inducing the three-forms in the fourfold projects down to the u ˜3 = 0 divisor of B3 . Recall that the locations of the seven-branes in a general F-theory model are given by the zeroes of the discriminant ∆. We can expand ∆ around u ˜3 = 0 to ∆ ≈ b22 (b2 b6 − b24 ) = u ˜53 (b′2 )3 g2 + O(˜ u63 ) .

(8.68)

This implies that in the weak coupling limit g2 describes the intersection of the D7-brane in the form of a Whitney-Umbrella explained in [116] with the O7-branch given by u ˜3 = 0. For our considerations, it is just important that a D7-brane is path connected, but the shape away from the O7-plane is irrelevant for our analysis of Wilson lines. Therefore, we ﬁnd that S ∩ σ(S) =

16 [

i=1

{pi } × P1 ,

g2 (pi ) = 0 .

(8.69)

129

8 Calabi-Yau fourfold examples The points pi can be interpreted as branching loci of the auxiliary hyperelliptic curve which is given by (8.64). Hence we ﬁnd h1,0 − (S ∪ σ(S)) = 7 .

(8.70)

Choosing a normalized basis α ˆ a , βˆa for the cocycles arising from this proce1,0 (S ∪ σ(S)) as dure we can give a basis for H− 1,0 γa = α ˆ a + ifˆab βˆb ∈ H− (S ∪ σ(S)) ,

(8.71)

with fˆab the normalized period matrix of the curve R discussed in section 7.1. The coupling of the corresponding ﬁelds, the Wilson moduli NA = Na , is given by the the normalized period matrix fAB = fˆab of R. Let us close by making one ﬁnal observation for this example geometry. We can also resolve the Z2 -singular points of the fourfold by blowing-up the ambient space A5 . This requires adding the exterior point ∗ ν10 = (0, −2, −2, −10, −15) .

(8.72)

This has, however, drastic consequences. As already mentioned before, there is no way to resolve the Z2 -singular points in a crepant way, i.e. preserving the anti-canonical bundle of the ambient-space. Closer inspection of the blow-up tells us that this blow-up is not crepant, but leads to a Calabi-Yau hypersurface in a new ambient-space that has a diﬀerent triangulation not compatible with the old triangulation structure. This leads to a change in topology, which can be seen from the Hodge-numbers h1,1 new = 5,

h2,1 new = 0,

h3,1 new = 3435,

χ = χold = 20688 ,

(8.73)

with the Euler number χ being preserved. This extremal transition between the two fourfolds follows a similar pattern as the conifold transition along curves described in [117]. The relations to the non-trivial three-form cohomology can also be made precise: the blow-up obstructs precisely the complex structure deformations described by g2 setting it to zero and hence also obstructing the three-form cohomology. This obstruction leads to a further gauge-enhancement to G2 along D3 and also the weak coupling limit is no longer singular, i.e. the O7-plane does no longer branch.

130

9 Conclusions and Outlook

In the ﬁrst part of this work we studied the two-dimensional low-energy eﬀective action obtained from Type IIA string theory on a Calabi-Yau fourfold with non-trivial three-form cohomology. The couplings of the threeforms were shown to be encoded by two holomorphic functions fAB and hA B , where the former depends on the complex structure moduli and the latter on the complexiﬁed Kähler structure moduli. Performing a large volume dimensional reduction of Type IIA supergravity, we were able to derive hA B explicitly as a linear function. We argued that fAB and hA B computed on mirror pairs of Calabi-Yau manifolds will be exchanged, at least, if one considers the theories at large volume and large complex structure. In order to show this, we investigated the non-trivial map between the three-form moduli arising from mirror geometries and argued that it involves a scalar ﬁeld dualization together with a Legendre transformation. This can be also motivated by the fact that chiral and twisted-chiral multiplets are expected to be exchanged by mirror symmetry. We thus established a linear dependence of the function fAB on the complex structure moduli near the large complex structure point and determined the constant topological pre-factor. In this part we also included a discussion of the superymmetry properties of the two-dimensional low-energy eﬀective action. This action is expected to be an N = (2, 2) supergravity theory, which we showed to extend the dilaton supergravity action of [44]. The bosonic action was brought to an elegant form with all kinetic and topological terms determined by derivatives of a single

131

9 Conclusions and Outlook ˜ = K + e2ϕ˜ S, where K and S can depend on the scalars in chiral function K and twisted-chiral multiplets, but are independent of the two-dimensional dilaton ϕ. ˜ In the Type IIA supergravity reduction the three-form scalars only appeared in the function S and are thus suppressed by e2ϕ˜ = e2φIIA . In this analysis the complex structure moduli and the three-form moduli were argued to fall into chiral multiplets, while the complexiﬁed Kähler moduli are in twisted-chiral multiplets. However, due to apparent shift symmetries of the three-form moduli and complexiﬁed Kähler moduli a scalar dualization accompanied by a Legendre transformation can be performed in two dimensions. This lead to dual descriptions in which certain chiral multiplets are replaced by twisted-chiral multiplets and vice versa. Remarkably, if one dualizes a subset of scalars appearing in K, we found that the requirement to bring the dual action back to the standard N = (2, 2) dilaton supergravity form imposes conditions on viable K. These constraints include a no-scale type condition on K. The emergence of such restrictions arose from general arguments about two-dimensional theories coupled to an overall e−2ϕ˜ factor. For Calabi-Yau fourfold reductions we checked that these conditions are indeed satisﬁed. It would be interesting to investigate this further and to get a deeper understanding of this result. Having shown that in the large complex structure limit the function fAB is linear in the complex structure moduli, we discussed the application of this result in an F-theory compactiﬁcation. By assuming that the Calabi-Yau fourfold is elliptically ﬁbered and that the three-forms exclusively arise from the base of this ﬁbration, we recalled that fAB is actually the gauge-coupling function of four-dimensional R-R vector ﬁelds. This gauge-coupling function was already evaluated in the weak string coupling limit in the orientifold literature. In this orientifold limit one can double-cover the base with a Calabi-Yau threefold. We found compatibility of the fourfold result with the expectation from mirror symmetry for Calabi-Yau threefold orientifolds. In this analysis we only included closed string moduli in the orientifold setting. Clearly, the results obtained from the Calabi-Yau fourfold analysis are more powerful and it would be interesting to further investigate the open string dependence in orientifolds using our results. Additionally we commented

132

brieﬂy on the case in which the three-forms have legs in the ﬁber of the elliptic ﬁbration. In this situation the inverse of RefAB sets the value of decay constants of four-dimensional axions [62]. Again we found compatibility in the closed string sector at weak string coupling in which fAB ∝ iτ . It would be interesting to include the open string moduli in the orientifold setting and derive corrections to fAB without restricting to the weak string coupling limit. The latter task requires to compute fAB away from the large complex structure limit for elliptically ﬁbered Calabi-Yau fourfolds. In the second part of this thesis we introduced a framework to explicitly derive the moduli dependence of non-trivial three-forms on Calabi-Yau fourfolds. Our focus was on geometries realized as hypersurfaces in toric ambient spaces for which we argued that properties of the non-trivial cohomology groups can be split into two parts, one arising from the ambient space and one part from the toric divisors using the Gysin-sequence. We have explicitly shown, how to obtain algebraic and non-algebraic complex structure deformations and also Kähler deformations that arise from toric and non-toric divisors. The special tool we used to do so was the homogeneous coordinate ring of the ambient space that enabled us to derive explicit expressions of non-trivial holomorphic forms using the chiral ring of the hypersurface and the Poincaré residue. We recovered in particular the well known formulas to calculate the spectrum of the eﬀective theories from toric data. After the general considerations we focused on the three-form cohomology, essentially inherited from one-forms on Riemann surfaces along which we have orbifold singularities supplemented by topological information about the corresponding resolution divisors. The three-form scalars were argued to parametrize the intermediate Jacobian of the Calabi-Yau fourfold which was shown to arise as a product of Jacobian varieties of Riemann surfaces. We derived how to obtain the Picard-Fuchs operators from the data of the chiral ring of a Riemann surface embedded in a toric variety which then lift to the Picard-Fuchs operators of the three-form cohomology of the fourfold. From this and the boundary conditions derived in the ﬁrst part of this work a calculation of the normalized period matrix of three-forms is possible. This normalized period matrix combined with topological intersection numbers

133

9 Conclusions and Outlook of a generalized sphere-tree used to resolve the singularities along the Riemann surface comprise the metric on the intermediate Jacobian and hence determine the couplings of the three-form scalars. In the following we discussed the three-form cohomology in hypersurfaces that are elliptic ﬁbrations in weighted projective spaces and explained the fate of these three-forms under Sen’s weak coupling limit. The three-forms give rise to two-form scalars or Wilson-line moduli in the weakly coupled IIB theory. We concluded with two explicit examples for the two types of threeforms on elliptically ﬁbered Calabi-Yau fourfolds and discussed in detail their weak coupling limit. In the ﬁrst example we found that the Riemann surface inducing a non-trivial three-form is the elliptic ﬁber over a base divisor. In this case the normalized period matrix of the three-form cohomology maps to the axio-dilaton which is constant over the same base divisor. In the second example the Riemann surface is a double cover over a P1 in the base and the normalized period matrix contains the information about the location of the branching points on this P1 . In the Calabi-Yau threefold of the weakly coupled description this normalized period matrix can be interpreted as as the coupling of Wilson-line moduli on D7 branes. This intricate second example has many non-trivial features like O3-planes and a non-Higgsable gauge-group and deserves further study. In the following we would like to point out several directions for future research. A ﬁrst interesting direction is to further extend and interpret the calculations outlined in chapter 7 in the context of mirror symmetry for Calabi-Yau fourfolds [76, 41, 57]. In particular, it would be desirable to derive a general expression for the Picard-Fuchs equations for three-form periods in terms of the toric data of the ambient space in analogy to the discussion of [89]. Furthermore, one striking observation to exploit mirror symmetry can be made by recalling the construction of the period matrix of the intermediate Jacobian. We note that mirror symmetry exchanges the two-dimensional faces θα with their duals θα∗ and hence maps the one-forms on the Riemann surface Rα to the resolution divisors Dl′α . Indeed the number of (1, 0)-forms, given by ℓ′ (θα ) in (6.87), and the number of resolution divisors, given by ℓ′ (θα∗ ) in (6.87), are exchanged. This implies that the relevant intersection data for

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the Dl′α must be captured by the period matrix of three-forms on the mirror geometry, at least at certain points in complex structure moduli space, as we have seen in the ﬁrst part of this work. This observation is further supported by basic facts from Landau-Ginzburg orbifolds [59, 58, 118], since in these constructions both the intersection data and periods are determined by the structure of the chiral rings of the fourfold and its mirror. One can thus conjecture that the complex structure dependent three-form periods calculate on the mirror geometry the Kähler moduli dependent quantum corrections to the intersection numbers between integral three-forms and two-forms. It is then evident to suggest that these Kähler moduli corrections already cover world-sheet instanton corrections to the three-form couplings, when using the Calabi-Yau fourfold as a string theory background. It would be very interesting to access these corrections directly on the Kähler moduli side and establish their physical interpretation. As the three-form cohomology was shown to be localized on divisors a consideration of local Calabi-Yau fourfolds and local mirror symmetry would be suﬃcient to do so. A second promising direction for future research is to apply our results in the duality between F-theory and the heterotic string theories. The relevance of three-forms in this duality was already pointed out, for example, in [119–121]. Indeed, in heterotic compactiﬁcations on elliptically ﬁbered Calabi-Yau threefolds with stable vector bundles, the moduli space of certain vector bundle moduli also admits the structure of a Jacobian variety. By duality this Jacobian turns out to be isomorphic to the intermediate Jacobian of the corresponding K3-ﬁbered Calabi-Yau fourfold. The described powerful techniques available for analyzing the three-form periods on fourfolds might help to shed new light on the derivations required in the dual heterotic setting. Our ﬁrst example describes a simple case of such an Ftheory compactiﬁcation with non-trivial intermediate Jacobian for which the comparison to its heterotic dual geometry can be performed explicitly. It is an interesting task to analyze several such dual settings in detail. The possibility of a direct calculation of the three-form metric also has immediate applications in string phenomenology. The scalars arising from the three-form modes can correspond to scalar ﬁelds in an F-theory compactiﬁca-

135

9 Conclusions and Outlook tion to four space-time dimensions. These scalars are naturally axions, since the shift-symmetry is inherited from the forms of the higher-dimensional theory. The axion decay constants are thus given by the three-form metric and determines the coupling to the Kähler and complex structure moduli and thus can be derived explicitly for a given fourfold geometry. Since these geometries might not be at the weak string coupling limit of F-theory, one might be lead to uncovered new possibilities for F-theory model building. For example, our second example is admitting, if at all, a very complicated weak string coupling limit, but can be analyzed nevertheless using the presented geometric techniques. In this example also non-Higgsable clusters and O3planes are present and it is interesting to investigate the physics of these objects in the presents of a non-trivial three-form cohomology. It is important to stress that consistency of Calabi-Yau fourfold compactiﬁcations generically require the inclusion of background ﬂuxes [122]. It is well-known that these are also relevant in most phenomenological applications. Therefore, it is of immediate interest to generalize our discussion to include background ﬂuxes. This will be particularly interesting in singular limits of the geometry, which are relevant in the construction of F-theory vacua. In particular, the intermediate Jacobian plays an important role in the computation of the spectrum of the eﬀective theory as, for example, suggested by the constructions of [123, 124]. The generalization to include ﬂuxes will also be relevant in discussing extremal transitions in Calabi-Yau fourfolds that change the number of three-forms [117]. To conclude this list of potential future directions, let us also mention the probably most obvious generalization of the discussions presented in this work and its immediate relevance for F-theory compactiﬁcation. In fact, in this thesis we have only considered hypersurfaces in toric ambient spaces. A generalization to complete intersections [125, 126], i.e. Calabi-Yau manifolds described by more then one equation, would be desirable. This is particularly evident when recalling that in F-theory compactiﬁcations on elliptically ﬁbered fourfolds, the non-trivial three-form cohomology of the base yields U(1)-gauge ﬁelds in the four-dimensional eﬀective theory [50] as we discussed in the ﬁrst part. Since the function fAB then corresponds to

136

the gauge coupling function, it is an interesting task to use geometric techniques for Calabi-Yau fourfolds to study setups away from weak coupling. For bases that are toric hypersurfaces the same techniques as we developed apply directly and enable a calcualtion of the Picard-Fuchs equations which we outlined in section 6.4.

137

A Three-dimensional N = 2 supergravity on a circle

In this appendix we consider N = 2 supergravity compactiﬁed on a circle of radius r. Our goal is to derive the resulting N = (2, 2) action. We also brieﬂy discuss the dualization of vector multiplets in three dimensions and point out the relation to B. We start with a three-dimensional N = 2 supergravity theory coupled to chiral multiplets with complex scalars φκ and vector multiplets with bosonic ﬁelds (LΣ , AΣ ). HereLΣ is a real scalar and AΣ a vector of an U (1) gauge theory. The bosonic part of the ungauged N = 2 action takes the form Z 1 (3) ˜ κ ¯λ dφκ ∧ ∗dφ¯λ + 1 K ˜ Σ Λ dLΣ ∧ ∗dLΛ R ∗1−K S (3) = φ φ 2 4 L L 1˜ Σ ˜ LΣ φκ dφκ ) + K (A.1) ∧ ∗dAΛ + dAΣ ∧ Im(K Σ Λ dA 4 L L where the kinetic terms of the vectors and scalars are determined by the ˜ single real kinetic potential K. We want to put this on a circle of radius r and period one, i.e. the background metric is of the form ds2(3) = gµν dxµ dxν + r 2 dy 2

(A.2)

where we already drop vectors, since in an un-gauged theory they do not carry degrees of freedom in two dimensions. Similarly, the vectors AΣ are

139

A Three-dimensional N = 2 supergravity on a circle only reduced to real scalars dAΣ = dbΣ ∧ dy. The resulting two-dimensional action thus reads Z 1 (2) ˜ κ ¯λ dφκ ∧ ∗dφ¯λ + 1 r K ˜ Σ Λ dLΣ ∧ ∗dLΛ rR ∗ 1 − r K S = φ φ 2 4 L L 1 ˜ Σ ˜ LΣ φκ dφκ ) , + K (A.3) ∧ ∗dbΛ − dbΣ ∧ Im(K Σ Λ db 4r L L with a two-dimensional R and Hodge star ∗. Note that the last term is topological and does not couple to the radius r of the circle. We can perform Weyl rescaling of the two-dimensional metric setting g˜µν = e2ω gµν . This transforms the Einstein-Hilbert term as Z Z 1 1 ˜ rR ˜ ∗1 = rR ∗ 1 + dω ∧ ∗dr , (A.4) 2 2 while leaving all other terms in the action (A.3) invariant. We then ﬁnd the action Z 1 (2) ˜ κ ¯λ dφκ ∧ ∗dφ¯λ R ∗ 1 + d log r ∧ ∗dω − K S = r φ φ 2 1˜ 1 ˜ Σ Λ Σ Λ + K dL ∧ ∗dL + db ∧ ∗db K Σ Λ Σ Λ 4 L L 4r 2 L L ˜ LΣ φκ dφκ ) (A.5) − dbΣ ∧ Im(K To make contact with the N = (2, 2) dilaton supergravity action (3.23) we set LΣ = r −1 v Σ , σ

Σ

Σ

r = e−2ϕ˜ ,

Σ

≡ b + iv .

Inserted into (A.5) we then obtain Z (2) −2ϕ ˜ 1 S = e R∗1 2 1˜ 1˜ Σ Λ Σ Λ ˜ − 2dϕ˜ ∧ ∗ dω − K vΣ vΛ v dv − KvΣ vΛ v v dϕ 2 2 Σ κ λ ¯ ˜ ˜ σΛ − Kφκ φ¯λ dφ ∧ ∗dφ + KσΣ σ¯ Λ dσ ∧ ∗d¯ ˜ vΣ φκ dφκ ) . −d Re σ Σ ∧ Im(K

140

(A.6) (A.7)

(A.8)

In order to match the action (3.23) one therefore has to ﬁnd an ω such that 1˜ 1˜ Σ Λ Σ Λ dω = −dϕ˜ + K ˜. vΣ vΛ v dv + KvΣ vΛ v v dϕ 2 2

(A.9)

˜ that is linear in To solve this condition, we ﬁrst notice that any term in K Σ ˜ v drops out from this relation, i.e. K can take the form ˜ = K + v Σ SΣ , K

(A.10)

¯ Furthermore, we can solve (A.9) by with an arbitrary function SΣ (φ, φ). ¯ and assuming that K = K1 + K2 splits into a v Σ -independent term K1 (φ, φ) Σ a term K2 (v) that only depends on v . Then (A.9) is satisﬁed if K2 (v) k , ω = −ϕ˜ + ϕ˜ − 2 2

v Σ KvΣ = −k ,

(A.11)

It is easy to check that the conditions (A.10) and (A.11) are actually satisﬁed ˜ One ﬁnds for the M-theory example (3.29) of K. Z ¯ , K2 (v) = log V , Ω∧Ω K1 (z) = − log 2ϕ

SΣ = e dΣ

Y4 AB¯

Re NA Re NB ,

(A.12)

such that k = −4. Finally, in order to show that (A.8) is indeed identical to the action (3.23), we still have to complete the last term in (A.8) to ˜ vΣ φκ dφκ ). In order to do that we use Im(dσ Σ ∧ K ˜ vΣ , ˜ vΣ φκ dφκ ) = 1 dIm σ Σ ∧ dK dIm σ Σ ∧ Re(K 2

(A.13)

˜ vΣ vΛ dv Λ . This ˜ vΣ φκ dφκ ) + K ˜ vΣ = 2Re(K which follows from the fact that dK implies that these terms simply yield a total derivative and shows that the reduction of N = 2 supergravity of the form (A.1) indeed yields the extended form of N = (2, 2) dilaton supergravity suggested in (3.23) coupled to the chiral multiplets with scalars φκ and twisted-chiral multiplets with scalars σ Σ . Interestingly, we had to employ the conditions (A.10) and (A.11), which hints to the fact that the action (3.23) might admit further interesting extensions.

141

A Three-dimensional N = 2 supergravity on a circle Let us end this appendix by pointing out that we could also have ﬁrst dualized the vectors AΣ to real scalars in three dimensions and then performed the circle reduction. The dual multiplets to the vector multiplets (LΣ , AΣ ) are three-dimensional chiral multiplets with bosonic parts being complex scalars TΣ given by ˜ + iρΣ . TΣ = ∂LΣ K

(A.14)

The metric is determined now from a proper Kähler potential given by K(T + T¯, M ) = K − Re TΣ LΣ , such that the ﬁnal action reads Z 1 (3) (3) ¯J , R ∗ 1 − KM I M¯ J dM I ∧ ∗dM S = 2

(A.15)

(A.16)

with M I = (TΣ , φκ ). We can again reduce this theory on a circle (A.2) and perform a Weyl-rescaling (A.4) to ﬁnd Z 1 (2) ¯J . S = (A.17) rR ∗ 1 + dr ∧ ∗dω − rKM I M¯ J dM I ∧ ∗dM 2 With the choices r = e−2ϕ˜ and ω = −ϕ˜ this reads Z I J (2) −2ϕ ˜ 1 ¯ R ∗ 1 + 2dϕ˜ ∧ ∗dϕ˜ − KM I M¯ J dM ∧ ∗dM . S = e 2

(A.18)

This result should also be obtainable from (A.8) by dualizing the chiral multiplets with scalars σ Σ . This is possible since bΣ appears only with its ﬁeldstrength dbΣ . The details of this dualization in two dimensions will be discussed in B.

142

B Twisted-chiral to chiral dualization in two dimensions

In this appendix we present the details of the dualization discussed in section 3.3 of a twisted-chiral multiplet to a chiral multiplet in two dimensions. The starting point is the action Z (2) −2ϕ ˜ 1 ˜ κ ¯λ dφκ ∧ ∗dφ¯λ R ∗ 1 + 2dϕ˜ ∧ ∗dϕ˜ − K SC-TC = e φ φ 2 ˜ φκ σ¯ Λ dφκ ∧ d¯ ˜ σΣ σ¯ Λ dσ Σ ∧ ∗d¯ σΛ σΛ − K +K ˜ Σ ¯λ dφ¯λ ∧ dσ Σ , (B.1) −K σ φ ˜ is given by where K ˜ = K + e2ϕ˜ S . K

(B.2)

In the following we use sub-scripts to indicate derivatives with respect to ˜ K ˜ depends on a number of chiral multiplets with ˜ φκ ≡ ∂φκ K. ﬁelds, e.g. K κ complex scalars φ and a number of twisted-chiral multiplets with complex scalars σ Σ . In order to perform a dualization, we assume that Re σ Σ has a shift symmetry and only appears via d Re σ Σ in (B.1). This implies that Re σ Σ can be dualized into a scalar ρΣ by the standard procedure. One ﬁrst replaces dRe σ Σ → F Σ in (B.1) and then adds a Lagrange multiplier term promotional to F Σ ∧ dρΣ . Then F Σ can be consistently eliminated from (B.1). 143

B Twisted-chiral to chiral dualization in two dimensions Denoting the imaginary part of σ Σ by v Σ = Im σ Σ the resulting action reads Z (2) −2ϕ ˜ 1 ˜ κ ¯λ dφκ ∧ ∗dφ¯λ R ∗ 1 + 2dϕ˜ ∧ ∗dϕ˜ − K SC = e φ φ 2 ˜ vΛ φλ dφλ ) ˜ vΣ vΛ e2ϕ˜ dρΣ − Im (K ˜ vΣ φκ dφκ ) ∧ ∗ e2ϕ˜ dρΛ − Im (K +K 1˜ Σ Λ + K dv ∧ ∗dv (B.3) Σ Λ 4 v v To compute the dualized action we make the following ansatz for the Legendre transformed variables TΣ TΣ = e−2ϕ˜

˜ ∂K ∂K ∂S + iρΣ = e−2ϕ˜ Σ + Σ + iρΣ , Σ ∂v ∂v ∂v

(B.4)

and the dual potential K ˜ − e2ϕ˜ Re TΣ v Σ . K=K

(B.5)

˜ under which the action (B.3) can be We want to derive the conditions on K brought to the form Z (2) I J −2ϕ ˜ 1 ¯ R ∗ 1 + 2dϕ˜ ∧ ∗dϕ˜ − KM I M¯ J dM ∧ ∗dM , (B.6) SC = e 2 with M I = (φκ , TΣ ). We ﬁrst determine from (B.4) and (B.5) that 1 ∂v Σ ˜ vΣ vΛ , = e2ϕ˜ K ∂TΛ 2 1 KTΣ = − e2ϕ˜ v Σ , 2

∂v Σ ˜ vΣ vΛ K ˜ v Λ φκ , = −K κ ∂φ

(B.7)

˜ φκ , Kφκ = K

˜ = 4K ˜ σΣ σ¯ Λ . Crucially, one ˜ vΣ vΛ ≡ ∂vΣ ∂vΛ K ˜ vΣ vΛ is the inverse of K where K also derives from (B.4) that ˜ vΣ φκ dφκ ) − 2KvΣ dϕ˜ . ˜ vΣ vΛ dv Σ + 2Re(K (B.8) dReTΣ = e−2ϕ˜ K

Note that there is the additional dϕ-term, ˜ which is absent in the standard ˜ dualization procedure. The conditions on K arise from demanding that the

144

dual action can be brought to the form (B.1) and no additional mixed terms involving dϕ˜ appear. To evaluate (B.1) one uses (B.7) to derive the identities 1 1 ˜ vΣ vΛ , ˜ vΣ vΛ K ˜ Λ ¯κ , KTΣ T¯Λ = − e4ϕ˜ K KTΣ φ¯κ = e2ϕ˜ K v φ 4 2 ˜ κ ¯λ − K ˜ φκ v Σ K ˜ vΣ vΛ K ˜ Λ ¯λ . Kφκ φ¯λ = K v φ φ φ

(B.9)

Inserting (B.8), (B.9) into (B.6) one ﬁnds the following terms involving dϕ˜ Z Σ (2) Sdϕ˜ = e−2ϕ˜ 2 + KvΣ Kv vB KvΛ dϕ˜ ∧ ∗dϕ˜ + KvΣ dv Σ ∧ ∗dϕ˜ . (B.10) These terms can be removed by a Weyl rescaling of the three-dimensional metric if certain conditions on K are satisﬁed. To see this, we perform a Weil rescaling g˜µν = e2ω gµν (B.11) which transforms the Einstein-Hilbert term as Z Z −2ϕ ˜ 1 −2ϕ ˜1 ˜ R˜ ∗1 = e R ∗ 1 − 2dω ∧ ∗dϕ˜ , e 2 2

(B.12)

while leaving all other terms invariant. Hence we can absorb the extra terms in (B.10) by a Weyl rescaling iﬀ − 2dω = KvΣ Kv

Σ vB

KvΛ dϕ˜ + KvΣ dv Σ .

(B.13)

Clearly, a simple solution to this equation is found if K satisﬁes KvΣ Kv

Σ vB

KvΛ = k ,

¯ + K2 (v) , K = K1 (φ, φ)

(B.14)

¯ independent of v Σ , and a function K2 (v) for a constant k, a function K1 (φ, φ) independent of φκ . In this case one can chose k 1 ω = − ϕ˜ − K2 (v) . 2 2

(B.15)

Note that (B.14) is satisﬁed for the result found in a Calabi-Yau fourfold reduction (3.26), i.e. k = −4 and K2 = log V.

145

Summary

In this summary we will give a short overview of the content of this thesis. The introduction chapter 1 explains the role of string theory in modern particle physics as a candidate for quantum gravity. We explain how to derive the eﬀective low-energy supergravity description of string theory and how to make contact with the observable world via dimensional reduction. A special class of such theories is given by F-theory that allows for an inclusion of non-perturbative eﬀects of stringy physics. F-theory is here understood as a decompactiﬁcation limit of M-theory on elliptically ﬁbered Calabi-Yau fourfolds. This elegant description of string theory is a powerful branch of string phenomenology and motivates the study of the geometry of CalabiYau fourfolds. We start the main body of this thesis by introducing the geometric properties of Calabi-Yau fourfolds and their harmonic forms in chapter 2. Here we focus especially on the harmonic three-forms whose physical properties are determined by the complex structure dependence of their so called normalized period matrix fAB , which was not well understood before. The next part of the thesis focuses on eﬀective theories of Calabi-Yau fourfolds. In chapter 3 we perform the dimensional reduction of Type IIA supergravity on a general Calabi-Yau fourfold and ﬁnd a N = (2, 2) dilaton-supergravity in two dimensions. Here we determine the massless spectrum and the kinetic potential of the resulting supergravity. We conjecture an extension of the usual Kählerpotential of this dilaton-supergravity by a term depending on the dilaton of the theory to account for the kinetic coupling of the novel three-form scalars. Subsequently we apply mirror symmetry in chapter 4, a duality of the same

147

Summary supergravity theory resulting from two diﬀerent Calabi-Yau fourfolds, to determine the structure of the normalized period matrix at the large complex structure point in moduli space. We see that it is mirror to a function hA B holomorphic in the mirror Kähler moduli at the large volume point. The linear leading order behavior of fAB at the large complex structure point is speciﬁed by topological intersection numbers MΣA B of the mirror. Following this discussion we extend our analysis to the dimensional reduction of eleven-dimensional supergravity in chapter 5. Afterwards we lift the resulting three-dimensional N = 2 supergravity to the eﬀective theory of F-theory on elliptically ﬁbered Calabi-Yau fourfolds leading to an eﬀective N = 1 supergravity description in four dimensions. We close this part by discussing the implications of non-trivial three-form cohomology and defer some technical details of the ﬁrst part into two appendices in A,B. Three-forms located on the base of the elliptically ﬁbered fourfold lead to a U (1)-gauge theory in the eﬀective four-dimensional theory whose coupling is determined by fAB . If the harmonic three-forms have a leg in the ﬁber they lead to scalars in the eﬀective theory whose coupling is determined by their normalized period matrix fAB . We also show consistency of our results with the known weakly coupled IIB orientifold description. In the second part of this thesis we construct explicit Calabi-Yau fourfold examples via hypersurfaces in toric varieties. We begin in chapter 6 with the general construction of toric spaces and their hypersurfaces. We then focus on Calabi-Yau fourfolds realized as so called semiample hypersurfaces avoiding the Lefschetz-hyperplane theorem that forbids non-trivial three-form cohomology on ample hypersurfaces. This is followed by a discussion of the origin of non-trivial harmonic forms on the hypersurface and we state well known formulas for the number of the various harmonic forms in terms of toric data. We also include in our discussion so called non-toric and non-algebraic deformations of the hypersurface that are easy to describe in the same framework. The derived description via chiral rings and Poincaré residues allows not only to determine the spectrum of the eﬀective theory upon compactiﬁcation, but also sets the stage for a calculation of the couplings. In chapter 7 we discuss the moduli space of the three-form scalars which is

148

called intermediate Jacobian of a Calabi-Yau fourfold whose complex structure dependence reduces to the complex structure dependence of Riemann surfaces. The normalized period matrices of these Riemann surfaces are the building blocks of the three-form normalized period matrix fAB of the CalabiYau fourfold. The Kähler dependence is captured by a so called generalized sphere-tree that can be computed in terms of the ambient space intersection theory and provides us with the aforementioned intersection numbers MΣA B . We determine in this situation the map between the geometrical quantities, necessary to derive the eﬀective theories discussed before. This part is completed by a lengthy discussion of two simple examples of Calabi-Yau fourfold geometries with non-trival three-form cohomology in chapter 8. Here we introduce the mathematical basics of elliptic ﬁbrations and discuss the corresponding F-theory physics including the weak coupling limit of Sen. The ﬁrst example contains one non-trivial three-form that gives rise to a so called two-form scalar in the weak coupling limit and in this case the normalized period matrix fAB reduces to the axio-dilaton τ of type IIB supergravity. In the second example we have a geometry with multiple non-trivial three-forms that we argue to give rise to so called Wilson-line scalars on a seven-brane. This example is in particular interesting as it includes many non-trivial features, like O3-planes and non-Higgsable clusters and deserves further investigation. We conclude the thesis in chapter 9 giving an outlook of possible future research directions. After recalling the results of our work, we point out that a further study of the N = (2, 2)-supergravity theory is necessary to fully settle the inclusion of three-form moduli in type IIA reductions on Calabi-Yau fourfolds, as we only derived a dilaton-dependent correction to the kinetic potential of the eﬀective theory which is therefore not Kähler in an obvious way. Furthermore, we argue that a better understanding of mirror symmetry of Calabi-Yau fourfold hypersurfaces with non-trivial three-form cohomology is desirable. This may enable a direct expression of the normalized periodmatrix fAB in toric terms. Furthermore, we argue that this matrix determines the couplings of axions in eﬀective theories and hence is relevant for string phenomenology. Therefore it is in particular necessary to determine

149

Summary fAB in more complicated geometries, for instance complete intersections in toric varieties.

150

Samenvatting

In deze samenvatting geven we een kort overzicht van de inhoud van dit proefschrift. In the introductie, sectie 1, leggen we uit wat de rol is van snaartheorie in de moderne deeltjesfysica als de kandidaat voor een theorie van kwantumzwaartekracht. We leggen uit hoe een eﬀectieve lage-energie supergravitatie beschrijving van snaartheorie kan worden afgeleid en hoe contact kan worden gemaakt met de wereld om ons heen via dimensie reductie. Een speciale klasse van zulke theorieën wordt gegeven door F-theorie dat ook niet-perturbatieve eﬀectieve eﬀecten van snaartheorie beschrijft. We zien F-theorie hier als een decompactiﬁcatie limiet van M-theorie op elliptisch gevezelde Calabi-Yau viervariëteiten. Deze elegante beschrijving van snaartheorie is een krachtige tak van snaarfenomenologie en geeft een motivatie voor de studie van de geometrie van Calabi-Yau viervariëteiten. Daarna introduceren we in sectie 2 de geometrische eigenschappen van CalabiYau viervariëteiten en hun harmonische diﬀerentiaalvormen. Hier focussen we ons op de harmonische drie-vormen waarvan de fysische eigenschappen bepaald worden door de afhankelijkheid van hun zogenoemde genormaliseerde periode matrix fAB van de complexe structuur. Deze afhankelijkheid is niet voldoende begrepen in de eerdere literatuur. Het volgende deel van het proefschrift richt zich op de eﬀectieve theorieën van Calabi-Yau viervariëteiten. In sectie 3 doen we de dimensionale reductie van Type IIA supergravitatie op een algemene Calabi-Yau viervariëteit en vinden we een N = (2, 2) dilatonsupergravitatie in twee dimensies. We bepalen het spectrum van massaloze deeltjes en de kinetische potentiaal van de resulterende supergravitatie theorie. Om de kinetische koppeling van de nieuwe drie-vorm scalairen te verklaren, doen we een voorstel voor een uitbreiding van de gebruikelijke

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Samenvatting Kählerpotentiaal van deze dilaton-supergravitatie door een term toe te voegen die afhangt van de dilaton. Daarna, in sectie 4, passen we spiegelsymmetrie toe, dat een dualiteit is van dezelfde theorie van supergravitatie op twee verschillende Calabi-Yau viervariëteiten, om de structuur van de genormaliseerde periode matrix te bepalen op het grote complexe structuur punt (Engels: large complex structure point) in de moduliruimte. We zullen zien dat het gespiegeld wordt naar een holomorfe functie hA B in de gespiegelde Kähler moduli op het grote volume punt (Engels: large volume point). Het lineaire hoogste orde gedrag van fAB op het grote complexe structuur punt wordt gespeciﬁceerd door de topologische intersectie getallen MΣA B van de spiegelvariëteit. Na deze discussie breiden we in sectie 5 onze analyse uit naar de de dimensionale reductie van elf-dimensionale supergravitatie. Daarna liften we de resulterende drie-dimensionale N = 2 supergravitatie op naar de eﬀectieve theorie van F-theorie op elliptisch gevezelde vier-variëteiten. Dit geeft een eﬀectieve N = 1 supergravitatie beschrijving in vier dimensies. We sluiten dit deel af met een discussie van de implicaties van niet-triviale drie-vorm cohomologieën. We verwijzen naar twee addendums in sectie 9 voor een aantal technische details van het eerste deel van de thesis. Drie-vormen die zich zich in de basis van de elliptisch gevezelde Calabi-Yau viervariëteit bevinden, geven een U (1)-ijktheorie in de eﬀectieve vier-dimensionale theorie. De koppeling wordt bepaald door fAB . Als de harmonische drie-vormen zich ook deels op de vezel bevinden, geven ze scalairen in de eﬀectieve theorie waarvan de koppeling bepaald is door hun genormaliseerde periode matrix fAB. We laten ook zien dat onze resultaten consistent zijn met de bekende zwakgekoppelde IIB orientifold beschrijving. In het tweede deel van dit proefschrift construeren we expliciete Calabi-Yau viervariëteit voorbeelden via hyperoppervlakken in torische variëteiten. We beginnen in sectie 6 met een algemene constructie van torische ruimten en hun hyperoppervlakken. We richten ons daarna op Calabi-Yau viervariëteiten die gerealiseerd worden als zogenoemde semiampel (Engels: semiample) hyperoppervlakken. Deze oppervlakken vermijden de Lefschetz-hypervlak stelling dat stelt dat het onmogelijk is een niet-triviale drie-vorm cohomologie op am-

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pel (Engels: ample) hyperoppervlakken te hebben. Dit wordt gevolgd door een discussie over de oorsprong van niet-triviale harmonische vormen op het hyperoppervlak en we geven bekende formules waarin het aantal harmonische vormen uitgedrukt wordt in torische data. In onze discussie behandelen we ook de zogenoemde niet-torische en niet-algebraïsche deformaties van het hyperoppervlak die gemakkelijk te beschrijven zijn in hetzelfde kader. De beschrijving die we aﬂeiden via chirale ringen en Poincaré residuen, geeft ons niet alleen het spectrum van de eﬀectieve theorie dat we krijgen na compactiﬁcatie, maar maakt ook de weg vrij voor een berekening van de koppelingen. In sectie 7 behandelen we de moduli ruimte van de drie-vorm scalairen dat de tussen-Jacobiaan van de Calabi-Yau viervariëteit genoemd wordt. De complexe structuur afhankelijkheid van de tussen-Jacobiaan reduceert naar de complexe structuur afhankelijkheid van Riemann-oppervlakken. De genormaliseerde periode matrices van deze Riemann oppervlakken zijn de bouwstenen van de drie-vorm genormaliseerde periode matrix fAB van de Calabi-Yau viervariëteit. De Kähler-afhankelijkheid wordt beschreven door een zogenoemde gegeneraliseerde sfeer-boom die uitgedrukt kan worden in de intersectietheorie van de omgevende ruimte. Dit geeft ons bovendien de eerder genoemde intersectiegetallen MΣA B . We bepalen ook de afbeelding tussen de geometrische grootheden die nodig is om de eﬀectieve theorieën, waar we het eerder over gehad hebben, af te leiden. We eindigen dit deel in sectie 8 met een lange discussie van twee simpele voorbeelden van Calabi-Yau viervariëteit geometrieën die een niet-triviale drievorm cohomologie hebben. Hier introduceren we de wiskundige basiskennis van elliptische vezelingen en bediscussiëren we de corresponderende F-theorie natuurkunde, inclusief de zwakke koppelings limiet van Sen. Het eerste voorbeeld heeft een niet-triviale drie-vorm dat een zogenoemde twee-vorm scalair geeft in de zwakke-koppelinglimiet. In dit geval reduceert de genormaliseerde periode matrix fAB naar de axio-dilaton τ van type IIB supergravitatie. In het tweede voorbeeld hebben we een geometrie met meerdere niet-triviale drie-vormen en we beargumenteren dat ze zogenoemde Wilson-lijn scalairen geven op een zeven-braan. Dit voorbeeld is interessant omdat het heel veel niet-triviale eigenschappen heeft, zoals O3-vlakken en clusters die je niet

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Samenvatting kunt Higgsen. Daarom zou het goed zijn als er meer onderzoek naar gedaan wordt. We eindigen dit proefschrift in sectie 9 door vooruit te kijken naar mogelijke toekomstige onderzoeksrichtingen. Na het samenvatten van de resultaten van dit werk, beargumenteren we dat een verdere studie van de N = (2, 2)supergravitatie theorie nodig is om de inclusie van drie-vorm moduli in type IIA reducties op Calabi-Yau viervariëteiten volledig te begrijpen. Dit omdat we alleen een dilaton-afhankelijke correctie voor de kinetische potentiaal van de eﬀectieve theorie afgeleid hebben, waardoor het niet duidelijk is of deze nog Kähler is. We beargumenteren ook dat het wenselijk is een beter begrip te krijgen van de spiegelsymmetrie van Calabi-Yau viervariëteit hyperoppervlakken met niet-triviale drie-vorm cohomologie. Dit zou het mogelijk kunnen maken om de genormaliseerde periode matrix fAB direct uit te drukken in torische data. Bovendien beargumenteren we dat deze matrix de koppelingen van axions in eﬀectieve theorieën bepaalt en daarom relevant is voor snaarfenomenologie. Daarom is het in het bijzonder nodig om fAB te bepalen in gecompliceerdere geometrieën, bijvoorbeeld complete intersecties in torische variëteiten.

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About the Author

I was born on September 9th 1990 in Günzburg, Germany, where I graduated from the Dossenberger Gymnasium in 2010. Afterwards I studied Mathematics at the University of Augsburg where I obtained a Bachelor degree in 2013. In parallel I completed the Elite-Bachelor "TopMath" of the TU Munich with honors. This program enables students to start their own research during the last year of their Bachelor studies. In my case I focused on algebraic topology under the supervision of Prof. Dr. Bernhard Hanke. My Bachelor thesis provided a proof of the Jordan separation theorem on smooth manifolds. After my undergraduate studies I went in 2013 to the LMU in Munich for the Elite-Master "Theoretical and Mathematical Physics" where I studied the geometry and physics of string theory. At the Max-Planck-Institute for Physics in Munich I wrote my Master thesis on three-forms on Calabi-Yau fourfolds and their eﬀective theories under the supervision of Dr. Thomas Grimm who later became the supervisor of my Ph.D. studies. After completion of my Master’s degree with honors in 2015, I started my research at MPI Munich building on previous work. Since Thomas became Associate Professor at Utrecht University in 2016, I followed to Utrecht in 2017 where I completed my studies towards a doctoral degree in 2018. This work is an account of the work completed during these studies.

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Aknowledgements

Most importantly I like to thank my supervisor Thomas Grimm for countless discussions and the freedom to follow my interests. Furthermore I thank Stefan Vandoren for making my studies possible and promoting me. I also like to thank my collaborators Ralph Blumenhagen and Irene Valenzuela from whom I learned a lot. The many great people that made my studies a wonderful experience I would also like to thank. My oﬃce-mates Andreas Kapfer and Michael Fuchs in Munich and Pierre Corvilain and Raﬀaele Batillomo in Utrecht. Florian Wolf, Daniel Kläwer, Daniela Herschmann, Nina Miekley, Kilian Mayer, Huibert het Lam, Aron Jansen, Markus Dierigl and Miguel Montero for many deep and not so deep conversations. In addition, I would like to thank all the exceptional people that I met during my travels to conferences and seminars of whom there are too many to thank individually. I also like to thank the Max-Planck-Institute for Physics in Munich where the ﬁrst half of my studies was completed. The International Munich Particle Research School (IMPRS) provided an inspirational environment for my research. The same holds for Utrecht University where the second half of my studies took place and I was welcomed with hospitality. Special thanks goes to my family for their never-ending support and understanding.

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