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TIT/HEP-632 SNUTP13-005 November 2013

Factorization of S 3/Zn partition function

Yosuke Imamura∗a , Hiroki Matsuno†a , and Daisuke Yokoyama‡b a

Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan b Center for Theoretical Physics, Seoul National University, Seoul 151-747, Korea

Abstract

We investigate S 3 /Zn partition function of 3d N = 2 supersymmetric field theories. In a gauge theory the partition function is the sum of the contributions of sectors specified by holonomies, and we should carefully choose the relative signs among the contributions. We argue that the factorization to holomorphic blocks is a useful criterion to determine the signs and propose a formula for them. We show that the orbifold partition function of a general non-gauge theory is correctly factorized provided that we take appropriate relative signs. We also present a few examples of gauge theories. We point out that the sign factor for the orbifold partition function is closely related to a similar sign factor in the lens space index and the 3d index.

∗

E-mail: [email protected] E-mail: [email protected] ‡ E-mail: [email protected] †

Contents 1 Introduction

1

2 Orbifold partition function 2.1 Naive projection . . . . . . . 2.2 Projection operator . . . . . 2.3 Factorization and sign factor 2.4 Gauge theories . . . . . . . 2.4.1 SQED . . . . . . . . 2.4.2 su(2) gauge theory .

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4 4 8 9 11 12 13

3 The sign factor to the lens space index and the 3d index

16

4 Conclusions

19

A Appendix A.1 Classical contribution of Chern-Simons terms

20 20

1

. . . . . . . . . . .

Introduction

In this paper, we discuss a rather technical issue concerning the sign of the partition function of 3d N = 2 supersymmetric theories. It is defined by the path integral Z DΦe−S ,

ZM =

(1)

where M is the background manifold and Φ collectively represents all dynamical fields in the theory. The overall factor is often ignored because it does not affect some observables such as correlation functions. However, in recent progress of supersymmetric field theories, the partition function itself plays an important role. For example, we can determine the superconformal R-charge of a 3d N = 2 theory at infra-red fixed point by maximizing the real part of the free energy F = − log ZS 3 [1, 2]. In order to compute the partition function including the overall factor unambiguously through the path integral (1) we need to fix the measure of the path integral carefully. A convenient way to do this is to exploit the fact that ZM is obtained from the supersymmetric index of a 4d N = 1 theory as the small radius limit. Let us consider the case of S 3 partition function. To obtain the 3d theory, we start from a 4d N = 1 theory in the background S 3 × S 1 . We define S 3 by |z1 |2 + |z2 |2 = 1, z1 , z2 ∈ C. (2)

1

If we regard the S 1 direction as a time, the path integral of the 4d theory is interpreted as the index i h −1 J1 + R −1 J2 + R Fa F D− R 2 2 2 (q p1 ) (q p2 ) za , (3) I(p1 , p2 , za ) = tr (−1) q

where F , R, D, and Fa are the fermion number, the U(1)R charge, the dilatation, and flavor charges, respectively. J1 and J2 are the angular momenta rotating z1 and z2 , respectively. The exponent of q in (3) 3 D − J1 − J2 − R = {Q, Q† } 2

(4)

is exact with respect to a supercharge Q, and (3) is independent of the variable q. Unlike the partition function ZS 3 , there is a natural normalization of I; in the trace over the Hilbert space, every gauge invariant state contributes to the index by weight 1, and there is no ambiguity of the normalization except for the signature. The S 3 partition function is obtained as the β → 0 limit of the index. In non-supersymmetric theories the small radius limit may in general diverge. However, in the reduction from a 4d N = 1 theory to a 3d N = 2 theory that we consider here, we can obtain a finite result due to the cancellation between bosonic and fermionic contributions.1 Therefore, once we obtain the 4d index, we can unambiguously obtain the partition function by the small radius limit [3, 4, 5]. r ω1 −βωi −βµa . (5) , za = e ZS 3 (b, µa ) = lim I(pi = e ), b = β→0 ω2 ωi and µa are interpreted in the 3d theory as squashing parameters and real mass parameters. The partition function depends on ωi through the single parameter b [5, 6, 7, 8, 9, 10, 11]. The ambiguity in the signature is due to the ambiguity in the statistics of the vacuum state. The statistics of states in the Hilbert space is fixed once that of the vacuum state is specified. However, there is no general rule to fix it, and we need an additional criterion to fix the overall sign. For some use of ZM , like F maximization, we only need the absolute value of ZM , and one may think that the sign ambiguity is not important. However, if the theory has multiple sectors, we should sum up their contributions, and we need to fix the relative signs among them. This is the case when we consider a gauge theory on a manifold with non-trivial fundamental group. In such a case there are degenerate vacua labeled by holonomies associated with non-trivial cycles. In this paper we focus on the orbifold S 3 /Zn defined from S 3 in (2) by the identification (z1 , z2 ) ∼ (ωz1 , ω −1z2 ) ω = e 1

2πi n

.

(6)

For this cancellation we should carefully include the zero-point contribution, which is often neglected.

2

The fundamental group of this manifold is Zn , and vacua are labeled by Zn (dyn) valued holonomies ha associated with dynamical U(1)a gauge symmetries as (dyn) well as continuous moduli parameters µa . It is also possible to introduce mass (ext) (ext) parameters µa and non-trivial holonomies ha for global U(1)a symmetries. (We label both gauge and global symmetries by a.) The partition function is (dyn) obtained by summing up the contribution from sectors with different ha Z X −S cl (ext) (ext) (7) f (h) dµ(dyn) e S3 /Zn ZS1−loop ZS 3 /Zn (b, µa , ha ) = 3 /Z (b, µa , ha ), n (dyn)

ha

where f (h) is the sign factor that we would like to determine, and the explicit form of the integrand will be given in the next section. A formula for the orbifold partition function has actually been already given in [12]. They derive the formula in two ways. One is the orbifold projection from S 3 partition function, and the other is reduction from the lens space index, which is obtained from S 3 × S 1 index by the orbifold projection. In both derivations, they do not take account of the possible emergence of non-trivial sign factors. The formula has been used for some applications, and works well. However, in some cases, we need to introduce extra sign factors. For example, it is demonstrated in [13] for a few examples of dual pairs that the matching of the orbifold partition function of dual theories requires non-trivial sign factors. A similar problem of relative weight also arises in the instanton sum in the S 4 partition function. It would be instructive to understand how we can determine the relative weights in that case, before we explain our strategy for the S 3 /Zn partition function. Let us consider an N = 2 supersymmetric gauge theory on S 4 . By equivariant localization, we can localize the dynamics of the theory at two poles of S 4 , and the partition function is written as [14, 15] Z=

∞ X

f (kN , kS )ZN (kN )ZS (kS ),

(8)

kN ,kS =0

where kN and kS are, respectively, the instanton number at the north pole and the anti-instanton number at the south pole. (Precisely, we need to perform the integral over the Coulomb branch parameterized by constant scalar fields. Here we focus only on the instanton sum, and consider the contribution from a specific point in the Coulomb branch.) We introduced the unknown phase factor f (kN , kS ). This phase factor is strongly restricted by assuming the locality of the theory [16]. If we assume the locality, the path integral for the localized modes at the two poles should be performed independently, and thus the partition function is factorized into contributions from the poles; Z=

∞ X

g(kN )ZN (kN )

kN =0

∞ X

kS =0

3

h(kS )ZS (kS ),

(9)

where g(kN ) and h(kS ) are unknown phase factors. Now we use the fact that disconnected components of the configuration space of the theory is labeled only by the total instanton number kN − kS . This means that two configurations la(1) (1) (2) (2) beled by (kN , kS ) and (kN , kS ) are in the same component of the configuration (1) (1) (2) (2) space if kN − kS = kN − kS . We can interpolate them by continuous deformation and the relative phase between the contributions from them can be in principle determined unambiguously by the continuity of the action functional. Here let us assume for simplicity that there are no relative phases. Namely, g(kN + n)h(kS + n) does not depend on n. Then, the relation g(kN + 1) h(kS ) = g(kN ) h(kS + 1)

(10)

holds. The left-(right-)hand side of this equation is independent of kS (kN ), and (10) is a constant independent of both kN and kS . Let c be the constant. We obtain g(kN ) = g(0)ckN , h(kS ) = h(0)c−kS , (11) and the total partition function is Z = f (0, 0)

∞ X

ckN −kS ZN (kN )ZS (kS ).

(12)

kN ,kS =0

Now we have determined the phase factor except the overall phase f (0, 0) and the constant c. The factor ckN −kS can be identified with the contribution of the topological θ term. In this way, the relative phases for instanton sectors of a 4d gauge theory can be fixed up to few parameters by the factorization of the partition function. We take the same strategy to determine the relative signs among holonomy sectors of the orbifold partition function. Actually, it is known that the S 3 and S 2 × S 1 partition functions are factorized into factors so-called holomorphic blocks [17, 18, 19, 20, 21], and a similar factorization is expected for the orbifold. Each block is identified with the vortex partition function on a solid torus [18, 20]. In the following, we determine the relative signs of holonomy sectors by requiring the factorization of the orbifold partition function.

2 2.1

Orbifold partition function Naive projection

Let us first summarize how the formula for the orbifold partition function ZS 3 /Zn is obtained from the S 3 partition function by naive Zn orbifold projection [12]. We consider a theory with general gauge group G and matter representation R. At a generic point in the Coulomb branch the gauge group G is broken to 4

its Cartan subgroup H. Let Va , Wα , and Φi denote the vector multiplets for the Cartan part, W-bosons, and chiral multiplets, respectively. For later convenience, we include external vector multiplets for global U(1)a symmetries in Va . (dyn) For distinction, we denote dynamical and non-dynamical components by Va (ext) (dyn) and Va , respectively. The scalar components µa for the dynamical vector multiplets parameterize the Coulomb branch while those for external vector (ext) multiplets, µa , are real mass parameters. By localization, we can reduce the path integral of an N = 2 supersymmetric theory on S 3 into a finite dimensional matrix integral with the integrand consisting of the classical and one-loop factors: Z cl (ext) ) = dµ(dyn) e−SS3 (µ) ZS1−loop (µ). (13) ZS 3 (µ 3 The integration measure is given by Z Z 1 Y ∞ (dyn) (dyn) dµ ≡ dµ |W | a −∞ a

(14)

where a runs over dynamical part of Va , and |W | is the order of the Weyl group of the gauge group. The one-loop factor is the product of the contributions of Wα and Φi 1 1 =Q ZS1−loop , (15) 3 iQ Q iQ α sb (µα + 2 ) i sb (µi − 2 (1 − ∆i ))

where Q = b+b−1 . ∆i is the Weyl weight of the scalar component of a chiral multiplet Φi . µα and µi are the scalar components of the vector multiplets coupling to Wα and Φi ; µα = qαa µa , µi = qia µa , (16)

where qαa and qia are the U(1)a charge of W-boson Wα and the chiral multiplet Φi , respectively. It is convenient to include R-charge in the charge matrix. We define qα0 and qi0 as the R-charges of the fermions in the multiplets Wα and Φi , qα0 = 1,

qi0 = ∆i − 1,

(17)

and we set the corresponding scalar parameter by µ0 ≡ µR =

iQ . 2

(18)

Including the contribution of the U(1)R symmetry, we define µ bα = µα +

iQ , 2

µ bi = µi −

iQ (1 − ∆i ). 2

(19)

Then the one-loop factor (15) is simply rewritten as 1 , µI ) I sb (b

ZS1−loop =Q 3 5

(20)

where I runs over both W-bosons and chiral multiplets. sb (z) is the double sine function, and can be expressed as an infinite product corresponding to the spherical harmonic expansion on S 3 . The contribution of a multiplet I is ∞ Y b(q + 21 ) + b−1 (p + 21 ) + ib µI 1 = . 1 1 sb (b µI ) p,q=0 b(p + 2 ) + b−1 (q + 2 ) − ib µI

(21)

The denominator and the numerator come from the bosonic and the fermionic modes with angular momenta (J1 , J2 ) = (p, q), respectively. The classical factor exists when the theory has Chern-Simons terms. If the action contains the Chern-Simons term Z ikab Aa dAb , (22) 4π the scalar field quadratic terms in the supersymmetric completion of (22) give the classical factor cl e−SS3 (µ) = e−πikab µa µb . (23) Let us move on to the orbifold partition function ZS 3 /Zn . The orbifold S 3 /Zn is defined from the S 3 in (2) by the identification (6). The vacua are parameterized by the scalar field µa and holonomies I n Aa , (24) ha = 2π γ where γ is a non-trivial loop in S 3 /Zn generating the fundamental group. ha is quantized to be integer, and ha and ha + n are identified because they are transformed to each other by a large gauge transformation. We define hI = {hα , hi }, holonomies coupling to Wα and Φi , in a similar way to (16): X hI = qIa ha . (25) a

We can turn on non-trivial holonomies for global symmetries as well as the dynamical gauge symmetries. Note that we will not turn on the holonomy for the R-symmetry because it breaks supersymmetry. (For a more general lens space L(p, q), we need to turn on non-trivial U(1)R holonomy to preserve supersymmetry.) After the orbifold projection, only modes compatible with the identification (6) contribute to the one-loop factor. The condition for Zn invariance for modes of multiplet I is p − q = hI mod n, (26) and the one-loop factor is obtained by restricting the product over p and q in (21) by the condition (26). We define the function sb,hI to express the contribution of 6

each multiplet by 1 = sb,hI (b µI )

Y

(p,q)∈Λ[hI ]

b(q + 12 ) + b−1 (p + 21 ) + ib µI , 1 1 −1 µI b(p + 2 ) + b (q + 2 ) − ib

(27)

where Λ[h] is the set consisting of (p, q) satisfying the condition (26): Λ[h] = {(p, q)|p, q ≥ 0, p − q = h

mod n}.

(28)

The one-loop factor (21) is replaced by ZS1−loop = 3 /Z n

Y I

1 . sb,hI (b µI )

(29)

For the classical part, the Zn orbifolding gives rise to the extra 1/n factor in the action, and the Chern-Simons term gives the holonomy dependent phase [22, 23, 24]: 2 πikab πi (µ,h) −S cl = e− n kab µa µb e− n (n−1)ha hb . e S3 /Zn (30) The integration measure is Z

dµ

(dyn)

1 Y = |W | a

Z

∞ −∞

(dyn)

dµa n

.

(31)

This is normalized by using the relation to the lens space index [12]. Combining these factors, we obtain the formula (7). Before ending this section, we would like to comment on a subtlety lurking in the formula (7). For the gauge invariance of the factor (30), the Chern-Simons levels kab must be integers. This is, however, not always the case. If the theory has parity anomaly, some components of the bare Chern-Simons level should be half odd integers to cancel the anomaly. In this case, the factor (30) itself is not gauge invariant. Namely, it may change its sign under a large gauge transformation that shifts ha by ha → ha + nca (ca ∈ Z). Of course, this is not an essential problem. For the consistency, we only need the gauge invariance of the whole integrand in (7) including the one-loop contribution. In the following, we propose a general formula for the sign factor f (h), which will give the sign of the partition function for each holonomy sector in a gauge invariant way. πi

We find a slightly different formula e n kab ha hb for the holonomy dependent phase in the literature. This is, however, not gauge invariant even for integer Chern-Simons levels. A simple derivation of the holonomy dependent phase in (30) is given in Appendix A.1. 2

7

2.2

Projection operator

As we mentioned in the introduction, we use the factorization to holomorphic blocks to determine the sign factor. For M = S 3 and S 2 ×S 1 , it is known that the partition function is written in terms of holomorphic blocks by [17, 18, 19, 20, 21] X ZM = B A (xa , q)B A (e xa , qe). (32) A

In the case of S 3 , the variables xa , q, x ea , and qe are given by 2

q = e2πib ,

xa = e2πbµa ,

−2

qe = e2πib ,

−1 µ a

x ea = e2πb

.

(33)

This factorization is also expected for the orbifold partition function with different definition for the variables xa , q, x ea , and qe. This factorization is naturally interpreted in Higgs branch localization, in which the index A labels Higgs vacua and the blocks are identified with the vortex partition functions[18, 20]. In order to obtain a factorized form of the orbifold partition function, it is convenient to rewrite sb,hI in (27) as 1 1 = P(bµI ,hI ) sb,hI (b µI ) sb (b µI )

(34)

where P(z,h) with z ∈ C and h ∈ Zn is the operator acting on a function of z defined by P(z,h) f (z) =

Y

(k,l)∈Lh

f (zk,l ),

zk,l ≡

z + ib(k −

n−1 ) 2

+ ib−1 (l − n

n−1 ) 2

,

(35)

where Lh = {(k, l)|0 ≤ k, l < n, k − l = h

mod n}.

(36)

An advantage of rewriting (27) with this operator is that the operator preserves the factorized form of the function. Namely, if a function f (z) is the product of two functions g(z) and h(z), the relation P(z,h) f (z) = (P(z,h) g(z))(P(z,h) h(z)) µ) of a theory is factorized holds. Therefore, if the S 3 partition function ZS 3 (b into holomorphic blocks as in (32), and if the orbifold partition function is obµ) by applying the operator P(bµ,h) , we can immediately obtain tained from ZS 3 (b the factorized form of the orbifold partition function by applying P(bµ,h) to the holomorphic blocks for S 3 . The operator P(z,h) is defined to simplify the expression of the Zn projection of the one-loop factor, and it is a priori not guaranteed that it correctly reproduces −S cl the classical factor e S3 /Zn in the orbifold partition function. Interestingly, up to the sign factor which we have not fixed yet, it reproduces the classical factor

8

(30) in the orbifold partition function from (23) for S 3 . Let us consider a ChernSimons term with factorized Chern-Simons level kab = κca cb . Z iκca cb Aa dAb . (37) 4π For S 3 , this gives the classical factor cl

2

e−SS3 = e−κπibµ ,

µ b = ca µ a .

Applying the operator P(bµ,h) on this function, we obtain −1 )2 πiκ (b+b πiκ 2 2 −1) −κπib µ2 (n 12 P(bµ,h) e =e n exp − µ b + [h](n − [h]) , n

(38)

(39)

where h = ca ha and [h] denotes the smallest non-negative integer in h + nZ. (We have not yet assumed that κ is an integer.) We rewrite this as πiκ 2 −κπib µ2 2κ 2 . (40) P(bµ,h) e = (b-dependent factor)ρ(h) exp − µ b + (n − 1)h n In this paper, the factorization is used simply as a criterion for the correct choice of sign factors, and we are not interested in the prefactor depending only on b. The exponential factor is nothing but the classical factor in (30), and ρ(h) is πi

πi

2

ρ(h) = e 2n [h](n−[h])e− 2n (n−1)h .

(41)

This function always takes +1 or −1 depending on h. As we will show shortly, we can compose the sign factor f (h) by using ρ(h).

2.3

Factorization and sign factor

Let us consider a general non-gauge theory on S 3 . As is pointed out in [20], it is convenient to decompose the Chern-Simons level into the part canceling the parity anomaly and the remaining part. kab =

X α

κα cαa cαb −

1X qIa qIb . 2 I

(42)

κα and cαa in the first term are integers, and the second term is the fractional contribution that cancels the parity anomaly. With this decomposition, we rewrite the partition function as Y Y 1 2 = (e−πiµα )κα Z∆ (b µI ) µI ) I sb (b α I

ZS 3 (µ) = e−πikab µa µb Q

9

(43)

where Z∆ (b µ) is the partition function of the “tetrahedron theory,”[25] and given by πi 2 e 2 µb . (44) Z∆ (b µ) = sb (b µ) The two kinds of factors in the product (43) are known to be factorized into holomorphic blocks [20] Z∆ (µ) = e−πi

b2 +b−2 24

2 −2 πi b +b 12

2

e−πiµ = e The blocks are given by B∆ (x; q) = (qx−1 ; q),

B∆ (x, q)B∆ (e x, qe),

(45)

BCS (x; q)BCS (e x; qe).

BCS (x; q) =

(46)

1

1 2

1

(−q x; q)(−q 2 x−1 ; q)

,

(47)

where (x; q) is the q-Pochhammer symbol (x; q) =

∞ Y

(1 − xq k ).

(48)

k=0

An important feature of the factorization is that the information of the background manifold is encoded in the definition of the arguments of holomorphic blocks. For S 3 they are given by (33), and the blocks for the orbifold should be given by the same functions with different arguments. We can confirm this by computing the holomorphic blocks for the orbifold by applying the operator P(z,h) to the holomorphic blocks for S 3 . Indeed, we can easily show P(µ,h) B∆ (x; q) = B∆ (x′ ; q ′ ), P(µ,h) B∆ (e x; qe) = B∆ (e x′ ; qe′ ), P(µ,h) BCS (x; q) = BCS (x′ , q ′ ), P(µ,h) BCS (e x; qe) = BCS (e x′ , qe′ ),

(49)

where the variables for the orbifold are 1

q ′ = ωq n ,

1

x′ = ω h x n ,

1

qe′ = ωe qn,

1

x e′ = ω −h x en .

(50)

The arguments above guarantee that we obtain the orbifold partition function that is correctly factorized into the holomorphic blocks by simply applying the projection operator to the factors in S 3 partition function (43). Y Y 2 (P(µα ,hα ) e−πiµα )κα ZS 3 /Zn (µ, h) ∝ P(bµI ,hI ) Z∆ (b µI ) α

∝

Y

I

e

πi κ (b µ2 +(n−1)h2α ) n α α

α

I

−S cl3

=e

πi (b µ2I +(n−1)h2I ) Y ρ(hI )e 2n

S /Zn

(µ,h)

Y ρ(hI ) sb,hI (b µI ) I 10

sb,hI (b µI )

(51)

“∝” means the ignorance of prefactors depending only on b. By comparing this to (7), we obtain Y f (h) = ρ(hI ). (52) I

(We cannot fix the overall sign factor independent of h, which we are not interested in.) We can absorb the sign factor by the redefinition of the orbifold double sine function simp (53) b,h (z) = ρ(h)sb,h (z), and then we can present the orbifold partition function in the same form with the original one. −S cl3

ZS 3 /Zn (µ, h) = e

S /Zn

(µ,h)

1

. imp µI ) I sb,hI (b

Q

(54)

In [13], a similar improvement of the orbifold double sine function sbb,h (z) = σh sb,h (z) is proposed for odd n = 2m + 1. The extra sign factor σh is related to ρ(h) through σh = (−)mh ρ(h). (55) In [13], only theories without parity anomaly are considered. ThisPmeans that the charge assignment qIa for every U(1) gauge symmetry satisfies I qIa ∈ 2Z. In such a case, the difference of σh and ρ(h) does not affect the partition function. In the case of even n, [13] did not succeed in finding such an improvement. The reason is that in [13] the sign factor σh is assumed to be a periodic function of h with period n. The function ρ(h) does not satisfy this condition. It may change its sign under the shift h → h + n. n(n−1) ρ(h + n) = (−1)(n−1)h+ 2 . ρ(h)

(56)

This means that the improved function simp b,h may change its sign in the large gauge transformation h → h + n. This, however, does not cause any problem. If the parity anomaly arising in the one-loop part is correctly canceled by the bare Chern-Simons term, the partition function (54) is invariant under the shift h → h + n.

2.4

Gauge theories

In the previous subsection we gave a prescription to fix the relative signs among holonomy sectors. It is implemented by the redefinition of the orbifold double sine function (53). The orbifold partition function of an arbitrary non-gauge theory with this improvement is correctly factorized into holomorphic blocks.

11

We expect that this improvement works for gauge theories, too. Namely, the orbifold partition function X Z 1 (µ,h) −S cl (ext) (ext) (57) ,h )= dµ(dyn) e S3 /Zn ZS 3 /Zn (µ Q imp s (b µ ) I b,h (dyn) I h

for a gauge theory is correctly factorized into holomorphic blocks as in (32). Unfortunately, we have not succeeded in proving this for an arbitrary gauge theory. We here consider two examples of gauge theories, SQED with Nf = 1, and an su(2) Chern-Simons theory with an adjoint chiral multiplet. 2.4.1

SQED

As the first example, let us consider SQED with one flavor (q, qe). This theory has four U(1) symmetries. One is a gauge symmetry U(1)G , and the others are global symmetries. See Table 1 for charge assignments. U(1)R is an R-symmetry. U(1)A Table 1: The charge assignments in the SQED and XYZ model. For U(1)R symmetry the charges of the fermion components are shown. q qe ∆−1 ∆−1 1 −1 0 0 1 1

U(1)R U(1)G U(1)V U(1)A

X −∆ 1 −1

Y −∆ −1 −1

Z 2∆ − 1 0 2

is a flavor symmetry acting on q and qe with charge +1. U(1)V is the topological symmetry, and the corresponding external gauge field AV couples to the U(1)G flux dAG through the Chern-Simons term Z 1 AV dAG . (58) 2π The partition function is S 3 /Z ZSQEDn

2πi 2πi n−1 Z X e n µV µG e n hV hG dµG = , imp imp n s (b µ )s (b µ ) q q e b,hq b,hqe hG =0

where µ bI and hI are defined by µ bq = (∆ − 1)

iQ + µA + µG , 2

µ bqe = (∆ − 1)

iQ + µA − µG . 2

hqe = hA − hG .

hq = hA + hG , 12

(59)

(60) (61)

Let us confirm that this orbifold partition function is factorized into holomorphic blocks. Although it would not be difficult to directly prove the factorization by performing the integral by using the residue theorem, we take another way. In [13], it is confirmed that the orbifold partition function of the SQED with an appropriate choice of the sign factor coincides with that of the XYZ model, the system consisting of three chiral multiplets X, Y , and Z interacting through the superpotential W = XY Z. Because XYZ model is a non-gauge theory and we have already proved the factorization of non-gauge theories, this duality relation guarantees the factorization of the partition function of the SQED. The charge assignments for XYZ model is also shown in Table 1, and the partition function is given by 1 (b µY )simp µZ ) b,hZ (b

S 3 /Z

ZXYZ n =

simp µX )simp b,hX (b b,hY

(62)

with the parameters µ bX = −∆

and

iQ −µA +µV , 2

µ bY = −∆

hX = −hA + hV ,

iQ −µA −µV , 2

hY = −hA − hV ,

µ bZ = (2∆−1) hZ = 2hA .

iQ +2µA . (63) 2 (64)

It is easy to numerically check the coincidence of (59) and (62). If we use the result of [13], what we have to do to confirm the relation ZXYZ = ZSQED is to show the sign factor determined in [13] by requiring ZXYZ = ZSQED is same as the extra sign factor introduced by replacing sb,h by simp b,h . Indeed, the product of the five sign factors corresponding to the five double sine functions in ZSQED and ZXYZ coincides with the factor given in [13]. σ(hG , hV , hA ) = ρ(hq )ρ(hqe)ρ(hX )ρ(hY )ρ(hZ ). 2.4.2

(65)

su(2) gauge theory

Next, as a simple example of non-Abelian gauge theory, we consider an su(2)1 gauge theory coupled to an adjoint chiral multiplet Φ. (We use su(2) instead of SU(2) to emphasize we do not specify the global structure of the gauge group.) Jafferis and Yin [26] proposed that this theory is dual to the theory consisting a single chiral multiplet X ∼ trΦ2 . If we can show the matching of the partition functions for the dual pair the factorization of the su(2) theory is guaranteed as the previous example. In general, we cannot completely specify a gauge theory only by local information. There may be different theories distinguished by global structure of the gauge group. In [27] importance of such distinction in four-dimensional supersymmetric gauge theories is pointed out, and it is investigated how such theories 13

are related to each other by Seiberg duality. The duality is checked in [28] by matching the lens space index, which is sensitive to the global structure of the gauge group. Similar aspects in three-dimensional gauge theories are studied in [29]. The su(2) theory we discuss here is also an example of such a theory. It has only an adjoint chiral multiplet as a matter field, and no elementary fields are transformed by the center of SU(2). Therefore, precisely speaking, there are two choices of the gauge group, SU(2) and SO(3). Although it is important problem to clarify how the different choices of the gauge group affect the duality and the factorization, we will not do any detailed analysis here. We only present the result of a preliminary analysis based on the numerical computation of the partition function. The symmetries and the charge assignments for the dual theories are shown in Table 2. In the su(2) theory, the action contains the Chern-Simons term Table 2: The symmetries and the charge assignments in Jafferis-Yin duality. The U(1)R charges in the table are those for fermion component of the multiplets.

SU(2)G U(1)R U(1)A

i 4π

Z

trfund

Φ adj ∆−1 1

X 2∆ − 1 2

2i AG dAG − A3G 3

for the dynamical gauge field, and Z Z i (−3/2)i ((∆ − 1)AR + AA )d((∆ − 1)AR + AA ) + AR dAR 4π 4π

(66)

(67)

for the external U(1)R and U(1)A gauge fields. On the other hand, in the chiral free theory contains the Chern-Simons term Z (−1/2)i ((2∆ − 1)AR + 2AA )d((2∆ − 1)AR + 2AA ) (68) 4π for the external gauge fields. We define the holonomy parameter h for the dynamical su(2) gauge group by ! I πih en U = exp i A = . (69) πih e− n γ 14

If the gauge group is SU(2), U n must be the unit matrix, and the holonomy is quantized by h ∈ 2Z, while for SO(3) gauge group h can be an arbitrary integer. The periodicity of h also depends on the global structure of the gauge group and the background manifold, and there are two possibilities, h ∼ h + 2n or h ∼ h + n. Here we will not argue which of these possibilities should be adopted. We simply compute the partition functions for all the holonomy sectors labeled by h = 0, 1, . . . , 2n − 1, and infer from the duality which sectors should be summed up. The orbifold partition function of each holonomy sector of the su(2) theory is Z 2 2 dµG − πi µ2G − πi (n−1)h2 πin 32 {(∆−1) iQ2 +µA } + Q4 3πi (n−1)h2A S 3 /Zn e e 2n Zsu(2) (h, hA ) = e 2n e 2n 2n 1 × imp , imp imp sb,hφ (b µφ+ )sb,hφ (b µφ0 )sb,hφ (b µφ− )simp µW + )simp µW − ) b,h + (b b,h − (b +

0

−

W

W

(70) where the parameters are given by µ bφ+ = (∆ − 1)

iQ + µA + µG , 2

hφ+ = hA + h,

iQ iQ + µA , µ bφ− = (∆ − 1) + µA − µG , 2 2 iQ iQ + µG , µ bW − = − µG , (71) µ bW + = 2 2 hφ0 = hA , hφ− = hA − h, hW + = h, hW − = −h. (72) µ bφ0 = (∆ − 1)

The orbifold partition function of the chiral free theory is S 3 /Zn

ZX

(hA ) = e 2n {(2∆−1) πi

iQ +2µA 2

2

(n−1)h2A } e 2πi n

1

, µX ) simp b,hX (b

(73)

where the parameters are given by µ bX = (2∆ − 1)

iQ + 2µA , 2

hX = 2hA .

(74)

Numerical results are divided to four cases: • In the case of n ∈ 4Z, only the even sector coincides up to a constant factor. X S 3 /Z S 3 /Z Zsu(2) n (h, hA ) = 2ZX n (hA ). (75) h=0,2,...,2n−2

• In the case of n ∈ 4Z + 1, both even and odd sector coincides up to a constant factor. X X √ πi S 3 /Z S 3 /Z S 3 /Z Zsu(2) n (h, hA ) = Zsu(2) n (h, hA ) = 2e 4 ZX n (hA ). h=0,2,...,2n−2

h=1,3,...,2n−1

(76)

15

• In the case of n ∈ 4Z + 2, only the odd sector coincides up to a constant factor. X S 3 /Z S 3 /Z Zsu(2) n (h, hA ) = −2iZX n (hA ). (77) h=1,3,...,2n−1

• In the case of n ∈ 4Z + 3, both even and odd sector coincides with different constant factor. X √ πi S 3 /Z S 3 /Z Zsu(2) n (h, hA ) = 2e 4 ZX n (hA ), (78) h=0,2,...,2n−2

X

h=1,3,...,2n−1

√ πi S 3 /Z S 3 /Z Zsu(2) n (h, hA ) = − 2e 4 ZX n (hA ).

(79)

In all cases, we do not need additional sign factor if we use the improved function simp b,h . These results strongly suggests that we should take the following holonomy sectors to sum up. • n ∈ 4Z: h = 0, 2, . . . , 2n − 2. • n ∈ 4Z + 1: h = 0, 2, . . . , 2n − 2 or h = 1, 3, . . . , 2n − 1. • n ∈ 4Z + 2: h = 1, 3, . . . , 2n − 1. • n ∈ 4Z + 3: h = 0, 2, . . . , 2n − 2 or h = 1, 3, . . . , 2n − 1. Unfortunately, we have no clear explanation for these non-trivial choices of the holonomy sectors. We hope we can return to this problem in future work.

3

The sign factor to the lens space index and the 3d index

In this section we discuss the effect of the sign factor f (h) derived in Subsection 2.2 to the 4d lens space index [12] and 3d superconformal index [30, 31, 32]. Since the orbifold partition function is related to the lens space index as well as the 3d superconformal index through dimensional reductions the same or similar sign factor should appear in those indices. Let us first summarize the lens space index and its reductions, and then, we discuss the sign factors. The lens space index can be obtained by the orbifold projection of the 4d index on S 3 × S 1 . The projection is performed along the Hopf fiber direction of the S 3 of the S 3 × S 1 , and it is realized by leaving the modes compatible

16

with the identification (6). Since the rotation along the Hopf fiber direction is characterized by (J1 − J2 ) the projection is realized by inserting the operator n−1 1 X 2πi m (J1 −J2 ) e n n m=0

(80)

to the index (3). Namely, the lens space index In is n−1 2πi 2πi 2πi 1X In (p1 , p2 , za ) = I(e n m p1 , e− n m p2 , e− n ha za ), n m=0

(81)

where we introduced the holonomies for global symmetries ha . Let us focus on a theory consisting of a single chiral multiplet Φ whose charge for a U(1) global symmetry is 1. The 4d index on S 3 × S 1 defined by (3) for the theory can be rewritten as follows [33]. IΦ (p1 , p2 , z) = Pexp′ [IΦsp ] ∞ X sp i+1 IΦ (p1 , p2 , z) = zpi1 pj2 − z −1 pj+1 1 p2 i,j=0

z = 1 − p1 p2

1 p2 + 1 − p1 1 − p2

(82) (83)

p1 p2 z −1 − 1 − p1 p2

p1 1 + 1 − p1 1 − p2

,

(84)

where z is a fugacity for the global symmetry, and the prime of the plethystic exponential denotes that it includes the zero point contributions; Pexp′ [x] = ex/2 Pexp[x]. In this form the insertion of the operator (80) is equivalent to leave the modes invariant under the condition i−j =h

mod n

(85)

in (83). Then, the lens space index for this theory can be written as follows. sp In,Φ (z, h) = Pexp′ [In,Φ (z, h)] sp In,Φ (z, h)

z = 1 − p1 p2

[h]

n−[h]

p p1 + 2 n n 1 − p1 1 − p2

!

p1 p2 z −1 − 1 − p1 p2

n−[h]

[h]

p p1 + 2 n n 1 − p1 1 − p2

!

(86) . (87)

Let us now consider the effect of the sign factor (41). The orbifold partition function is derived in the small radius limit (5) of the lens space index. Since the sign factor is independent of the radius β it can be uplifted to the lens space index, and we define the improved index as ! Y Inimp(z, h) = ρ(hI ) In (z, h). (88) I

17

For the theory considered above the improved lens space index is written as iπ

imp,sp imp (z, h)] (89) In,Φ (z, h) = e− 2 h(1−h) Pexp′ [In,Φ h n−h n−h −1 h z p p1 p2 z p2 p1 p1 imp,sp In,Φ (z, h) = + 2 n − + . n n 1 − p1 p2 1 − p1 1 − p2 1 − p1 p2 1 − p1 1 − pn2 (90)

Note that [h] in In,Φ is now replaced by h. Although both In,Φ and ρ(h) include [h] the combination of those (88) is analytic in terms of h. In order to illustrate the effect of the phase factor we obtained to the 3d index, we first describe the 3d index for the tetrahedron theory, which is written as follows " |h| # |h| 1+ 2 −1 2 z |h|−h 1 |h|−h q z q 1 3d Pexp . (91) − I∆ = (−1)− 2 |h| (−q 2 ) 2 z − 2 1−q 1−q Later, it is noticed in [19] that the 3d index can be written in an analytic form by adding a simple phase factor: # " h 1+ h2 −1 2z z q q 3d,imp 3d I∆ . (92) = i|h| I∆ = Pexp − 1−q 1−q The phase factor is needed for the 3d index to be factorized. As is pointed out in [12], the 3d index is obtained from the 4d lens space index by taking n → ∞ limit. When we take this limit h is kept finite and is identified with the magnetic flux in S 2 × S 1 . In this limit the sign factor (41) can be rewritten as πi

2

ρ(h) = e 2 (|h|−h ) .

(93)

The contribution of the classical factor in (30) in the large n limit becomes e−πikab ha hb .

(94)

As the tetrahedron theory has Chern-Simons term with level −1/2 its contribution combined with the extra sign factor (93) is πi

2

πi

2

e 2 h × e 2 (|h|−h ) = i|h| .

(95)

This is precisely the factor introduced in (91) to improve the 3d index. Note that the improved 3d index is analytic in terms of h, which is inherited from the analytic structure of the improved 4d index.

18

4

Conclusions

We focused on the sign of the orbifold partition function. We proposed a formula that systematically determine the relative signs among holonomy sectors. We use the factorization to the holomorphic blocks as a criterion to determine correct signs. For non-gauge theories we proved that the partition function with signs determined by the formula is correctly factorized into the holomorphic blocks. In the case of gauge theories, as a simple example, we considered two theories: SQED with Nf = 1 and su(2) gauge theory with an adjoint flavor. For the former we confirmed the factorization of the orbifold partition function. In the case of the su(2) theory, we have not fully understood the summation over the holonomy sectors. We guessed which holonomy sector contributes to the partition function with the help of the duality to a non-gauge theory, which is known as Jafferis-Yin duality. We found that we have to choose an appropriate subset of the holonomy sectors to obtain the partition function consistent to the duality. The formula for the sign factor gives the correct relative phases for the contributing sectors. Therefore, up to the subtlety for the choice of the holonomy sector, which is probably related to the global aspects of the gauge bundle on the orbifold, the formula seems to give appropriate signs. We also discussed the sign factors in the lens space index of the 4d theories, and the index for 3d theories. The formulae for these indices have been known, and contain non-trivial sign factors, which are often implicit in the literature. We showed that they are closely related to the sign factor for the orbifold partition function.

Acknowledgments We would like to thank S. Terashima for useful information and S. Kim for valuable discussion. Y. I. is partially supported by Grant-in-Aid for Scientific Research (C) (No.24540260), Ministry of Education, Science and Culture, Japan. H. M. acknowledges the financial support from the Center of Excellence Program by MEXT, Japan through the “Nanoscience and Quantum Physics” Project of the Tokyo Institute of Technology. The work of D.Y. is supported in part by the National Research Foundation of Korea Grant NRF-2012R1A2A2A02046739, the Global Center of Excellence Program by MEXT Japan through the “Nanoscience and Quantum Physics” Project of the Tokyo Institute of Technology, and Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation.

19

A

Appendix

A.1

Classical contribution of Chern-Simons terms

Let us compute the Chern-Simons term Z Z ik ik AdA = F ∧ F, SCS = 4π M 4π X

(96)

for a flat U(1) connection. X is a manifold whose boundary is M = S 3 /Zn . To compute e−SCS unambiguously for an arbitrary integer k, X should be a spin manifold [34]. We represent M = S 3 /Zn as the Hopf fibration over B = S 2 . Let us represent B as the boundary of the solid cylinder C: r ≤ 1, 0 ≤ z ≤ n + 1, where (r, φ, z) is the cylindrical coordinate system in three-dimensional space. We divide M into n + 1 regions: M0 , M1 , . . . , Mn . The ℓ-th region Mℓ is defined by ℓ ≤ z ≤ ℓ + 1. Let 0 ≤ ψℓ < 2π be the fiber coordinate in the region Mℓ . On the boundary between adjacent regions, the fiber coordinates are related by ψℓ |z=ℓ = (ψℓ+1 + φ)|z=ℓ . We want to realize a flat connection with the holonomy I n h= A, 2π γ

(97)

(98)

where γ is a cycle along an S 1 fiber. Let us consider the following gauge potential in Mℓ . h (99) Aℓ |Mℓ = dψℓ + cℓ dφ, n For the gauge field on the top (z = n + 1) and the bottom (z = 0) of the cylinder to be smooth and flat, we need to set c0 = cn = 0. The gauge field jumps on the boundaries by h Aℓ − Aℓ−1 |z=ℓ = cℓ − cℓ−1 − dφ. n

(100)

(101)

For this to be a gauge symmetry, the coefficients cℓ should satisfy cℓ − cℓ−1 −

h ∈ Z. n

(102)

To compute the Chern-Simons action for this gauge potential, we need to define the manifold X and to extend the gauge potential into X. We adopt X 20

which is topologically n-centered Taub-NUT. We represent X as an S 1 fibration over C with n centers placed on the axis at z = 1, 2, . . . , n. X is also divided into n + 1 regions X0 , X1 , . . . , Xn at z = 1, 2, . . . , n. We take the following ansatz for the extension of the gauge field in the ℓ-th region. h − B(z)(1 − f (r)) dψℓ + cℓ f (r)dφ, (103) Aℓ = n where B(z) and f (r) are continuous functions satisfying B(0) = B(n + 1) = 0,

f (0) = 0,

f (1) = 1.

(104)

These conditions guarantee that (103) coincides to (99) on M. Let us think about the smoothness of the gauge field at the center at z = ℓ. The center is located on the boundary between Mℓ and Mℓ−1 . In each region the gauge field near the center is given by Aℓ = bℓ dψℓ ,

Aℓ−1 = bℓ dψℓ−1 ,

bℓ =

h − B(ℓ). n

(105)

If we assume bℓ ∈ Z,

(106)

these gauge field can be eliminated by the gauge transformation Aℓ → A′ℓ = Aℓ − bℓ dψℓ ,

Aℓ → A′ℓ−1 = Aℓ−1 − bℓ dψℓ−1 ,

(107)

and then the gauge field is smooth at the center. For the gauge potential after the gauge transformation (107) the jump of the gauge field on the boundary is A′ℓ − A′ℓ−1 |z=ℓ = (cℓ − cℓ−1 − B(ℓ))f (r)dφ.

(108)

and A′ℓ and A′ℓ−1 are smoothly connected if cℓ − cℓ−1 = B(ℓ) (1 ≤ ℓ ≤ n).

(109)

Note that the condition (106) follows from this and (102). By using relations we obtained above, we can easily compute the integral of the instanton density over Xℓ . 1 Z 1 ℓ+1 2 2 f (r) − f (r) Fℓ ∧ Fℓ = 8π cℓ [B(z)]ℓ 2 Xℓ 0 = 4π 2 cℓ (cℓ+1 − 2cℓ − cℓ−1 ).

(110)

This depends on the constants cℓ that satisfy the conditions (100) and (102). Although there are infinitely many solutions to these conditions and the integral (110) depends on the choice of a solution, the classical action is uniquely 21

determined up to the unphysical shift by 2πiZ. Z n−1 X 1−n 2 ik F ∧ F = πik cℓ (cℓ+1 − 2cℓ + cℓ−1 ) = πik h S= 4π X n

mod 2πi.

ℓ=1

(111)

We can easily extend this result to general Chern-Simons term for multiple abelian gauge fields Z ikab 1−n S= Aa dAb = πi kab ha hb mod 2πi. (112) 4π S 3 /Zn n

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