ABJ fractional brane from ABJM Wilson loop

Abstract

We present a new Fermi gas formalism for the ABJ matrix model. This formulation identifies the effect of the fractional M2-brane in the ABJ matrix model as that of a composite Wilson loop operator in the corresponding ABJM matrix model. Using this formalism, we study the phase part of the ABJ partition function numerically and find a simple expression for it. We further compute a few exact values of the partition function at some coupling constants. Fitting these exact values against the expected form of the grand potential, we can determine the grand potential with exact coefficients. The results at various coupling constants enable us to conjecture an explicit form of the grand potential for general coupling constants. The part of the conjectured grand potential from the perturbative sum, worldsheet instantons and bound states is regarded as a natural generalization of that in the ABJM matrix model, though the membrane instanton part contains a new contribution.

Published for SISSA by Springer

Received: December 6, 2013

Accepted: February 19, 2014

Published: March 17, 2014

Sho Matsumotoa and Sanefumi Moriyamaa,b,c

aGraduate School of Mathematics, Nagoya University,

Nagoya 464-8602, Japan

bKobayashi Maskawa Institute, Nagoya University,

Nagoya 464-8602, Japan

cYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We present a new Fermi gas formalism for the ABJ matrix model. This formulation identies the e ect of the fractional M2-brane in the ABJ matrix model as that of a composite Wilson loop operator in the corresponding ABJM matrix model. Using this formalism, we study the phase part of the ABJ partition function numerically and nd a simple expression for it. We further compute a few exact values of the partition function at some coupling constants. Fitting these exact values against the expected form of the grand potential, we can determine the grand potential with exact coe cients. The results at various coupling constants enable us to conjecture an explicit form of the grand potential for general coupling constants. The part of the conjectured grand potential from the perturbative sum, worldsheet instantons and bound states is regarded as a natural generalization of that in the ABJM matrix model, though the membrane instanton part contains a new contribution.

Keywords: Matrix Models, Wilson, t Hooft and Polyakov loops, Nonperturbative E ects, M-Theory

ArXiv ePrint: 1310.8051

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP03(2014)079

Web End =10.1007/JHEP03(2014)079

ABJ fractional brane from ABJM Wilson loop

JHEP03(2014)079

Contents

1 Introduction 1

2 ABJ fractional brane as ABJM Wilson loop 52.1 Proof of the formula for the partition function 82.2 Proof of the formula for the half-BPS Wilson loop 9

3 Consistency with the previous works 103.1 Perturbative sum 113.2 Worldsheet instanton 113.3 Cancellation mechanism 12

4 Phase factor 134.1 Phase factor 15

5 Grand potential 155.1 Grand potential at certain coupling constants 155.2 Grand potential for general coupling constants 19

6 Discussions 19

A A useful determinantal formula 20

B Expansion of Fredholm determinant 21

1 Introduction

An explicit Lagrangian description of multiple M2-branes [1] has opened up a new window to study M-theory or non-perturbative string theory. It was proposed that N multiple M2-branes on C4/Zk are described by N = 6 supersymmetric Chern-Simons-matter theory

with gauge group U(N) U(N) and levels k and k. Due to supersymmetry, partition

function and vacuum expectation values of BPS Wilson loops in this theory on S3 were reduced to a matrix integration [25], which is called the ABJM matrix model. Here the coupling constant of the matrix model is related to the level k inversely.

The ABJM matrix model has taught us much about M-theory or stringy nonperturbative e ects. Among others, we have learned [6] that it reproduces the N3/2 be

havior of the degrees of freedom when N multiple M2-branes coincide, as predicted from the gravity dual [7]. Also, as we see more carefully below, it was found in [8] that all the divergences in the worldsheet instantons are cancelled exactly by the membrane instantons. This reproduces the lesson we learned in the birth of M-theory or non-perturbative strings:

1

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string theory is not just a theory of strings. It is only after we include non-perturbative branes that string theory becomes safe and sound.

After the pioneering paper [6] which reproduced the leading N3/2 behavior, the main interest in the study of the ABJM matrix model was focused on the perturbative sum [9, 10] and instanton e ects [6, 11]. All of the computations in these papers were done in the t Hooft limit, N with the t Hooft coupling = N/k held xed, though for approach

ing to the M-theory regime with a xed background, we have to take a di erent limit. Namely, we have to consider the limit N with the parameter k characterizing M-

theory background xed [12, 13]. To overcome this problem, in [14] the matrix model was rewritten, using the Cauchy determinant formula, into the partition function of a Fermi gas system with N non-interacting particles, where the Planck scale is identied with the level: ~ = 2k. This expression separates the roles of k from N, which enables us to take the M-theory limit. Note that the M-theory limit probes quite di erent regimes from the t Hooft limit. Especially, using the WKB expansion in the M-theory limit, we can study the k expansion of the membrane instantons systematically.

Using the Fermi gas formalism, we can also compute several exact values of the partition function with nite N at some coupling constants [15, 16]. We can extrapolate these exact values to the large N regime and read o the grand potential [8]. The grand potential reproduces perfectly the worldsheet instanton e ects predicted by its dual topological string theory on local P1

P1 when instanton number is smaller than k/2, though serious discrepancies appear beyond it. Namely, the worldsheet instanton part of the grand potential is divergent at some values of the coupling constant, while the partition function of the matrix model is perfectly nite in the whole region of the coupling constant. By requiring the cancellation of the divergences and the conformance to the nite exact values of the partition function at these coupling constants, we can write down a closed expression for the rst few membrane instantons for general coupling constants [8, 17], which also matches with the WKB expansion. Furthermore, using the exact values, we can study the bound states of the worldsheet instantons and the membrane instantons [18]. We also nd that the instanton e ects consist only of the contributions from the worldsheet instantons, the membrane instantons and their bound states, and no other contributions appear. Finally in [19] we relate the membrane instanton to the quantization of the spectral curve of the matrix model, which is further related to the rened topological strings on local P1

the Nekrasov-Shatashivili limit [2022].

From the exact solvability viewpoints, we could say that the ABJM matrix model belongs to a new class of solvable matrix models besides that of the Gaussian ones and that of the original Chern-Simons ones. As we have seen, this class of matrix models can be rewritten into a statistical mechanical model using the Cauchy determinant formula and contains an interesting structure of pole cancellations between worldsheet instantons and membrane instantons. The ABJM matrix model is the only example satisfying these properties so far.

The most direct generalization of the ABJM theory is the ABJ theory [23] with the inclusion of fractional branes. It was proposed that N = 6 supersymmetric Chern-Simons-

matter theory with gauge group U(N1)U(N2) and the levels k, k describes min(N1, N2)

2

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P1 in

C4/Zk. The partition function and the vacuum expectation values of the BPS Wilson loops in the ABJ theory are also reduced to matrix models. Without loss of generality we can assume M = N2 N1 0 and k 0 for expectation values of hermitian operators. The unitarity constraint requires M

to satisfy 0 M k.

The integration measure of the ABJM matrix model preserves the super gauge group

U(N|N) while that of the ABJ matrix model preserves U(N1|N2) [24, 25]. In the language

of the topological string theory, the ABJM matrix model corresponds to the background geometry local P1

M2-branes with |N1 N2| fractional M2-branes on

P1 with two identical Kahler parameters, while the ABJ matrix model corresponds to a general non-diagonal case. Hence, the ABJ matrix model is a direct generalization also from this group-theoretical or topological string viewpoint.

In this paper we would like to study how the nice structures found in [8, 14, 15, 18, 19] are generalized to the ABJ matrix model. We start our project by presenting a Fermi gas formalism for the ABJ matrix model. Our formalism shares the same density matrix as that of the ABJM matrix model and hence the same spectral problem [26]. The e ects of fractional branes are encoded in a determinant factor which takes almost the same form as that of the half-BPS Wilson loops in the ABJM matrix model [27].

Another interesting Fermi-gas formalism was proposed previously by the authors of [28].1 Compared with their formulation, our formalism has an advantage in the numerical analysis since the density matrix is the same and all the techniques used previously can be applied here directly.

In the formalism of [28], they found that the formula with integration along the real axis is only literally valid for 0 M k/2. For k/2 < M k, additional poles get across

the real axis and we need to deform the integration contour to avoid these poles. Here we nd that the same deformation is necessary in our formalism. Besides, we have pinned down the origin of this deformation in the change of variables in the Fourier transformation.

We believe that our Fermi gas formalism has also cast a new viewpoint to the fractional branes. In string theory, it was known that graviton sometimes pu s up into a higher-dimensional object, which is called giant graviton [30]. In the gauge theory picture, this object is often described as a determinant operator. Our Fermi gas formalism might suggest an interpretation of the fractional branes in the ABJ theory as these kinds of composite objects, though the precise identication needs to be elaborated. Later we will see that the derivation of our Fermi gas formalism relies on a modication of the Frobenius symbol (see gure 1). Since the hook representation has a natural interpretation as fermion excitations, this modication can be regarded as shifting the sea level of the Dirac sea. This observation may be useful for giving a better interpretation of our formula.

Using our new formalism we can embark on studying the instanton e ects. First of all, we compute rst several exact or numerical values of the partition function. From these studies, we nd that the phase part of the partition function has a quite simple expression.

1There were some points in [28] which need justication. This is another motivation for our current proposal. After we nished establishing this new formalism and proceeded to studying the grand potential, we were informed by M. Honda of his interesting work [29].

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The grand potential dened by the partition function after dropping the phase factors

Jk,M() = log

XN=0eN|Zk(N, N + M)| , (1.1)

can be found by tting the coe cients of the expected instanton expressions using these exact values. We have found that they match well with a natural generalization of the expression for the perturbative sum, the worldsheet instantons and the bound states of the worldsheet instantons and the membrane instantons in the ABJM matrix model. However, the membrane instanton part contains a new kind of contribution.

Finally, we conjecture that the large chemical potential expansion of the grand potential is given by

Jk,M() = Ck3 3e + Bk,Me + Ak +

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Xm=1 d(m)k,Me4me /k

+

X=1 (1)M

eb()ke +

ec()k

M2Ck e()k

e2e . (1.2)

Here the perturbative coe cients are

Ck = 22k , Bk,M =

13k +

k24

M2

2k ,

M

2 +

Ak =

16 log

k4 + 2(1)

(3)

82 k2 +

1 3

Z

dx

ekx 1

3x sinh2 x

3x3 +

1 x

, (1.3)

while the worldsheet instanton coe cients are

d(m)k,M =

Xg=0

2 sin 2m kd

2g2, (1.4)

Xd|m

Xd1+d2=d (1)d1m/d()d2m/dngd1,d2 m/d

with ngd1,d2 being the Gopakumar-Vafa invariants of local P1

P1 and = e2iM/k. Aside from the sign factor (1)M, the membrane instanton coe cients are the same as in the

ABJM case [18, 19]

eb()k =

2

Xg=0

Xd|

Xd1+d2=d eik(d1d2)/2d(1)gngd1,d2 (/d)2

(2 sin k/4d)2g sin k/2d ,

eb()k2k , (1.5)

and the bound states are incorporated by

e = + 1

Ck

ec()k = k2

d dk

X=1(1)Ma()ke2. (1.6)

Note that ngd1,d2 in (1.5) is di erent from ngd1,d2 in (1.4). In terms of the rened topological string invariant ngL,gRd1,d2, both of them are given as follows [19]:

ngd1,d2 = ng,0d1,d2, ngd1,d2 =

XgL+gR=g(1)gngL,gRd1,d2. (1.7)

4

It should be noticed that, compared with the ABJM result, our formula (1.2) has a nontrivial term multiplied by e()k, which is related to a()k by

X=1(1)Ma()ke2 =

X=1(1)Me()ke2e . (1.8)

The coe cients a()k and e()k are determined from the quantum mirror map and their explicit form is given in [19]. If we restrict ourselves to the case of integral k, a()k can be read from the following explicit relation between e and :

e =

(1)k/2M2e24F3

1, 1, 32, 32; 2, 2, 2; (1)k/2M16e2

1, 1, 32, 32; 2, 2, 2; 16e4 , for odd k.(1.9)

The organization of this paper is as follows. In the next section, we shall rst present our Fermi gas formalism for the partition function and the vacuum expectation values of the half-BPS Wilson operator. After giving a consistency check for the conjecture in section 3, we shall proceed to the study of exact and numerical values of partition function and large chemical potential expansion of the grand potential using our Fermi gas formalism in sections 4 and 5. Finally we conclude this paper by discussing future problems in section 6. We present two lemmas in the appendices to support the proof of our formalism in section 2.

2 ABJ fractional brane as ABJM Wilson loop

Let us embark on studying the ABJ matrix model, whose partition function is given by

Zk(N1, N2) = (1)

1

2 N1(N11)+ 12 N2(N21)

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, for even k,

+ e44F3

Z

dN1

(2)N1

dN2

(2)N2

N1!N2!

Qi<j 2 sinh ij2

Qa<b 2 sinh ab2

Qi,a 2 cosh ia2

!2eik4 ([summationtext]i 2i[summationtext]a 2a). (2.1)

We shall rst summarize the main results and prove them in this section.

If we dene the grand partition function by

k,M(z) =

XN=0zNZk(N, N + M), (2.2)

it can be expressed in a form very similar to the vacuum expectation values of the half-BPS Wilson loops in the ABJM matrix model [27] (see also [3133]),

k,M(z)

k,0(z) = det HMp,M+q1(z)

1pM1qM , (2.3)

with Hp,q(z) dened by

Hp,q(z) = Ep()

1 + zQ(, ) P (, )

1Eq(). (2.4)

5

Here various quantities

P (, ) = 1

2 cosh 2

, Q(, ) = 1

2 cosh 2

, Ej() = e(j+

1

2 ), (2.5)

are regarded respectively as matrices or vectors with the indices , and multiplication

between them is performed with the measure

Z

d 2 e

ik4 2,

Z

d 2 e

ik4 2, (2.6)

as in [27].

For the vacuum expectation values of the half-BPS Wilson loops in the ABJ matrix model, we can combine the results of the ABJ partition function (2.3) and the ABJM half-BPS Wilson loop [27] in a natural way. As in the ABJM case, the half-BPS Wilson loop in the ABJ matrix model is characterized by the representation of the supergroup U(N1|N2)

whose character is given by the supersymmetric Schur polynomial

s((e1, . . . , eN1 )/(e1, . . . , eN2 )). (2.7)

Here is a partition and we assume that N1+1 N2 (otherwise, s(x/y) = 0). The

vacuum expectation values are dened by inserting this character into the partition function

hsik(N1, N2) =

(1)

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Z

dN1

(2)N1

dN2

(2)N2 s((e1,. . ., eN

1

2 N1(N11)+ 12 N2(N21)

1 )/(e1,. . ., eN2 ))

N1!N2!

Qi<j 2 sinh ij2

Qa<b 2 sinh ab2

Qi,a 2 cosh ia2

!2eik4 ([summationtext]i 2i[summationtext]a 2a). (2.8)

Our analysis shows that the grand partition function dened by

hsiGCk,M(z) =

XN=0zNhsik(N, N + M), (2.9)

is given by

hsiGCk,M(z)

k,0(z) = det

e

Hlp,M+q1(z)

1pM+r 1qM

Hlp,aq(z)

1pM+r 1qr

, (2.10)

where Hp,q(z) is the same as that dened in (2.4) while

eHp,q(z) is dened by

1 + zQ(, ) P (, ) 1Q(, ) Eq(). (2.11)

In (2.10), the arm length aq and the leg length lp are the non-negative integers appearing in the modied Frobenius notations (a1a2 ar|l1l2 lr+M) of the Young diagram . In the ABJM case, the (ordinary) Frobenius notation (a1a2 ar|l1l2 lr) of Young diagram [12 ] = [12 ]T in the partition notation was dened by aq = q q, lp = p p

with r = max{s|s s 0} = max{s|s s 0} and explained carefully in gure 1 of [27].

In the ABJ case, we dene the modied Frobenius notation (a1a2 ar|l1l2 lr+M) by

aq = q q M, lp = p p + M, (2.12)

6

eHp,q(z) = zEp()

with

r = max{s|s s M 0} = max{s|s s + M 0} M. (2.13)

Diagrammatically, the arm length and the leg length are interpreted as the horizontal and vertical box numbers counted from the shifted diagonal line. This is explained further by an example in gure 1.

Our rst observation is the usage of a combination of the Cauchy determinant formula and the Vandermonde determinant formula2

QN1i<j(xi xj)

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QN2a=1(xi + ya)

= (1)N1(N2N1) det

1 x1+y1

1 x1+yN2

QN2a<b(ya yb)

QN1 i=1

... ... ...

1xN1+y1

1 xN1+yN2

yN2N111 yN2N11N2

... ... ...y01 . . . y0N2

. (2.14)

Here on the right hand side, the upper N1 N2 submatrix and the lower (N2 N1) N2

submatrix are given respectively by

1xi + ya

1iN11aN2, yN2N1pa

1pN2N11aN2 . (2.15)

The determinantal formula (2.14) can be proved without di culty by considering the N2 N2 Cauchy determinant and sending the extra N2 N1 pieces of xi to innity.

Here comes the main idea of our computation. Without the extra monomials yN2N1pa,

as emphasized in [14, 27], the partition function can be rewritten into traces of powers of the density matrices. In the study of the ABJM half-BPS Wilson loop [27], the monomials of the Wilson loop insertion play the role of the endpoints in this multiplication of the density matrices. This can be interpreted as follows: the partition function is expressed by closed strings of the density matrix while the Wilson loops are expressed by open strings. This implies that the ABJ partition function, after rewritten by using (2.14), can also be expressed by powers of the density matrices with monomials yN2N1pa in the both

ends, similarly to the case of the ABJM Wilson loop. The only problem is to count the combinatorial factors correctly.

We can also prove this relation by counting the combinatorial factors explicitly. However, it is easier to present the proof by using various determinantal formulas. In the following subsections we shall provide proofs for the results (2.3) and (2.10) in this way. Readers who are not interested in the details of the proofs can accept the results and jump to section 3.

2We are informed by M. Honda that this formula already appeared in [34].

7

(a) (b)

Figure 1. Frobenius notation for the ABJM case (a) and for the ABJ case (b). The same Young diagram [1234567] = [7766421] or [1234567] = [7655442] is expressed as (a1a2a3a4|l1l2l3l4) = (6532|6421) in the ABJM case while (a1a2a3|l1l2l3l4l5l6) = (320|975421) in

the ABJ case (M = 3). It is also convenient to regard the rst three horizontal arrows in (b) as additional arm lengths (1, 2, 3).

2.1 Proof of the formula for the partition function

In this subsection, we shall present a proof for (2.3). Let us plug xi = ei and ya = ea or xi = ei and ya = ea into (2.14). Multiplying these two equations side by side, we nd

(1)

1

2 N1(N11)+ 12 N2(N21)

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!2

Qi<j 2 sinh ij2

Qa<b 2 sinh ab2

Qi,a 2 cosh ia2

(P (i, j))1iN1

1jN2

(EMp(j))1pM 1jN2

= det

(Q(j, i))1iN1

1jN2

(EM+p1(j))1pM 1jN2

det

, (2.16)

where Q, P and E are dened in (2.5). In order to evaluate the integration of the product (2.16) of two N2 N2 determinants, we apply the formula (A.1) with r = 0. Then

we obtain

(1)

1

2 N1(N11)+ 12 N2(N21) 1

N2!

Z

Ya=1 da 2

Qi<j 2 sinh ij2

Qa<b 2 sinh ab2

Qi,a 2 cosh ia2

!2eik4

[summationtext]a 2a

N2

= det

((P Q)(i, j))1i,jN1 ((P EM+q1)(i))1iN1

1qM

((EMp Q)(j))1pM

1jN1 (EMp EM+q1)1p,qM

, (2.17)

8

where the explicit expression for each component in the determinant is given by

(P Q)(, ) = Z

d2 P (, )Q(, )e

ik4 2, (P Eq)() = Z

d2 P (, )Eq()e

ik4 2,

(Ep Q)() =

Z

d2 Ep()Q(, )e

ik4 2, Ep Eq = Z

d2 Ep()Eq()e

ik4 2.

(2.18)

Therefore the grand partition function (2.2) becomes

k,M(z) =

XN=0zN N!

Z

Yi=1eik4 2i di2 det

((P Q)(i, j))NN ((P EM+q1)(i))NM

((EMp Q)(j))MN (EMp EM+q1)MM

N

!,

(2.19)

by appendix B. Using the formula

Det A B

C D

!

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which can be expressed as the Fredholm determinant Det of the form

k,M(z) = Det 1 + zP Q zP E E Q E E

!, (2.20)

= Det A Det(D CA1B), (2.21)

and simplifying the components by

Ep Eq zEp Q 1 + zP Q

1P Eq = Ep

1 + zQ P

1Eq, (2.22)

we nally arrive at (2.3).

2.2 Proof of the formula for the half-BPS Wilson loop

In this subsection we shall present a proof for (2.10). The discussion is parallel to that of the previous subsection. From the formula due to Moens and Van der Jeugt [35], we have

s((e1, . . . , eN1 )/(e1, . . . , eN2 ))

= (1)r det

(P (i, j))1iN1

1jN2 (Eaq(i))1iN11qr

(Elp(j))1pM+r

1jN2 (0)(M+r)r

det

(P (i, j))1iN1

1jN2

(EMp(j))1pM 1jN2

, (2.23)

where (a1a2 ar|l1l2 lM+r) is the modied Frobenius notation of given in (2.12).

Combining this determinantal expression with (2.16), we have

(1)

1

2 N1(N11)+ 12 N2(N21)s((e1, . . . , eN1 )/(e1, . . . , eN2 ))

!2

Qi<j 2 sinh ij2

Qa<b 2 sinh ab2

Qi,a 2 cosh ia2

(2.24)

= (1)r det

(P (i, j))1iN1

1jN2 (Eaq(i))1iN11qr

(Elp(j))1pM+r

1jN2 (0)(M+r)r

det

(Q(j, i))1iN1

1jN2

(EM+p1(j))1pM

1jN2

.

9

Integrating this with the formula (A.1), we see that

(1)

1

2 N1(N11)+ 12 N2(N21) 1

N2!

Z

N2

!2

Ya=1eik4 2a da 2

Qi<j 2 sinh ij2

Qa<b 2 sinh ab2

Qi,a 2 cosh ia2

s((e1, . . . , eN

1 )/(e1, . . . , eN2 )) (2.25)

= (1)r det

((P Q)(i, j))1iN1

1jN1 ((P EM+q1)(i))1iN11qM (Eaq(i))1iN11qr

((Elp Q)(j))1pM+r

1jN1 (Elp EM+q1)1pM+r1qM (0)(M+r)r

.

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Now the denition (2.9) of hsiGCk,M(z) and appendix B give

hsiGCk,M(z) = (1)r

XN=0zN N!

N

Yi=1eik4 2i di 2

det

((P Q)(i, j))NN (P EM+q1(i))NM (Eaq(i))Nr

((Elp Q)(j))(M+r)N (Elp EM+q1)(M+r)M (0)(M+r)r

!

= (1)r Det

1 + zP Q zP E zEa

El Q El E 0

!. (2.26)

Finally, using (2.21) and (2.22), we nd

hsiGCk,M(z) k,0(z)

= (1)rDet

El EEl Q (1+ zP Q)1zP E

El Q (1 + zP Q)1zEa

= Det

El (1 + zQ P )1E

, (2.27)

which is the desired formula (2.10). In the last determinant, the rows are determined by modied legs l1, l2, . . . , lM+r, whereas the columns are determined by (M, . . . , 2, 1)

and modied arms a1, a2, . . . , ar.

3 Consistency with the previous works

In the subsequent sections, we shall use our Fermi gas formalism (2.3) to evaluate several values of the partition function and proceed to conrm our conjecture of the grand potential in (1.2). However, obviously only the values of the partition function at several coupling constants are not enough to x the whole large expansion in (1.2). Hence, before starting our numerical studies, we shall rst pause to study the consistency between our conjecture of the perturbative part and the worldsheet instanton part in (1.2) with the corresponding parts in the t Hooft expansion [6]. After xing the worldsheet instanton contribution, we easily see that it diverges at some coupling constants. As in the case of the ABJM matrix model [8], since the matrix model is nite for any (k, M) satisfying 0 M k (at

least 0 M k/2, as we shall see in the next section), the divergences in the worldsheet

instantons have to be cancelled by the membrane instantons and their bound states. We

10

zEl (1 + zQ P )1Q Ea

shall see that, for this cancellation mechanism to work for d(m)k,M, we need to introduce the phase (1)M for

eb()k and

ec()k in (1.2).33.1 Perturbative sum

The perturbative part of the grand potential in (1.2) implies that the perturbative sum of the partition function reads

Zpertk(N, N + M) = eAkC1/3k Ai[C1/3k(N Bk,M)]. (3.1)

The argument of the Airy function is proportional toN Bk,Mk =

13k2 . (3.2)

It was noted in [6, 36] that the renormalized t Hooft coupling constant

= N

k

124, (3.3) in the ABJM case has to be modied to

= N1 + N22k

(N1 N2)22k2

124, (3.4) in the ABJ case. We have changed Bk,0 into Bk,M to take care of this modication.

3.2 Worldsheet instanton

Let us see the validity of our conjecture on the worldsheet instanton d(m)k,M. First note that the worldsheet instanton can be summarized into a multi-covering formula

JWS() =

Xg=0

JHEP03(2014)079

2g2 (e4k 1)nd1(e4k )nd2n . (3.5)

This naturally corresponds to shifting the two Kahler parameters by 2iM/k.

Next, we shall see that the expression of the worldsheet instanton (1.4) reproduces the genus-0 free energy of the matrix model [6]. As in [8], the rst few worldsheet instanton terms of the free energy Fk,M = log Zk,M with abbreviation Zk,M = Zk(N, N + M) are given by

F WS(1)k,M = ZWS(1)k,M,

F WS(2)k,M = ZWS(2)k,M

1

2(ZWS(1)k,M)2, (3.6)

where the partition functions are

ZWS(1)k,M = d(1)k,M

Ai[C1/3k(N + 4k Bk,M)] Ai[C1/3k(N Bk,M)]

Xn,d1,d2 ngd1,d2

2 sin 2n k

,

Ai[C1/3k(N + 8k Bk,M)]

Ai[C1/3k(N Bk,M)], (3.7)

3The contents of this section are based on a note of Sa.Mo. during the collaboration of [19]. Sa.Mo. is grateful to the collaborators for various discussions.

ZWS(2)k,M =

d(2)k,M + (d(1)k,M)2 2

11

and we have assumed that the worldsheet instantons are given by (1.4),

d(1)k,M =

n0101 + n001 4 sin2 2k

,

d(2)k,M =

n0102 + n0012 8 sin2 4k

+ n0202 + n011 + n00224 sin2 2k

. (3.8)

From the asymptotic form of the Airy function

Ai[z] = e

2

3 z3/2

2z1/4

1 548z3/2 + O(z3) , (3.9)

we nd

Ai[C1/3k(N + 4mk Bk,M)] Ai[C1/3k(N Bk,M)]

JHEP03(2014)079

= e22

1 22m(m 16) k2

p

m k2

+ O(k4)

. (3.10)

Hence, the free energy is given by

F WS(1)k,M = e2

2

g2s 14(n0101 + n001) + O(g0s) ,

F WS(2)k,M = e4

132(n0102 + n0012) 14(n0202 + n011 + n0022)

+ 1

16(n0101 + n001)2x

+ O(g0s)

, (3.11)

2

g2s

with x = 1/(

p2).

After plugging the Gopakumar-Vafa invariants [37, 38],

n010 = n001 = 2, n020 = n002 = 0, n011 = 4, (3.12)

this reproduces the genus-0 free energy

Fg=0 = 432

3

3/2 + 23i

3

M k

3+ const (3.13)

2

+

1

2( + 1)e2

116(2 + 16 + 2) +

x

4 ( + 1)2

e42

+ O(e6

2

),

which was found in subsection 5.3 of [6].

3.3 Cancellation mechanism

In the preceding subsections, we have presented a consistency check with previous studies for the perturbative part and the worldsheet instanton part of our conjecture (1.2). Note that these worldsheet instantons contain divergences at certain coupling constants. (See (3.8).) As in the case of the ABJM matrix model [8], since there should be no divergences in the matrix integration for 0 M k, the divergences have to be cancelled by the

membrane instantons and the bound states. Corresponding to the extra phases from 1

12

in d(m)k,M, we have found that the singularity of the worldsheet instanton (1.4) is cancelled if we introduce the extra sign factor (1)M in the membrane instantons. Namely, we have

checked that the singularity in

d(m)k,Me4me /k + (1)M

eb()

k e +

ec()k

e2e , (3.14)

at k = 2m/ is canceled for several values. The extra sign factor (1)M can also be

understood by the shift of the Kahler parameters in the ABJ matrix model as pointed out below (3.5).

4 Phase factor

After the consistency check of the perturbative sum, the worldsheet instantons and the cancellation mechanism in the previous section, let us start to compute the grand partition function k,M(z) in (2.3). Since the grand partition function k,0(z) of the ABJM matrix model was studied carefully in our previous paper [8], we shall focus on the computation of the components of the matrix (2.4). After expanding in z, we nd

Hm,n(z) =

XN=0(z)NH(N)m,n, (4.1)

where each term H(N)m,n is simply given by a 2N + 1 multiple integration.For N = 0 we easily nd (~ = 2k)

H(0)m,n = Z

dy

~ e

JHEP03(2014)079

i

2

[planckover2pi1][notdef](m+ 12 )ye

i2[planckover2pi1] y2e

2

[planckover2pi1][notdef](n+ 12 )y = e

k e

4 2i

2k (mn)2, (4.2)

while for N 6= 0 we nd

H(N)m,n = Z

dy0

dx1

dy1

~ ~ ~

dxN ~

dyN ~ e

2

[planckover2pi1][notdef](m+ 12 )y0e

i2[planckover2pi1] y20 1

i2[planckover2pi1] x21 1

2 cosh y0x12k

e

2 cosh x1y12k

2

[planckover2pi1][notdef](n+ 12 )yN . (4.3)

e

i2[planckover2pi1] y21

1

2 cosh yN1xN2k

e

i2[planckover2pi1] x2N 1

2 cosh xNyN2k

e

i2[planckover2pi1] y2N e

Introducing the Fourier transformation,

1

2 cosh yi1xi2k

=

Z

dpi

2

eipi(yi1xi)/~

2 cosh pi2

,

1

2 cosh xiyi2k

=

Z

dqi

2

eiqi(xiyi)/~

2 cosh qi2

, (4.4)

and integrating over y1, x1, , yN+1, we nd

H(N)m,n = e

i

k e

4 2i

2k (m+ 12 )2e

2i

2k (n+ 12 )2

Z

dp1dq1

2~

dpNdqN

2~ (4.5)

e

1[planckover2pi1] 2(m+ 12 )p1 1

2 cosh p12

e

i[planckover2pi1] p1q1 1

2 cosh q12

e

i[planckover2pi1] q1p2

e

i[planckover2pi1] pN qN 1

2 cosh qN2

e

1[planckover2pi1] 2(n+ 12 )qN .

13

Using further the formulas

Z

dp1

2 e

1[planckover2pi1] 2(m+ 12 )p1 1

2 cosh p12

e

i[planckover2pi1] p1q1 = 1

2 cosh q1+2i(

m+ 12 )

2k

,

Z

i[planckover2pi1] pi(qiqi1)

dpi

2

e

, (i = 2, 3, , N 1) (4.6)

to carry out the p-integrations, we nally arrive at the expression

H(N)m,n = e

i

2 cosh pi2

= 1

2 cosh qi1qi2k

k e

4 2i

2k (m+ 12 )2e

2i

2k (n+ 12 )2

Z

dq1

dq2

~ ~

dqN

1

~ 2 cosh q1+2i(

m+ 12 )

2k

JHEP03(2014)079

1

2 cosh q12

1

2 cosh q1q22k

1

2 cosh q22

1

2 cosh qN1qN2k

1

2 cosh qN2

e

1k (n+ 12 )qN . (4.7)

As in the case of the Wilson loops, we can express H(N)m,n (N 6= 0) as

H(N)m,n = e

i

k e

4 2i

2k (m+ 12 )2e

2i

2k (n+ 12 )2

Z

dx

1

~ 2 cosh x+2i(

m+ 12 )

2k

1

2 cosh x2

(N1)n(x), (4.8)

where the functions (N)n(x) are dened by

(N)n(x) =

r2 cosh x 2

Z

dy

~ N(x, y)

e

1k (n+ 12 )y

q2 cosh y2, (4.9)

with

(x, y) = 1

p2 cosh x2

1

2 cosh xy2k

1

q2 cosh y2. (4.10)

In (4.9), the multiplication among the density matrices (x, y) is dened with a measure 1/~,

N(x, y) =

Z

dz

~ (x, z) N1(z, y). (4.11)

The functions (N)n(x) can be determined recursively by

(N)n(x) =

r2 cosh x 2

Z

(N1)n(y)

q2 cosh y2, (4.12)

with the initial condition (0)n(x) = e

dy

~ (x, y)

1k (n+ 12 )x.

2k has poles aligning on the imaginary axis. The pole with the smallest positive imaginary part is at x = i k 2 m + 12

Note that, in (4.8), the function 1/ cosh x+2i(

m+ 12 )

for

M in the range 0 M < (k + 1)/2 since m runs from 0 to M 1. Hence, for M in this

range, the relative position between the pole and the real axis is the same as the ABJM

14

case M = 0 and we can trust the formula (4.8) literally. However, for (k + 1)/2 M k

the above pole comes across the real axis and we need to deform the integration contour of (4.8), which is originally along the real axis, to the negative imaginary direction. This phenomenon and the contour prescription rule were already pointed out in [28]. In their work, they proposed this prescription by requiring the continuity at M = (k + 1)/2 and the Seiberg duality. They also checked that this prescription gives the correct values of the partition function (2.1) for small N and k. Our above analysis further pins down the origin of this deformation of the integration contour. The deformation comes from changing the integration variables from (4.3) to (4.8). For simplicity, hereafter, we shall often refer to the validity range as 0 M k/2 instead of 0 M < (k + 1)/2.

4.1 Phase factor

Unlike the case of the Wilson loops, the complex phase factor looks very non-trivial and needs to be studied separately. Using our Fermi gas formalism (2.3), we have found from numerical studies that the phase factor is given by a rather simple formula:

1

2 arg Zk(N, N + M) =

18M(M 2) +

1 4MN

112k (M3 M). (4.13)

We have checked this formula numerically for N = 0, 1, 2, 3. The results are depicted in gure 2. As noted in the above paragraph, our numerical studies are valid not only for 0 M k/2 but also slightly beyond k/2; 0 < M < (k + 1)/2. In fact, we believe that

our phase formula (4.13) is valid for the whole region of 0 M k because we can show

that this phase reproduces a phase factor appearing in the Seiberg duality

1

2 arg

Zk(N, N + M)

[Zk(N, N + k M)]

JHEP03(2014)079

4 (4.14)

as was conjectured in [39] and further interpreted as a contact term anomaly in [40].

5 Grand potential

After studying the phase factor of the partition function in the previous section, let us turn to their absolute values and study the grand potential dened by these absolute values (1.1).

5.1 Grand potential at certain coupling constants

As was found in [8, 15, 16] the computation of the ABJM partition functions becomes particularly simple for k = 1, 2, 3, 4, 6. Also, as we have seen in section 4, the formula (4.8) with integration along the real axis is literally valid only for 0 M k/2. Hence, we can

compute various values of the partition function for

(k, M) = (2, 1), (3, 1), (4, 1), (6, 1), (4, 2), (6, 2), (6, 3). (5.1)

The results of their absolute values are summarized in gure 3.4 As discussed in [39], the case of k/2 M k is related to that of 0 M k/2 by the Seiberg duality.

4Some of the values were already found in [41]. Comparing our results with theirs is a very helpful check of our formalism. We are grateful to M. Shigemori for sharing his unpublished notes with us.

15

= k2

24 +

112 +

k(N 1)

0.4

0.4

0.2

0.2

2 4 6 8 10

0.0

0.0

[Minus]0.2

[Minus]0.2

2 4 6 8 10

[Minus]0.4

[Minus]0.4

(a) N = 0 (b) N = 1

JHEP03(2014)079

0.4

2 4 6 8 10

0.2

0.2

0.0

0.0

[Minus]0.2

[Minus]0.2

[Minus]0.4

0 1

2 3 4 5 6 7

(c) N = 2 (d) N = 3

Figure 2. Numerical studies of the phase factor of the partition function. The horizontal axis denotes k while the vertical axis shows the phase normalized by 2. Numerical data are depicted by points and our expectations (4.13) mod 1 are expressed by curves. Each picture corresponds to di erent values of N and each curve in the picture starting from k = 2M 1 corresponds di erent

values of M.

Let us consider the grand potential dened with the absolute values of the partition function (1.1). Our strategy to determine the grand potential from the partition function is exactly the same as that of [8] and we shall explain only the key points here. Since the grand potential with the sum truncated at nite N always contains some errors, it is known that tting with the partition function itself gives a result with better accuracy. First we can compare the values found in gure 3 with the perturbative sum (3.1). This already shows a good concordance. For the m-th instanton e ects, after subtracting the perturbative sum and the major instanton e ects, we t the partition function against the linear combinations of

(N)nC1/3keAkAih

C1/3k

N + 4mk Bk,M

i

. (5.2)

Finally we reinterpret the result in terms of the grand potential. Our results are summarized in gure 4.

Compared with our study in [8, 18] we have much smaller number of exact values of the partition function. The lack of data causes quite signicant numerical errors (about 1%). Nevertheless, since we have already known the rough structure of the instanton expansion, we can nd the exact instanton coe cient without di culty.

16

|Z2(0, 1)| =

12, |Z2(1, 2)| =

142 , |Z2(2, 3)| =

2 8 12822 ,

|Z2(3, 4)| =

52 48460823 , |Z2(4, 5)| =

814 8482 + 480

29491224 ,

2 3

12 , |Z3(2, 3)| =

|Z3(0, 1)| =

13, |Z3(1, 2)| =

(93 14) + 33

432 ,

|Z3(3, 4)| =

14 18 153

1728 ,

|Z4(0, 1)| =

1

2, |Z4(1, 2)| =

2

32 ,

JHEP03(2014)079

|Z4(2, 3)| = 0.00003473909952494269119117566353230112859310233773233

7261807934218890234955828380992634025931149937612,

|Z6(0, 1)| =

16, |Z6(1, 2)| =

33

1082 ,

|Z6(2, 3)| = 3.76773027707758200049183186585155883429506373384028699

96374213997516824024006754651401031813928511 106,

|Z6(3, 4)| = 5.26914099452731795482041046853051131744637477848566664

22916096253100787064300949345207528685791 1010,

|Z4(0, 2)| =

1

22, |Z4(1, 3)| =

4

322 ,

|Z4(2, 4)| = 0.00001506227428345380302357520499270222421841701033492

362553063511451195968480813607610027807404966983,

|Z6(0, 2)| =

16, |Z6(1, 3)| =

7 123

432 ,

|Z6(2, 4)| = 4.77900663573206185466590506879892353173666149000261702

495431896753514231026609667127826160173459 107,

|Z6(0, 3)| =

162, |Z6(1, 4)| =

452 86

1296 ,

|Z6(2, 5)| = 2.34333487780752843368477720747976341731283580616750538

345879256373591282194222350629426352014176 107.

Figure 3. Some exact or numerical values of partition functions.

Note that the instanton coe cients of (k, M) = (k, k/2) are similar to those of (k, M) = (k, 0) for even k and those of (k, M) = (6, 1), (6, 2) are similar to those of (k, M) = (3, 1). Due to this similarity, we have to confess that we only really t the values of the partition function for (k, M) = (3, 1) and (k, M) = (4, 1) up to seven instantons. For other cases, after tting for about three instantons, the patterns become clear and we can bring the results from the known ones and simply conrm the validity.

17

Jnpk=2,M=1 =

42 + 2 + 1 2

e2 +

522 + + 9/422 + 2

e4

+

7362 304/3 + 154/932 + 32

e6

+

27012 13949/24 + 11291/1922 + 466

e8

+

1618242 634244/15 + 285253/7552 + 6720

e10

JHEP03(2014)079

+

12274402 5373044/15 + 631257/2032 + 292064 3

e12 + O(e14),

Jnpk=3,M=1 =

2

3e4/3 e8/3 +

42 + + 1/4

32

34 e4 + 2518e16/3 +68 15e20/3

+

522 + /2 + 9/1662 +296 9

e8 1894189 e28/3 + O(e32/3),

Jnpk=4,M=1 =

42 + 2 + 122 2

e2 +

522 + + 9/442 + 18

e4

+

7362 304/3 + 154/962 608 3

e6 + O(e8),

Jnpk=6,M=1 =

2

3e2/3 e4/3 +

42 + 2 + 1

32 +

34 e2 + 2518e8/3 68 15e10/3

+

522 + + 9/462 +296 9

e4 + 1894189 e14/3 + O(e16/3),

Jnpk=4,M=2 = e +

42 + 2 + 1

22

e2 163 e3 + 522 + + 9/442 + 2

e4

256

5 e5 +

7362 304/3 + 154/9

62 + 32

e6 40967 e7 + O(e8),

Jnpk=6,M=2 =

2

3e

2

3

e

43 +

42 + 2 + 132 34 9

e2 + 2518e83 + 68 15e103

+

522 + + 9/462 +296 9

e4 1894189 e143 + O(e16/3),

Jnpk=6,M=3 =

4 3e

2

3

2e

43 +

42 + 2 + 132 20 9

e2 889 e83 1085 e103

+

e4 25208189 e143 + O(e16/3).

Figure 4. Grand potential obtained by tting the exact or numerical values of partition function.

18

522 + + 9/462 298 9

5.2 Grand potential for general coupling constants

Now let us compare the grand potential in gure 4 with a natural generalization of our instanton expansion in the ABJM matrix model. We rst observe a good match for the m-th pure worldsheet instanton e ects for m < k/2. Secondly, we nd that we have to modify signs by the factor (1)M for the functions a()k,

eb()k,

ec()k characterizing the membrane instantons. This is important not only for ensuring the cancellation of the divergences as we noted in subsection 3.3, but also for reproducing the correct coe cients of 2. Thirdly, we conrm that the prescription of introducing the sign factor (1)M reproduces correctly

the bound states, where there are no pure membrane instanton e ects.

As for the constant term in the membrane instanton, there is an ambiguity as long as it does not raise any singularities. There are two candidates for it: one is of course to take exactly the same constant term as in the ABJM case when expressed in terms of the chemical potential . Another choice is to dene

ec()k by respecting the derivative relation. Namely, in the ABJM matrix model it was observed that, when the grand potential Jk()

is expressed in terms of the e ective chemical potential e , the constant term is the derivative of the linear term (1.5). These two choices give di erent answers because of the change in Bk,M. Comparing these two candidates with our numerical results in gure 4, we have found that neither of them gives the correct answer. Instead, the di erence with the latter one is always k/M times bigger than the former one. From this observation, we can write down a closed form for our conjecture in (1.2). We have checked this conjecture up to seven worldsheet instantons and four membrane instantons.

Although we restrict our analysis to the case 0 M k/2, we believe our nal

conjecture (1.2) is valid for the whole region of 0 M k because of the consistency with

the Seiberg duality. Though the expression (1.2) does not look symmetric in the exchange between M and k M, if we pick up a pair of integers whose sum is k, we nd two identical

instanton expansion series after cancelling the divergences.5 We have checked this fact for all the pairs whose sums are k = 1, 2, 3, 4, 6.

6 Discussions

In this paper we have proposed a Fermi gas formalism for the partition function and the half-BPS Wilson loop expectation values in the ABJ matrix models. Our formalism identies the fractional branes in the ABJ theory as a certain type of Wilson loops in the ABJM theory. Hence, our formalism shares the same density matrix as that of the ABJM matrix model, which is suitable for the numerical studies. We have continued to study the exact or numerical values of the partition function using this formalism. Based on these values, we can determine the instanton expansion of the grand potential at some coupling constants k = 2, 3, 4, 6 and conjecture the expression (1.2) for general coupling constants.

Let us raise several points which need further clarications.

The rst one is the phase factor of our conjecture. As we have seen in gure 2, we have checked this conjecture for N = 0, 1, 2, 3 carefully. However when N 3 the numerical

5We are grateful to S. Hirano, K. Okuyama, M. Shigemori for valuable comments on it.

19

JHEP03(2014)079

errors become signicant and it is di cult to continue the numerical studies with high accuracy for large k. It is desirable to study it more extensively.

The second one is the relation to the formalism of [28], which looks very di erent from ours. As pointed out very recently in [29] it was possible to rewrite the formalism of [28] into a mirror expression where the physical interpretation becomes clearer. We would like to see the exact relation between theirs and ours.

Thirdly, we have found an extra term in (1.2) proportional to the quantum mirror map e()k [19]. We have very few data to identify its appearance and it would be great to check it also from the WKB expansion [14, 17], though we are not sure whether the restriction 0 M k/2 gives any di culty in the WKB analysis. Furthermore, we cannot identify

its origin in the rened topological strings or the triple sine functions as proposed in [19]. We hope to see its origin in these theories. It may be a key to understand the gravitational interpretation [42] of the membrane instantons.

The fourth one is about the Wilson loop in the ABJ theory. After seeing that there are only new terms appearing in the membrane instantons, we expect that the instanton expansion of the vacuum expectation values of the Wilson loop should be expressed similarly as that in the ABJM case [27]. However, we have not done any numerical studies to support it. Also, it is interesting to see how our study is related to other recent works on the ABJ Wilson loops [4345].

Finally, one of the motivation to study the ABJ matrix model is its relation to the higher spin models. Since we have written down the grand potential explicitly, it is possible to take the limit proposed in [46]. We would like to see what lessons can be learned for the higher spin models.

Acknowledgments

We are grateful to Jaemo Park and Masaki Shigemori for very interesting communications and for sharing their private notes with us. Sa.Mo. is also grateful to H. Fuji, H. Hata,Y. Hatsuda, S. Hirano, M. Honda, M. Marino and K. Okuyama for valuable discussions since the collaborations with them. The work of Sh.Ma. was supported by JSPS Grant-in-Aid for Young Scientists (B) 25800062.

A A useful determinantal formula

Lemma A.1. Let (i)1in+r and (j)1jn be functions on a measurable space and let (iq)1in+r

1qr be an array of constants. Then we have

1 n!

Z

n

Yk=1dxk det

JHEP03(2014)079

(i(xk))1in+r1kn (iq)1in+r1qr det(j(xk))1j,kn

= det

(mij)1in+r1jn (iq)1in+r1qr , (A.1)

with mij =

R

dxi(x)j(x).

20

Proof. Expand two determinants on the left hand side with respect to columns:

1 n!

Z

n

Yk=1dxk det

(i(xk))1in+r1kn (iq)1in+r1qr det(j(xk))1j,kn

= 1

n!

Z

n

Yk=1 dxk

XSn+r sgn()

n

Yk=1 (k)(xk)

r

Yq=1 (n+q),q

XSn sgn()

n

Yk=1 (k)(xk)

= 1

n!

XSn+r sgn()

r

Yq=1(n+q),q

XSn sgn()

n

Yk=1

Z

dx(k)(x)(k)(x)

= 1

n!

XSn sgn()

JHEP03(2014)079

XSn+r sgn()

r

Yq=1(n+q),q

n

Yk=1 m(k),(k)

(mi,(j))1in+r1jn (iq)1in+r1qr . (A.2)

It follows from the alternating property for determinants that this equals to

1 n!

XSn det

(mi,j)1in+r1jn (iq)1in+r1qr = det (mi,j)1in+r1jn (iq)1in+r1qr . (A.3)

= 1

n!

XSnsgn() det

B Expansion of Fredholm determinant

Although we have used an innite-dimensional version, we shall give a nite-dimensional version of the identity below. For a positive integer n, we let [n] = {1, 2, . . . , n}.

Lemma B.1. Let N, L be non-negative integers. Let A = (aij), B = (biq), C = (cpj), and

D = (dpq) be matrices of nite sizes N N, N L, L N, and L L, respectively. Let

1NL be the (N + L) (N + L) diagonal matrix whose the rst N diagonal entries are 1

and other entries are 0. Then the following identity holds.

det 1NL+ A B

C D

!!

=

N

Xn=01 n!

N

Xk1,...,kn=1det (aki,kj)1i,jn (bki,q)1in,1qL

(cp,kj)1pL,1jn D

!. (B.1)

Proof. Put A = (aij)1i,jN+L = A B

C D

!. Expanding the determinant with respect to

rows, we have

det(1NL + A) =

XSN+L sgn()

N

Yi=1(i,(i) + ai,(i))

L

Yp=1aN+p,(N+p). (B.2)

Divide the product for i: for each SN+L,

N

Yi=1(i,(i) + ai,(i)) = XI[N]

YiIai,(i)

Yi[N]\Ii,(i). (B.3)

21

Here the product

Qi[N]\I i,(i) vanishes unless (i) = i for all i [N] \ I, i.e., unless the support supp() of is a subset of I {N + 1, . . . , N + L}. In that case, the permutation

can be seen as a permutation on I {N + 1, . . . , N + L}. Denoting by SI{N+1,...,N+L}

the permutation group consisting of such permutations,

det(1NL + A) =

XI[N] XSI[{N+1,...,N+L} sgn()

!. (B.4)

It is immediate to see that this identity presents the desired identity.

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References

[1] O. Aharony, O. Bergman, D.L. Ja eris and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, http://dx.doi.org/10.1088/1126-6708/2008/10/091

Web End =JHEP 10 (2008) 091 [arXiv:0806.1218] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0806.1218

Web End =INSPIRE ].

[2] V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, http://dx.doi.org/10.1007/s00220-012-1485-0

Web End =Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0712.2824

Web End =INSPIRE ].

[3] A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, http://dx.doi.org/10.1007/JHEP03(2010)089

Web End =JHEP 03 (2010) 089 [arXiv:0909.4559] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0909.4559

Web End =INSPIRE ].

[4] D.L. Ja eris, The exact superconformal R-symmetry extremizes Z, http://dx.doi.org/10.1007/JHEP05(2012)159

Web End =JHEP 05 (2012) 159 [arXiv:1012.3210] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3210

Web End =INSPIRE ].

[5] N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, http://dx.doi.org/10.1007/JHEP03(2011)127

Web End =JHEP 03 (2011) 127 [arXiv:1012.3512] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3512

Web End =INSPIRE ].

[6] N. Drukker, M. Mario and P. Putrov, From weak to strong coupling in ABJM theory, http://dx.doi.org/10.1007/s00220-011-1253-6

Web End =Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.3837

Web End =INSPIRE ].

[7] I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, http://dx.doi.org/10.1016/0550-3213(96)00295-7

Web End =Nucl. Phys. B 475 (1996) 164 [http://arxiv.org/abs/hep-th/9604089

Web End =hep-th/9604089 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9604089

Web End =INSPIRE ].

[8] Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton e ects in ABJM theory from Fermi gas approach, http://dx.doi.org/10.1007/JHEP01(2013)158

Web End =JHEP 01 (2013) 158 [arXiv:1211.1251] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.1251

Web End =INSPIRE ].

[9] H. Fuji, S. Hirano and S. Moriyama, Summing up all genus free energy of ABJM matrix model, http://dx.doi.org/10.1007/JHEP08(2011)001

Web End =JHEP 08 (2011) 001 [arXiv:1106.4631] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1106.4631

Web End =INSPIRE ].

[10] M. Hanada et al., Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant, http://dx.doi.org/10.1007/JHEP05(2012)121

Web End =JHEP 05 (2012) 121 [arXiv:1202.5300] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1202.5300

Web End =INSPIRE ].

[11] N. Drukker, M. Mario and P. Putrov, Nonperturbative aspects of ABJM theory, http://dx.doi.org/10.1007/JHEP11(2011)141

Web End =JHEP 11 (2011) 141 [arXiv:1103.4844] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.4844

Web End =INSPIRE ].

[12] C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-Matrix Models and Tri-Sasaki Einstein Spaces, http://dx.doi.org/10.1103/PhysRevD.83.046001

Web End =Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1011.5487

Web End =INSPIRE ].

22

YiIai,(i)

L

Yp=1 aN+p,(N+p)

=

XI[N]det (ai,j)i,jI (ai,N+q)iI,q[L]

(aN+p,j)p[L],jI (aN+p,N+q)p,q[L]

JHEP03(2014)079

[13] K. Okuyama, A Note on the Partition Function of ABJM theory on S3,

http://dx.doi.org/10.1143/PTP.127.229

Web End =Prog. Theor. Phys. 127 (2012) 229 [arXiv:1110.3555] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1110.3555

Web End =INSPIRE ].

[14] M. Mario and P. Putrov, ABJM theory as a Fermi gas, http://dx.doi.org/10.1088/1742-5468/2012/03/P03001

Web End =J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1110.4066

Web End =INSPIRE ].

[15] Y. Hatsuda, S. Moriyama and K. Okuyama, Exact results on the ABJM Fermi gas, http://dx.doi.org/10.1007/JHEP10(2012)020

Web End =JHEP 10 (2012) 020 [arXiv:1207.4283] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.4283

Web End =INSPIRE ].

[16] P. Putrov and M. Yamazaki, Exact ABJM Partition Function from TBA, http://dx.doi.org/10.1142/S0217732312502008

Web End =Mod. Phys. Lett. A 27 (2012) 1250200 [arXiv:1207.5066] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.5066

Web End =INSPIRE ].

[17] F. Calvo and M. Mario, Membrane instantons from a semiclassical TBA, http://dx.doi.org/10.1007/JHEP05(2013)006

Web End =JHEP 05 (2013) 006 [arXiv:1212.5118] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1212.5118

Web End =INSPIRE ].

[18] Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton bound states in ABJM theory, http://dx.doi.org/10.1007/JHEP05(2013)054

Web End =JHEP 05 (2013) 054 [arXiv:1301.5184] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1301.5184

Web End =INSPIRE ].

[19] Y. Hatsuda, M. Mario, S. Moriyama and K. Okuyama, Non-perturbative e ects and the rened topological string, arXiv:1306.1734 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.1734

Web End =INSPIRE ].

[20] A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, http://dx.doi.org/10.1007/JHEP04(2010)040

Web End =JHEP 04 (2010) 040 [arXiv:0910.5670] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0910.5670

Web End =INSPIRE ].

[21] A. Mironov and A. Morozov, Nekrasov Functions from Exact BS Periods: The Case of SU(N), http://dx.doi.org/10.1088/1751-8113/43/19/195401

Web End =J. Phys. A 43 (2010) 195401 [arXiv:0911.2396] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.2396

Web End =INSPIRE ].

[22] M. Aganagic, M.C. Cheng, R. Dijkgraaf, D. Kre and C. Vafa, Quantum geometry of rened topological strings, http://dx.doi.org/10.1007/JHEP11(2012)019

Web End =JHEP 11 (2012) 019 [arXiv:1105.0630] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1105.0630

Web End =INSPIRE ].

[23] O. Aharony, O. Bergman and D.L. Ja eris, Fractional M2-branes, http://dx.doi.org/10.1088/1126-6708/2008/11/043

Web End =JHEP 11 (2008) 043 [arXiv:0807.4924] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.4924

Web End =INSPIRE ].

[24] N. Drukker and D. Trancanelli, A supermatrix model for N = 6 super Chern-Simons-matter theory, http://dx.doi.org/10.1007/JHEP02(2010)058

Web End =JHEP 02 (2010) 058 [arXiv:0912.3006] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0912.3006

Web End =INSPIRE ].

[25] M. Mario and P. Putrov, Exact results in ABJM theory from topological strings, http://dx.doi.org/10.1007/JHEP06(2010)011

Web End =JHEP 06 (2010) 011 [arXiv:0912.3074] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0912.3074

Web End =INSPIRE ].

[26] J. Kallen and M. Mario, Instanton e ects and quantum spectral curves, arXiv:1308.6485 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1308.6485

Web End =INSPIRE ].

[27] Y. Hatsuda, M. Honda, S. Moriyama and K. Okuyama, ABJM Wilson loops in arbitrary representations, http://dx.doi.org/10.1007/JHEP10(2013)168

Web End =JHEP 10 (2013) 168 [arXiv:1306.4297] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.4297

Web End =INSPIRE ].

[28] H. Awata, S. Hirano and M. Shigemori, The partition function of ABJ theory, http://dx.doi.org/10.1093/ptep/ptt014

Web End =Prog. Theor. Exp. Phys. (2013) 053B04 [arXiv:1212.2966] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1212.2966

Web End =INSPIRE ].

[29] M. Honda, Direct derivation of mirror ABJ partition function, http://dx.doi.org/10.1007/JHEP12(2013)046

Web End =JHEP 12 (2013) 046 [arXiv:1310.3126] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1310.3126

Web End =INSPIRE ].

[30] J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from Anti-de Sitter space, http://dx.doi.org/10.1088/1126-6708/2000/06/008

Web End =JHEP 06 (2000) 008 [http://arxiv.org/abs/hep-th/0003075

Web End =hep-th/0003075 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0003075

Web End =INSPIRE ].

[31] A. Borodin, G. Olshanski and E. Strahov, Giambelli compatible point processes, Adv. Appl. Math. 37 (2006) 209 [http://arxiv.org/abs/math-ph/0505021

Web End =math-ph/0505021 ].

[32] A. Klemm, M. Mario, M. Schiereck and M. Soroush, ABJM Wilson loops in the Fermi gas approach, arXiv:1207.0611 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.0611

Web End =INSPIRE ].

23

JHEP03(2014)079

[33] A. Grassi, J. Kallen and M. Mario, The topological open string wavefunction, arXiv:1304.6097 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1304.6097

Web End =INSPIRE ].

[34] E.L. Basor and P.J. Forrester, Formulas for the evaluation of Toeplitz determinants with rational generating functions, http://dx.doi.org/10.1002/mana.19941700102

Web End =Math. Nachr. 170 (1994) 5 .

[35] E.M. Moens and J. Van der Jeugt, A determinantal formula for supersymmetric Schur polynomials, http://dx.doi.org/10.1023/A:1025048821756

Web End =J. Algebr. Combinator. 17 (2003) 283 .

[36] O. Aharony, A. Hashimoto, S. Hirano and P. Ouyang, D-brane charges in gravitational duals of 2 + 1 dimensional gauge theories and duality cascades, http://dx.doi.org/10.1007/JHEP01(2010)072

Web End =JHEP 01 (2010) 072 [arXiv:0906.2390] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0906.2390

Web End =INSPIRE ].

[37] R. Gopakumar and C. Vafa, M theory and topological strings. 1., http://arxiv.org/abs/hep-th/9809187

Web End =hep-th/9809187 [http://inspirehep.net/search?p=find+EPRINT+hep-th/9809187

Web End =INSPIRE ].

[38] M. Aganagic, M. Mario and C. Vafa, All loop topological string amplitudes from Chern-Simons theory, http://dx.doi.org/10.1007/s00220-004-1067-x

Web End =Commun. Math. Phys. 247 (2004) 467 [http://arxiv.org/abs/hep-th/0206164

Web End =hep-th/0206164 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0206164

Web End =INSPIRE ].

[39] A. Kapustin, B. Willett and I. Yaakov, Tests of Seiberg-like Duality in Three Dimensions, arXiv:1012.4021 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.4021

Web End =INSPIRE ].

[40] C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons contact terms in three dimensions, http://dx.doi.org/10.1007/JHEP09(2012)091

Web End =JHEP 09 (2012) 091 [arXiv:1206.5218] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1206.5218

Web End =INSPIRE ].

[41] M. Shigemori, unpublished notes.

[42] S. Bhattacharyya, A. Grassi, M. Mario and A. Sen, A One-Loop Test of Quantum Supergravity, http://dx.doi.org/10.1088/0264-9381/31/1/015012

Web End =Class. Quant. Grav. 31 (2014) 015012 [arXiv:1210.6057] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1210.6057

Web End =INSPIRE ].

[43] V. Cardinali, L. Griguolo, G. Martelloni and D. Seminara, New supersymmetric Wilson loops in ABJ(M) theories, http://dx.doi.org/10.1016/j.physletb.2012.10.051

Web End =Phys. Lett. B 718 (2012) 615 [arXiv:1209.4032] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1209.4032

Web End =INSPIRE ].

[44] M. Bianchi, G. Giribet, M. Leoni and S. Penati, The 1/2 BPS Wilson loop in ABJ(M) at two loops: the details, http://dx.doi.org/10.1007/JHEP10(2013)085

Web End =JHEP 10 (2013) 085 [arXiv:1307.0786] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.0786

Web End =INSPIRE ].

[45] L. Griguolo, G. Martelloni, M. Poggi and D. Seminara, Perturbative evaluation of circular 1/2 BPS Wilson loops in N = 6 Super Chern-Simons theories, http://dx.doi.org/10.1007/JHEP09(2013)157

Web End =JHEP 09 (2013) 157

[arXiv:1307.0787] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.0787

Web End =INSPIRE ].

[46] C.-M. Chang, S. Minwalla, T. Sharma and X. Yin, ABJ Triality: from Higher Spin Fields to Strings, http://dx.doi.org/10.1088/1751-8113/46/21/214009

Web End =J. Phys. A 46 (2013) 214009 [arXiv:1207.4485] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.4485

Web End =INSPIRE ].

24

JHEP03(2014)079