Apr 18, 2014 - Masazumi Honda. High Energy Accelerator Research Organization (KEK),. Tsukuba, Ibaraki 305-0801, Japan,. Yukawa Institute for Theoretic...

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Preprint typeset in JHEP style - PAPER VERSION

KEK-TH-1671,YITP-13-95

Direct derivation of “mirror” ABJ partition function

Masazumi Honda High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan [email protected]

Abstract: We study the partition function of the three-dimensional N = 6 U (N )k × U (N + M )−k superconformal Chern-Simons matter theory known as the ABJ theory. We prove that the ABJ partition function on S 3 is exactly the same as a formula recently proposed by Awata, Hirano and Shigemori. While this formula was previously obtained by an analytic continuation from the L(2, 1) lens space matrix model, we directly derive this by using a generalization of the Cauchy determinant identity. We also give an interpretation for the formula from brane picture. Keywords: Matrix Models, M-Theory, Supersymmetry and Duality.

Contents 1. Introduction

1

2. Proof 2.1 For 2M ≤ |k| + 1 2.2 For 2M ≥ |k| − 1

3 5 7

3. Discussion

8

1. Introduction Recently there has been significant progress in understanding low-energy effective theories of multiple M2-branes. The simplest case of such theories is the so-called ABJM theory [1], which is the 3d N = 6 supersymmetric Chern-Simons matter theory (CSM) with the gauge group U (N )k × U (N )−k . The authors in [1] discussed that this theory describes N M2branes on C4 /Zk in the low-energy limit. Furthermore it has been shown by the localization method [2, 3] (see also [4]) that a class of supersymmetric observables in N = 2 theory on S 3 have representations in terms of certain matrix integrals. Thanks to the localization technique, several works have extensively studied the partition function and BPS Wilson loops in the ABJM theory on S 3 [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Especially a breakthrough was caused by a seminal paper [10], which rewrites the ABJM partition function as an ideal Fermi gas system (see also [19, 7, 9, 20]). Based on this formalism, recent studies have revealed structures of the partition function [15] and half-BPS Wilson loop [17] including worldsheet and membrane (D2-brane) instanton effects [21, 8]. In this paper we study the N = 6 CSM with more general gauge group U (N )k ×U (N + M )−k known as the ABJ theory [22]. This theory has been expected to arise when we have N M2-branes on C4 /Zk , together with M fractional M2-branes sitting at the singularity. The authors in [22] also argued that the ABJ theory has good approximations by the 11d SUGRA on AdS4 × S 7 /Zk with discrete torsion for N 1/5 ≫ k, and the type IIA SUGRA on AdS4 ×CP3 with nontrivial B-field holonomy for N 1/5 ≪ k ≪ N , respectively. Furthermore the recent works [23] have also conjectured that the ABJ theory is dual to N = 6 parityviolating Vasiliev theory on AdS4 with a U (N ) gauge symmetry when M, k ≫ 1 with M/k and N kept fixed. Thus it is worth studying the ABJ theory in detail. Here we study the partition function of the U (N )k × U (N + M )−k ABJ theory on S 3 . By using the localization method, the partition function is given by [3] 1

(N,N +M ) (k) ZABJ

2

2

i− 2 (N −(N +M ) )sign(k) = (N + M )!N !

Z

∞ −∞

PN 2 2 dN +M µ dN ν − ik (PN+M j=1 µj − a=1 νa ) 4π e N +M N (2π) (2π)

–1–

×

"Q

µ −µl

2 sinh j 2 QN +M QN

1≤j

j=1

Q

1≤a

b 2 sinh νa −ν 2

#2

.

(1.1)

For M 6= 0, this representation is not suitable for the Fermi gas approach since the Cauchy identity is not helpful in contrast to the ABJM case [10]. Nevertheless, Awata, Hirano and Shigemori (AHS) recently proposed [24] that the ABJ partition function is equivalent to 1

(N,N +M ) ZAHS (k)

=

i− 2 (N

2 +(N +M )2 )sign(k)+N + M 2

N

(−1) 2 (N −1)

2N kN +M/2 N ! Z

i∞−2πη −i∞−2πη

(1 − q)

sa N (q 2π +1 )M dN s Y 1 s (−2πi)N 2 sin s2a (−q 2πa +1 )M a=1

M (M −1) 2

Y

1≤a

G2 (M + 1; q)

(1 − q

sb −sa 2π

)2

(1 + q

sb −sa 2π

)2

, (1.2)

where − 2πi k

q=e

, (a)n =

n−1 Y

− z2 (z−1)

(1−aq m ), G2 (z+1; q) = (1−q)

m=0

# " m ∞ Y 1 − q z+m (1−q m )z . 1 − qm

m=1

Here η specifies the integral contour. The authors in [24] have determined η for N = 1 as ( 0+ for |k| 2 −M ≥0, η= (1.3) |k| |k| − 2 + M + 0+ for 2 − M ≤ 0

to be consistent with the Seiberg-like duality [22] but not for general N . One expects that the AHS formula (1.2) gives a generalization of the “mirror” description of the ABJM partition function. One of the strongest evidence is that ZAHS |M =0,k=1 is the same as the partition function of the N = 4 super QCD with one adjoint and fundamental hypermultiplets [19] related through 3d mirror symmetry [25, 26, 1]. There is also an interpretation for general k from the S-dual brane construction [27, 28]. The AHS representation also has several advantages. First, this is suitable for the Fermi gas approach and Tracy-Widom theorem [29], which reduces the grand canonical analysis to Thermodynamic Bethe Ansatz-like equation. Second, it is easier to perform Monte Carlo simulation as in the ABJM case [11] than the original formula (1.1). Finally, the AHS formula highly simplifies analysis in the Vasiliev limit. Despite of the advantages, the AHS proposal is still conjecture in the following senses: • The derivation of ZAHS started with an analytic continuation [5] from the partition function of the L(2, 1) lens space matrix model [30]. The analytic continuation has not been rigorously justified in spite of much strong evidence [5, 8, 11, 15, 31]. • While we can represent the partition function of the L(2, 1) matrix model in terms of a convergent series, its analytic continuation to the ABJ theory yields a non-convergent series. The AHS formula corresponds to its well-defined integral representation and reproduces the series order by order in the perturbative expansion by 2πi/k although there would be non-perturbative ambiguity generically. In this paper we prove the AHS conjecture ZABJ = ZAHS and determine the integral contour for arbitrary parameters as discussed in section 2. It will turn out that the choice (1.3) of η is still correct even for general N . Section 3 is devoted to discussion.

–2–

2. Proof In this section we prove ZABJ = ZAHS . Let us start with the localization formula (1.1). For M = 0, the Cauchy determinant identity is quite useful to derive its “mirror” description [19, 9, 10], but not for M 6= 0. Here instead we use a generalization1 of the Cauchy identity (corresponding to Lemma. 2 of [32]) : Q Q j

1

N+M

2

N+M

N

where j, l, a and b run 1 ≤ j, l ≤ N + M , 1 ≤ a, b ≤ N . One of easiest way to prove this identity is to use Boson-Fermion correspondence in 2d CFT. More concretely, the left-hand side corresponds to a correlation function on P1 of a bc system in “Boson” representation: hM |b(x1 ) · · · b(xN +M )c(yN ) · · · c(y1 )|0i,

(2.2)

where we have identified as b ↔: eϕ : and c ↔: e−ϕ : by using a free boson satisfying X an z −n , [am , q˜] = δm,0 , [am , an ] = mδm+n,0 , ϕ(z) = q˜ + a0 log z − n n6=0

an≥0 |0i = 0,

h0|˜ q = h0|an<0 = 0,

h0|0i = 1,

|M i = eM q˜|0i,

(2.3)

while the right-hand side is the one in “Fermion” representation with identifications: b ↔ ψ¯ and c ↔ ψ in terms of charged fermions satisfying X X 1 1 ¯ ψ¯n z −n− 2 , ψ(z) = ψ(z) = ψn z −n− 2 , n∈Z+1/2

n∈Z+1/2

{ψ¯m , ψ¯n } = {ψm , ψn } = 0, {ψ¯m , ψn } = δm+n,0 , ψ¯n>0 |0i = ψn>0 |0i = 0, h0|ψ¯n<0 = h0|ψn<0 = 0,

|M i = ψ¯−M + 1 · · · ψ¯− 1 |0i. (2.4) 2

2

If we take xj = eµj and ya = −eνa in the determinant identity (2.1), then we find Q =

µj −µl 2

Q

a

j

e

j=1

Q

e

a=1

det

νa −νb 2

θN,l 2 cosh

1

µj −νl 2

+ e(

N +M + 21 −j

)µj θ

l,N +1

!

,

(2.5)

We thank Sanefumi Moriyama to tell us about this identity and suggest that the identity would be useful for analyzing the ABJ matrix model (1.1). He showed us a different proof for the identity in April 2012. In the meanwhile, we remembered the identity when we read a (Japanese) textbook on conformal field theory written by Yasuhiko Yamada. Therefore we are also grateful to Yasuhiko Yamada.

–3–

where θj,l

( 1 for j ≥ l = . 0 for j < l

(2.6)

Plugging this into (1.1), we find (N,N +M ) (k) ZABJ

ik 2 Z ∞ N +M N +M N NABJ X d µ dN ν Y − ik µ2j −M µj Y e 4π νa +M νa σ = (−1) e 4π N +M (2π)N a N! σ 2 cosh µa −ν −∞ (2π) 2 a=1 j=1 ! NY +M NY +M 1 1 θ N,j (N +M + 2 −j )µσ(j) θ e(N +M + 2 −l)µl j,N +1 , µσ(j) −νj + e 2 cosh j=1 l=N +1 2

where

1

NABJ = i− 2 (N

2 −(N +M )2 )sign(k)

.

(2.7)

e π px , dx 2 cosh x −∞

(2.8)

Making a Fourier transformation 1 1 p = 2 cosh 2 π

Z

∞

i

and introducing auxiliary variables yN +1 , · · · , yN +M constrained by 1 π with l = N + 1, · · · , N + M, N +M + −l yl = i 2

(2.9)

the partition function becomes ZABJ

Z ∞ N N +M N +M d xd y dN +M µ dN ν Y − ik µ2j −M µj NABJ X σ (−1) e 4π = N π N +M (2π)N +M (2π)N N! σ −∞ π j=1

× ×

N Y

e

a=1 NY +M

ik 2 ν +M νa + πi xa (µa −νa )+ πi (yσ(a) µa −ya νa ) 4π a

2 cosh xa · 2 cosh ya # 1 i π πδ yl − (N + M + 1/2 − l) e(N +M + 2 −l)µl + π yσ(l) µl . i

"

l=N +1

(2.10)

PN +M PN +M Here we used j=1 yj µσ(j) = j=1 yσ−1 (j) µj and redefined the permutation symbol as −1 σ → σ. Performing the Fresnel integrals over µi and νa allows us to find ZABJ =

2 2i N dN x dN +M y Y e− kπ xa (ya −yσ(a) )+ k M (ya −yσ(a) ) M N π N +M 2 cosh xa · 2 cosh ya −∞ π N !|k|N + 2 σ a=1 # " NY +M i 2 2 2 1 1 iπ π πe− k (N + 2 −l) δ yl − (N + M + 1/2 − l) e kπ yl + k (N + 2 −l)yσ(l) . (2.11) i

e−

iM π sign(k) 4

NABJ X

(−1)σ

Z

∞

l=N +1

2 Im(ya − yσ(a) ) ≤ 1. Since Note that the integration over xa is convergent only for2 kπ yσ(a) would be −iπ(M − 1/2) depending on the permutation, the integration is always 2

2 2 Im(ya − yσ(a) ) = 1 is understood as a limit from kπ Im(ya − yσ(a) ) < 1. The saturated case kπ

–4–

safe for 2M ≤ |k| + 1. On the other hand, a part of the integrations is divergent for 2M > |k| + 1. However, these divergences must be apparent and cancel out after summing over the permutation since this case is equivalent to the safe case through the Seiberg-like duality [22] between the ABJ theories with gauge groups U (N )k × U (N + M )−k

and U (N + |k| − M )k × U (N )−k ,

(2.12)

which has been proven for the S 3 partition functions [33, 34]. Although we could regularize the divergences, instead we will adopt another way as discussed in sec. 2.2. 2.1 For 2M ≤ |k| + 1 We can continue our computation straightforwardly for this case. Integrating over xa leads us to 2 iM π Z ∞ N +M Y N e− 4 sign(k) NABJ X d y e k M (ya −yσ(a) ) (N,N +M ) σ ZABJ (k) = (−1) M ya −yσ(a) N +M −∞ π · 2 cosh ya N !|k|N + 2 σ a=1 2 cosh k " # NY +M i 2 2 2 iπ 1 1 π πe− k (N + 2 −l) δ yl − (N + M + 1/2 − l) e kπ yl + k (N + 2 −l)yσ(l) . i l=N +1

Noting

P

a (ya

− yσ(a) ) = −

ZABJ =

e−

iM π sign(k) 4

PN +M

− yσ(l) ) and rescaling ya as ya → ya /2, one finds

l=N +1 (yl

NABJ M

Z

∞

N dN +M y Y 1 N +M (2π) 2 cosh y2a

−∞ N !|k|N + 2 a=1 # " NY +M i 2 M 2 y 1 π l − iπ N + −l × πe k ( 2 ) δ − (N + M + 1/2 − l) e 4kπ yl − k yl 2 i l=N +1 ! 1 θN,l (N +M +1/2−l)y j k × det θl,N +1 . (2.13) y −y + e 2 cosh j2k l

If we use the identity (2.5) again and integrate over yN +1 , · · · , yN +M , then we obtain (N,N +M )

ZABJ = Z

(k)

sign(k) − 2 (N 2 +(N +M )2 )

i

∞

−∞

N

M

M

(−1) 2 (N −1)+ 2 (M −1)+N M iN + 2

M (M 2 −1) 12

Y

q M N !2N kN + 2 1≤l

a=1

h

π(l − m) i 2i sin k (2.14)

l=0

Taking account of Y

1≤l

h

2i sin

M M M π(l − m) i 2 = (−1) 2 (M −1) q − 12 (M −1) (1 − q) 2 (M −1) G2 (M + 1; q), k i(xb −xa )

1 − q 2π xa − xb = , tanh i(xb −xa ) 2k 1 + q 2π

M −1 Y l=0

ixa

1

xj + 2πi(l + 1/2) (q 2π + 2 )M = (−1)M , tanh ixa 1 2k (−q 2π + 2 )M

–5–

and making a transformation sa = iya − π, the partition function takes the form of 1

ZABJ =

i− 2 (N

2 +(N +M )2 )sign(k)+N + M 2

1

(−1) 2 N (N −1)

2N kN +M/2 N ! Z

i∞−2πη −i∞−2πη

(1 − q)

sa N (q 2π +1 )M dN s Y 1 s (−2πi)N 2 sin s2a (−q 2πa +1 )M a=1

M (M −1) 2

Y

1≤a

G2 (M + 1; q)

(1 − q

sb −sa 2π

)2

(1 + q

sb −sa 2π

)2

.

(2.15)

Although η is naively η = 1/2, we can change η in a range 0 < η < 1 by the Cauchy integration theorem. If we take η = 0+ for 2M ≤ |k| and η = 21 + 0+ for 2M = |k| + 1, then this is nothing but the AHS formula for 2M ≤ |k| and 2M = |k| + 1 (as a special case of 2M ≥ |k|), respectively. Note that the choice (1.3) of η is still correct for general N . For a later convenience, we introduce the U (M )k pure Chern-Simons partition function (without level shift) on S 3 as [30, 35] (M )

M

= |k|− 2

M2 2

M

M

M

2

M

i 2 k− 2 q − 12 (M −1) (1 − q) 2 (M −1) G2 (M + 1; q) M −1 Y πl M −l . 2 sin |k|

ZCS (k) = i−sign(k)

(2.16)

l=1

In terms of this, the ABJ partition function can be rewritten as N

ZABJ

i−sign(k)N (N −1) (−1) 2 (N −1) M (M 2 −1) (M ) q 12 = ZCS (k) N !2N |k|N Z i∞−2πη N M Y 1 s + 2lπ dN s Y tan sa N 2|k| −i∞−2πη (−2πi) a=1 2 sin 2 l=1

Y

1≤a

(1 − q

sb −sa 2π

)2

(1 + q

sb −sa 2π

)2

. (2.17)

Remark We can also express the ABJ partition function as Q M iM π yj −yl Q Z 2 ya −yb e− 4 sign(k) q 12 (M −1) NABJ ∞ dN +M y a

l=N +1

Each factor in this integrand has an interpretation from the brane picture. Recall that the type IIB brane construction for the U (N )k × U (N + M )−k ABJ theory consists of N circular D3-branes, (1, −k)5-brane, NS5-brane and M D3-branes suspended between the two 5-branes [22]. Taking S-transformation, the (1, −k)5-brane and NS5-brane become (−k, 1)5-brane and D5-brane, respectively. First, note that the second factor N Y

a=1

1 2 cosh y2a

agrees with the contribution from a bi-fundamental hypermultiplet [3]. This multiplet comes from strings ending on the D5-brane and D3-branes. Next, the first factor Q yj −yl Q ya −yb 1 1≤j

–6–

is a bit nontrivial. For M = 0, the authors in [27] argued that this factor comes from a system of the (1, −k)5-brane and D3-branes (see also [28]). Hence we can interpret (2.19) as natural generalization of this contribution. Finally, the last factor NY +M

l=N +1

2π δ yl − (N + M + 1/2 − l) i

reflects a fact that the M suspended D3-branes are locked into position by the two 5-branes. 2.2 For 2M ≥ |k| − 1 The integration (2.11) is apparently divergent for this case. Instead of imposing some regularizations, we use the Seiberg-like duality (2.12) for the ABJ theory [22]. Note that this duality has been already proven for the S 3 partition functions because the duality comes [34] from the Giveon-Kutasov duality [36] proven in [33]. Since the dual ABJ partition (N,N +M ) (k) in terms of function is given by (2.17) for 2M ≥ |k| − 1, we can express ZABJ (N +|k|−M,N ) (−k) through the duality. ZABJ Let us show ZABJ = ZAHS for3 2M ≥ |k| − 1. Via the Seiberg-like duality (2.12) as mathematical identity, the ABJ partition function is given by (N,N +M )

ZABJ

(k) = (−1) e

N+M 2

(N +M −1)+ N+|k|−M (N +|k|−M −1) − 21 (2N 2 −(N +M )2 −(N +|k|−M )2 )sign(k) 2

i

πi (k 2 +6N |k|−6|k|+2)sign(k) 12

(N,N +|k|−M )

ZABJ

(−k).

(2.20)

Here the phase factor has been determined4 in [34]. Plugging (2.17) into this leads us to N

(N,N +M ) (k) ZABJ

i−sign(k)N (N −1) (−1) 2 (N −1) M (M 2 −1) (|k|−M ) = q 12 (−k) ZCS N !2N |k|N Z i∞−0+ |k|−M N Y 1 dN s Y s + 2lπ Y tan N 2 sin s2a 2|k| −i∞−0+ (−2πi) a=1

1≤a

l=1

(|k|−M )

By using the level-rank duality5 for the pure CS theory: ZCS eq. (E.2) in [24]: M M 1 Y 1 Y s + 2πl s + 2πl = tan tan sin 2s 2|k| sin 2s 2|k| |k| l=1

l=1

s→s+2π −

2

(1 − q

sb −sa 2π

)2

(1 + q

sb −sa 2π

)2

.

(M )

(−k) = ZCS (k) and

,

(2.21)

+M ,M →|k|−M

we obtain

1

(N,N +M )

ZABJ

(k) =

i− 2 (N

2 +(N +M )2 )sign(k)+N + M 2

1

(−1) 2 N (N −1)

2N kN +M/2 N !

3

(1 − q)

M (M −1) 2

G2 (M + 1; q)

For 2M = |k| and 2M = |k| + 1, we can also apply the argument in sec. 2.1. We set the condition 2M ≥ |k| − 1 such that the dual ABJ partition function is given by (2.17). 4 Note that our normalization for the partition function differs from the original paper [34] by N +M N (N,N+M ) (N,N+M ) (k). (k) = (−1) 2 (N−1)+ 2 (N+M −1) NABJ ZKWY Zours 5 See e.g. appendix. B of [34] for a proof.

–7–

Z

sa N (q 2π +1 )M dN s Y 1 s (−2πi)N 2 sin s2a (−q 2πa +1 )M a=1

i∞−2πη −i∞−2πη

where η=−

Y

1≤a

(1 − q

sb −sa 2π

)2

(1 + q

sb −sa 2π

)2

|k| + M + 0+ . 2

,(2.22)

(2.23)

For 2M ≥ |k|, this exactly agrees with ZAHS . For 2M = |k| − 1, we can show that making a transformation sa → −sa with a choice η = 0− and using the periodicity of the integrand: sa ∼ sa + 4kπi give the AHS formula. Thus we find again that the choice (1.3) of the integral contour is valid for general N . Remark As already discussed in [24], the ABJ partition function vanishes for M > k since the pure CS partition function in the prefactors vanishes for this case. This manifests an expectation that the supersymmetries are spontaneously broken in this case [22, 37] (see also [38]).

3. Discussion In this paper we have proven that the ABJ partition function on S 3 is exactly the same as the formula (1.2) recently proposed by Awata, Hirano and Shigemori [24], which can be interpreted as the “mirror” description of the ABJ partition function. It has also turned out that the choice (1.3) of the integral contour, previously determined only for N = 1, is still correct for general N . Our proof heavily relied on the determinant identity (2.1) and the following illuminating structure: Z i2 ik P 2 P 2 h ZABJ ∼ dN +M µdN ν hM |b(eµ1 ) · · · b(eµN+M )c(−eνN ) · · · c(−eν1 )|0i e− 4π ( i µi − a νa ) , (3.1) which might be reminiscent of the AGT relation [39]. This would imply that somehow the Boson-Fermion correspondence knows how to simplify the ABJ partition function on S 3 . One can see similar structure also in general N = 3 quiver CSM. It is interesting if we find any physical origin of this structure. A recent work [40] in three dimensions and topological string perspective [41] might provide valuable insights along this direction. For 2M > |k| + 1, we have to change the integral contour. We can also express the partition function for this case as ZABJ |2M >|k|+1 ∼

Z

∞

N +M

d −∞

NY +M

"

l=N +1

y

Q

j

Q

yj −yl 2k

Q

a

ya −yb 2k

# 2π (N + M + 1/2 − l) 2πδ yl − i

N Y

1 2 cosh y2a a=1 .

|k| ya →ya +2πi − 2 +M

The similar representation (2.18) was useful to find the brane interpretation for 2M ≤ |k| + 1. Formally this shift is similar to anomalous R-charge, imaginary mass and voertex

–8–

loop [3]. It would be intriguing to interpret this shift in the integrand from the brane picture. As mentioned in sec. 1, the AHS representation (1.2) is suitable for the Fermi gas approach and Monte Carlo simulation, and easier to study the higher spin limit than the original representation (1.1). Therefore it is illuminating to apply these approaches to the ABJ partition function and investigate wide parameter region. Finally we comment on a relation between the ABJ theory and L(2, 1) matrix model. The authors in [24] obtained (1.2) by an analytic continuation from the partition function of the L(2, 1) matrix model represented by a convergent series. Because its analytic continuation to the ABJ theory yields an ill-defined series, the AHS formula corresponds to its well-defined integral representation and correctly reproduces the formal series order by order in the perturbative expansion. Since here we have proven ZABJ = ZAHS , our argument combined with the one in [24] might give key idea for rigorous proof of the analytic continuation. Acknowledgment We would like to thank Hidetoshi Awata, Shinji Hirano, Keita Nii and Masaki Shigemori for responses to our questions about the work [24] and another related work in progress. We would also like to thank Kazuo Hosomichi, Marcos Mari˜ no, Sanefumi Moriyama and Kazumi Okuyama for valuable discussions and comments. We are grateful to Yukawa Institute for Theoretical Physics for hospitality, where part of this work was done.

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