# Exact Results in ABJM Theory from Topological Strings

**ABSTRACT** Recently, Kapustin, Willett and Yaakov have found, by using localization techniques, that vacuum expectation values of Wilson loops in ABJM theory can be calculated with a matrix model. We show that this matrix model is closely related to Chern-Simons theory on a lens space with a gauge supergroup. This theory has a topological string large N dual, and this makes possible to solve the matrix model exactly in the large N expansion. In particular, we find the exact expression for the vacuum expectation value of a 1/6 BPS Wilson loop in the ABJM theory, as a function of the 't Hooft parameters, and in the planar limit. This expression gives an exact interpolating function between the weak and the strong coupling regimes. The behavior at strong coupling is in precise agreement with the prediction of the AdS string dual. We also give explicit results for the 1/2 BPS Wilson loop recently constructed by Drukker and Trancanelli Comment: 18 pages, two figures, small misprints corrected and references added, final version to appear in JHEP

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**ABSTRACT:**We study U(N|M) character expectation value with the supermatrix Chern-Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determinantal formula, which gives U(N|M) character expectation values in terms of U(1|1) averages for a particular type of character representations. This means that the U(1|1) character expectation value is a building block for all the U(N|M) averages, and in particular, by an appropriate limit, for the U(N) invariants. In addition to the original model, we introduce another supermatrix model obtained through the symplectic transform, which is motivated by the torus knot Chern-Simons matrix model. We obtain the Rosso-Jones-type formula and the spectral curve for this case.07/2014; - SourceAvailable from: Marcos Marino[Show abstract] [Hide abstract]

**ABSTRACT:**We show that, in ABJ(M) theories with N=8 supersymmetry, the non-perturbative sector of the partition function on the three-sphere simplifies drastically. Due to this simplification, we are able to write closed form expressions for the grand potential of these theories, which determines the full large N asymptotics. Moreover, we find explicit formulae for the generating functionals of their partition functions, for all values of the rank N of the gauge group. They involve Jacobi theta functions on the spectral curve associated to the planar limit. We also show that the partition function, as a function of N, can be extended in a natural way to an entire function on the full complex plane, and we explore some possible consequences of this fact for the quantum geometry of M-theory and for putative de Sitter extensions.09/2014; - SourceAvailable from: Valentina Forini[Show abstract] [Hide abstract]

**ABSTRACT:**We compute the null cusp anomalous dimension of ABJM theory at strong coupling up to two-loop order. This is done by evaluating corrections to the corresponding superstring partition function, weighted by the $AdS_4\times \mathbb{CP}^3$ action in AdS light-cone gauge. We compare our result, where we use an anomalous shift in the $AdS_4$ radius, with the cusp anomaly of $\mathcal{N}=4$ SYM, and extract the two-loop contribution to the non-trivial integrable coupling $h(\lambda)$ of ABJM theory. It coincides with the strong coupling expansion of the exact expression for $ h(\lambda)$ recently conjectured by Gromov and Sizov. Our work provides thus a non-trivial perturbative check for the latter, as well as evidence for two-loop UV-finiteness and quantum integrability of the Type IIA $AdS_4\times \mathbb{CP}^3$ superstring in this gauge.07/2014;

Page 1

Preprint typeset in JHEP style - PAPER VERSION

Exact results in ABJM theory from topological strings

Marcos Mari˜ noa,band Pavel Putrovb

aD´ epartement de Physique Th´ eorique etbSection de Math´ ematiques,

Universit´ e de Gen` eve, Gen` eve, CH-1211 Switzerland

marcos.marino@unige.ch,

pavel.putrov@unige.ch

Abstract: Recently, Kapustin, Willett and Yaakov have found, by using localization techniques,

that vacuum expectation values of Wilson loops in ABJM theory can be calculated with a matrix

model. We show that this matrix model is closely related to Chern–Simons theory on a lens

space with a gauge supergroup. This theory has a topological string large N dual, and this

makes possible to solve the matrix model exactly in the large N expansion. In particular, we find

the exact expression for the vacuum expectation value of a 1/6 BPS Wilson loop in the ABJM

theory, as a function of the ’t Hooft parameters, and in the planar limit. This expression gives an

exact interpolating function between the weak and the strong coupling regimes. The behavior at

strong coupling is in precise agreement with the prediction of the AdS string dual. We also give

explicit results for the 1/2 BPS Wilson loop recently constructed by Drukker and Trancanelli.

arXiv:0912.3074v4 [hep-th] 12 May 2010

Page 2

Contents

1. Introduction1

2. Matrix model for the ABJM Wilson loop2

3. The lens space matrix model and its large N dual4

4. Exact results for the ABJM model

4.1 General results

4.2 The 1/6 BPS Wilson loop

4.3 The ABJM slice

9

9

11

13

5. Conclusions 16

1. Introduction

Localization techniques in supersymmetric gauge theories have produced in recent years explicit

expressions for a variety of correlation functions. In [1], they were used to prove the longstanding

conjecture [2, 3] that the vev of a 1/2 BPS Wilson loop in N = 4 super Yang–Mills theory can

be computed by a correlator in a Gaussian matrix integral. This gives the celebrated formula

?W ? =

2

√λI1

?√

λ

?

(1.1)

for the planar limit of the Wilson loop in the fundamental representation. In this formula, I1is

a modified Bessel function and λ = g2N is the ’t Hooft coupling. (1.1) gives an exact expression

in λ that interpolates between strong and weak coupling (see [4] for a review).

More recently, Kapustin, Willett and Yaakov [5] have applied localization techniques to

ABJM theories [6, 7]. These are superconformal field theories in three dimensions based on a

U(N1) × U(N2) Chern–Simons theory coupled to matter, and they have large N AdS4 duals.

In [5], it was shown that the calculation of vevs of 1/6 BPS Wilson loops in these theories

can be reduced to a calculation in a matrix model, and they verified that the Wilson loop vev

reproduces the calculations of [8, 9, 10]. However, the matrix model they obtained is highly

nontrivial, and the comparison with the gauge theory calculation was done by performing a

perturbative expansion up to two loops. Some partial results on the planar limit of the matrix

model have been obtained in [11].

In this note we point out that the matrix model of [5] can be solved by relating it to the

Chern–Simons matrix models introduced in [12], and in particular to the lens space matrix model

studied in detail in [13, 14]. In [13] it was shown that the lens space Chern–Simons matrix model

is the large N dual of topological string theory on a certain class of local Calabi–Yau geometries,

providing in this way a nontrivial generalization of the Gopakumar–Vafa duality [15]. This means

in particular that the large N limit of the model can be studied by using standard techniques

– 1 –

Page 3

in mirror symmetry. It turns out that the matrix model of [5] can be regarded as a supergroup

version of the lens space matrix model studied in [12, 13, 14]. Since large N duals describe matrix

models as well as their supergroup extensions, we can use topological string theory on a local

Calabi–Yau geometry to obtain exact results in ABJM theory.

In this paper we use the solution of [13, 14] to obtain an exact expression for the planar limit

of the vev of the 1/6 BPS Wilson loop constructed in [8, 10, 9], as a function of the two ’t Hooft

parameters of the theory. This analytic expression is relatively complicated but it can be written

down explicitly in terms of elliptic functions, see for example (4.40), (4.43) for the case in which

the two gauge groups have the same rank. It interpolates smoothly between weak and strong

coupling, and at strong coupling it agrees precisely with the AdS theory prediction obtained in

[8].

The 1/6 BPS Wilson loop operator that we study in the planar limit involves only one of

the gauge connections in the U(N1) × U(N2) quiver, and it is not the most natural operator

from the point of view of topological string theory. It has been recently shown in [16] that it

is possible to construct a 1/2 BPS Wilson loop operator in the ABJM theory which localizes

precisely to the natural Wilson loop operator for Chern–Simons theory on L(2,1) (extended

to a supergroup). The vev of such an operator can be calculated to all orders in 1/N, as an

exact function of the ’t Hooft parameters, by combining the solution of [13, 14] with the matrix

model-inspired techniques of [17, 18]. In the topological string dual, the vevs of these 1/2 BPS

operators correspond to open topological string amplitudes.

Our exact formulae for the vevs of Wilson loops in ABJM theory have the flavour of mirror

symmetry. There are two sets of coordinates for the parameter space: the “bare” coordinates

and the flat coordinates. The flat coordinates are identified with the ’t Hooft parameters. They

are computed by period integrals and they can be related to the bare coordinates through the

mirror map. The vev of the Wilson loop is naturally expressed in terms of bare coordinates, and

one has to invert the mirror map in order to re-express it in terms of ’t Hooft parameters. In

retrospect, one could say that the result of [3, 2, 1] for the 1/4 BPS Wilson loop in N = 4 super

Yang–Mills theory is comparatively easier since in their case the relevant matrix model is the

Gaussian one, with a simpler moduli space, and where the ’t Hooft parameter is simply equal to

the bare coordinate.

The organization of this paper is as follows. In section 2 we review very briefly the matrix

model obtained in [5]. In section 3 we present the solution of the lens space matrix model building

on [13, 14]. In section 4 we explain the relation between the two matrix models, we derive the

exact results for the 1/6 BPS Wilson loop in the planar limit, and we study its behavior both at

strong and at weak coupling. Finally, some conclusions are presented in section 5.

2. Matrix model for the ABJM Wilson loop

The ABJM theory is a quiver Chern–Simons–matter theory in three dimensions with gauge

group U(N)k× U(N)−kand N = 6 supersymmetry. The Chern–Simons actions have couplings

k and −k, respectively, and the theory contains four bosonic fields CI, I = 1,··· ,4, in the

bifundamental representation of the gauge group. One can consider an extension [7] with a

more general gauge group U(N1)k× U(N2)−k. A family of Wilson loops in this theory has been

constructed in [8, 9, 10], with the structure

? ?

WR=

1

dR(N1)TrRP exp

iAµ˙ xµ+2π

k|˙ x|MI

JCI¯CJ

?

ds (2.1)

– 2 –

Page 4

where Aµis the gauge connection in the U(N1)kgauge group, dR(N1) is the dimension of the

represenation R of U(N1), xµ(s) is the parametrization of the loop, and MJ

mined by supersymmetry. It can be chosen so that, if the geometry of the loop is a line or a

circle, four real supercharges are preserved. Therefore, we will call (2.1) the 1/6 BPS Wilson

loop. A similar construction exists for a loop based on the other gauge group,

? ?

where Aµis the U(N2)−kgauge connection. The planar limit of the vev of (2.1) was computed in

[8, 9, 10], for N1= N2= N, in the fundamental representation R = , and in the weak coupling

regime λ ? 1, where

λ =N

Iis a matrix deter-

?

WR=

1

dR(N2)TrRP expi? Aµ˙ xµ+2π

k|˙ x|MI

J¯CICJ

?

ds (2.2)

k

(2.3)

is the ’t Hooft parameter. The result is

?W ? = 1 +5π2

6

λ2+ O?λ3?. (2.4)

On the other hand, in the strong coupling regime λ ? 1, the Wilson loop vev can be calculated

by using the large N string dual, i.e. type IIA theory on AdS4× P3[8, 9, 10]. This gives the

prediction1

?W ? ∼ eπ√2λ.

As in the case of the 1/2 BPS Wilson loop in N = 4 Yang–Mills theory, the exact answer for

the planar limit of this vev should interpolate between the weak coupling behavior (2.4) and the

strong coupling prediction of the large N string dual, (2.5).

A crucial step in finding such an exact answer was taken in the paper [5]. It was shown

there, through a beautiful application of the localization techniques used in [1], that the vev of

(2.1) can be computed as a correlation function in a matrix model. This matrix model is defined

by the partition function

i<jsinh2?µi−µj

(2.5)

ZABJM(N1,N2,gs) =

?

N1

?

i=1

dµi

N2

?

j=1

dνj

?

2

?

sinh2?νi−νj

2

2

?

?

i,jcosh2?µi−νj

?

e−

1

2gs(?

iµ2

i−?

jν2

j),

(2.6)

where the coupling gsis related to the Chern–Simons coupling k of the ABJM theory as

gs=2πi

k.

(2.7)

One of the main results of [5] is that

?WR? =

1

dR(N1)?TrR(eµi)?ABJM, (2.8)

1To be precise, the dual calculation is made by considering a fundamental string in AdS4, which is a 1/2 BPS

object. In [8, 10] it was argued that the the strong coupling behavior obtained in this way should apply to the

symmetric Wilson loop Wsymdefined in (2.12) below. It should also apply to the 1/2 BPS Wilson loop constructed

in [16]. However, as we will show in this paper, the leading exponential behavior is common to all these Wilson

loops.

– 3 –

Page 5

i.e. the vev of the 1/6 BPS Wilson loop (2.1) can be obtained by calculating the vev of the

matrix eµiin the matrix model (2.6). This was explicitly checked in [5] by computing the vev

in the r.h.s. of (2.8) in the matrix model, for the fundamental representation. Notice that the

Wilson loop for the other gauge group,

??

WR? =

1

dR(N2)?TrR(eνi)?ABJM

(2.9)

is obtained from (2.8) simply by exchanging N1 ↔ N2 and changing the sign of the coupling

constant g → −g.

The Wilson loop (2.1) breaks the symmetry betwen the two gauge groups. Recently, a class

of 1/2 BPS Wilson loops has been constructed in the ABJM theory [16] which treats the two

gauge groups in a more symmetric way. These loops have a natural supergroup structure in

which the quiver gauge group U(N1)×U(N2) is promoted to U(N1|N2), and they can be defined

in any super-representation R. In [16] it has been argued that this 1/2 BPS loop, which we will

denote by SWR, localizes to the matrix model correlator

1

sR

in the ABJM matrix model. Here,

?SWR? =

?

StrR

?eµi

0

0 −eνj

??

ABJM

(2.10)

sR= StrR

?1 0

0 −1

?

. (2.11)

When R = , we have the simple relationship

?SW ? =

1

N1+ N2

?

N1?W ? + N2??

W ?

?

. (2.12)

In general, as it is clear from (2.10), the vev of the 1/2 BPS Wilson loop can be obtained if we

know the vevs of the 1/6 BPS Wilson loop, but we expect it to be simpler.

The work of [5] reduced the computation of vevs of Wilson loops in ABJM theory to the

computation of matrix model correlators in the matrix model (2.6). Perturbative calculations

are now straightforward. However, in order to obtain exact interpolation functions, we have to

resum all double-line diagrams at fixed genus in the matrix model. This is straightforward in

the Gaussian matrix model which computes the vev of the 1/2 BPS Wilson loop in N = 4 SYM

[3, 2, 1], but it is not for the model (2.6). As we will see in this paper, the most efficient way to

solve this matrix model in the 1/N expansion is to relate it to the lens space matrix model of

[12] and to its large N string dual [13].

3. The lens space matrix model and its large N dual

The lens space matrix model was introduced in [12, 13] in order to compute the partition function

of Chern–Simons theory on lens spaces of the form L(p,1) = S3/Zp. In this paper we will be

particularly interested in the case p = 2. In this case, the matrix model has the structure

?

i=1j=1 i<j

?

ZL(2,1)(N1,N2,gs) =

N1

?

dµi

N2

?

dνj

?

sinh2

?µi− µj

?µi− νj

2

?

?

sinh2

?νi− νj

2

?

×

i,j

cosh2

2

e−

1

2gs(?

iµ2

i+?

jν2

j)

(3.1)

– 4 –

Page 6

and it describes the expansion of the Chern–Simons partition function around a generic non-

trivial flat connection, corresponding to the symmetry breaking pattern

U(N) → U(N1) × U(N2). (3.2)

The model has a large N expansion of the form

F = logZ =

∞

?

g=0

Fg(t1,t2)g2g−2

s

(3.3)

where

ti= gsNi

(3.4)

are the partial ’t Hooft parameters.The genus zero free energy has the structure

F0(t1,t2) = FG

0(t1) + FG

0(t2) + Fp

0(t1,t2). (3.5)

where FG

0(t) is the Gaussian matrix model genus zero amplitude,

FG

0(t) =1

2t2?

log t −3

4

?

. (3.6)

and Fp

0(t1,t2) is the contribution from fatgraphs of genus zero. The first nontrivial terms are

Fp

0(t1,t2) =

1

288(t4

1

345600(4t6

1+ 6t3

1t2+ 18t2

1t2

2+ 6t1t3

2+ t4

2)

−

1+ 45t5

1t2+ 225t4

1t2

2+ 1500t3

1t3

2+ 225t2

1t4

2+ 45t1t5

2+ 4t6

2) + ···

(3.7)

Higher genus free energies can be computed analogously.

πi + B

πi − B

1/a

a

−1/b

−b

z

Z = ez

A

−A

Figure 1: The cuts for the CS lens space matrix model in the z plane and in the Z = ezplane.

The matrix model (3.1) can be studied from the point of view of the 1/N expansion. This was

done in detail in [13], where this expansion was identified with the genus expansion of topological

– 5 –

Page 7

string theory on a local Calabai–Yau manifold. Many results of [13] were rederived in the paper

[14, 19] by using standard matrix model techniques, which we now review.

At large N, the two sets of eigenvalues, µi, νj, condense around two cuts centered around

z = 0, z = πi, respectively. We will write them as

C1= (−A,A),

C2= (πi − B,πi + B), (3.8)

in terms of the endpoints A,B. It is also useful to use the exponentiated variable

Z = ez, (3.9)

In the Z plane the cuts (3.8) get mapped to

(1/a,a),(−1/b,−b), (3.10)

which are centered around Z = 1, Z = −1, respectively, and

a = eA,b = eB, (3.11)

see Fig. 1. We will use the same notation C1,2for the cuts in the Z plane. An important quantity

introduced in [14] is the total resolvent of the matrix model, ω(z). It is defined as

?N1

i=1

2

ω(z) = gs

?

coth

?z − µi

??

+ gs

?N2

j=1

?

tanh

?z − νj

2

??

. (3.12)

In terms of the Z variable, it is given by

ω(z)dz = −(t1+ t2)dZ

Z

+ 2gs

?N1

i=1

?

dZ

Z − eµi

?

+ 2gs

?N2

j=1

?

dZ

Z + eνj

?

(3.13)

and it has the following expansion as Z → ∞

ω(z) → t1+ t2+2gs

Z

?N1

i=1

?

eµi−

N2

?

j=1

eνj

?

+ ···

(3.14)

From the total resolvent it is possible to obtain the density of eigenvalues at the cuts. In the

planar approximation, we have that

?

where ρ1(µ), ρ2(ν) are the densities of eigenvalues on the cuts C1, C2, respectively, normalized as

?

The standard discontinuity argument tells us that

ω0(z) = −(t1+ t2) + 2t1

C1

ρ1(µ)

Z

Z − eµdµ + 2t2

?

C2

ρ2(ν)

Z

Z + eνdν,(3.15)

C1

ρ1(µ)dµ =

?

C2

ρ2(ν)dν = 1. (3.16)

ρ1(X)dX = −

1

4πit1

1

4πit2

dX

X

(ω0(X + i?) − ω0(X − i?)),X ∈ C1,

ρ2(Y )dY =

dY

Y

(ω0(Y + i?) − ω0(Y − i?)),Y ∈ C2.

(3.17)

– 6 –

Page 8

The planar resolvent (3.15) was found in explicit form in [14]. It reads,

?e−t/2

where

ω0(z) = 2log

2

??

(Z + b)(Z + 1/b) −

?

(Z − a)(Z − 1/a)

??

, (3.18)

t = t1+ t2

(3.19)

is the total ’t Hooft parameter. It is useful to introduce the variables

α = a +1

a,

β = b +1

b.

(3.20)

as well as

ζ =α − β

2

,ξ =α + β

2

. (3.21)

The expansion (3.14) implies then

ξ = 2et. (3.22)

As it is standard in matrix models, the ’t Hooft parameters turn out to be period integrals with

a nontrivial relation to a, b:

?

The derivatives of these periods can be calculated in closed form by adapting a trick from [20].

If we write

?e−t

with

f(Z) = Z2− ζZ + 1,

it follows that

∂t1,2

∂ζ4πi

C1,2

where K(k) is the complete elliptic integral of the first kind, and its modulus is given by

t1=

1

4πi

C1

ω0(z)dz,t2=

1

4πi

?

C2

ω0(z)dz.(3.23)

ω0(z) = log

2

?

f(Z) −

?

f(Z)2− ξ2Z2??

(3.24)

(3.25)

= −1

?

dZ

?f(Z)2− ξ2Z2= ±

√ab

π(1 + ab)K(k), (3.26)

k2=(a2− 1)(b2− 1)

(1 + ab)2

. (3.27)

The above relationships determine in principle the planar content of the theory. However,

this matrix model solution is further clarified by considering its large N dual, which was in

fact discovered before [13]. This dual is given by topological string theory on the anti-canonical

bundle of the Hirzebruch surface F0= P1× P1. The mirror geometry is encoded in a family of

elliptic curves Σ, which can be written as

y =z1x2+ x + 1 −?(1 + x + z1x2)2− 4z2x2

Here, z1,z2 parametrize the moduli space of complex structures, which is the mirror to the

enlarged K¨ ahler moduli space of local F0. This moduli space has a very rich structure discussed

in [13, 21].

2

. (3.28)

– 7 –

Page 9

The mirror geometry (3.28) is nothing but the spectral curve of the lens space matrix model,

and it is closely related to the resolvent ω0(Z). Indeed, one finds that ω0(Z) ∼ log y(x) provided

we identify the variables as

x = −Zz−1/2

and

1

√z1,

1

, (3.29)

ζ =

ξ = 2

?z2

z1. (3.30)

In order to make further contact with the matrix model, one has to look at the moduli space

of (3.28) near the orbifold point discovered in [13]. This is defined as the point x1= x2= 0 in

terms of the variables:

x1= 1 −z1

z2,x2=

1

√z2

?

1 −z1

z2

?. (3.31)

Using mirror symmetry, we can calculate the periods of ω0(z) along the cycles of the spectral

curve as solutions to a Picard–Fuchs equation. In terms of the coordinates x1,2, the Picard–Fuchs

system is given by the two operators

L1=1

+ (−1 + x1)x2

L2= (2 − x1)x2∂x2+?−1 + (1 − x1)x2

+ 2(−1 + x1)x1x2∂x1∂x2+ (1 − x1)x2

4

?8 − 8x1+ x2

1

?x2∂x2+1

1∂x1− x1

4

?−4 + (−2 + x1)2x2

1

?∂2

x1.

2

?∂2

x2

?2 − 3x1+ x2

?x2∂x1∂x2+ (−1 + x1)2x2

1∂x1

1∂2

x1,

2x2− x2

1∂2

(3.32)

A basis of periods near the orbifold point was found in [13]. It reads,

σ1= −log(1 − x1) =

?

Fσ2= σ2log(x1) +

?

m

cm,0xm

1,

σ2=

m,n

cm,nxm

1xn

2,

?

m,n

dm,nxm

1xn

2,

(3.33)

where the coefficients cm,nand dm,nare determined by the following recursions relations

cm,n=cm−1,n

(n + 2 − 2m)2

4(m − n)(m − 1),

n(n − 1)(cm,n−2(n − m − 1)(n − m − 2) − cm−1,n−2(n − m − 1)2),

dm,n=dm−1,n(n + 2 − 2m)2+ 4(n + 1 − 2m)cm,n+ 4(2m − n − 2)cm−1,n

4(m − n)(m − 1)

dm,n=

n(n − 1)(dm,n−2(n − m − 1)(n − m − 2) − dm−1,n−2(n − m − 1)2

+ (2n − 2 − 2m)cm−1,n−2+ (2m + 3 − 2n)cm,n−2).

cm,n=

1

,

1

(3.34)

– 8 –

Page 10

The ’t Hooft parameters of the matrix model are related to the periods above as

t1=1

4(σ1+ σ2),t2

=1

4(σ1− σ2).

(3.35)

The remaining period in (3.33) might be used to compute the genus zero free energy of the matrix

model. Notice that x1,2(or equivalently ζ,ξ as defined in (3.21)) are “bare” coordinates, while

σ1,2(and therefore t1,2) are flat coordinates, annihilated by the Picard–Fuchs operators.

It is now a matter of (computer) routine to calculate the different quantities, like the end-

points of the cuts, as an expansion in the ’t Hooft parameters. One obtains, for example,

a = 1 + 2√t1+ 2t1+1

2

√t1(3t1+ t2) + t1(t1+ t2) + ···

(3.36)

The expansion for b is obtained from this one simply by exchanging t1↔ t2.

4. Exact results for the ABJM model

4.1 General results

We will now use our knowledge of the solution of the lens space matrix model to solve the ABJM

model, at least at the planar level. It is clear that the matrix model (2.6) is closely related to

(3.1), but there are some obvious differences: in (2.6) the interaction between the µ and the

ν eigenvalues is in the denominator, and the Gaussian action for the νs has the opposite sign.

These ingredients are precisely the ones needed to make (2.6) a supergroup extension of (3.1). We

will now quickly review some results on supermatrix models, following [22, 23, 24]. A Hermitian

supermatrix has the form

?A Ψ

where A (C) are N1× N1(N2× N2) Hermitian, Grassmann even matrices, and Ψ is a complex,

Grassmann odd matrix. The supermatrix model is defined by the partition function

?

where we consider a polynomial potential V (Φ), and Str is the supertrace

Φ =

Ψ†C

?

(4.1)

Zs(N1|N2) =

DΦe−1

gsStrV (Φ)

(4.2)

StrΦ = TrA − TrC. (4.3)

There are two types of supermatrix models with supergroup symmetry U(N1|N2): the ordinary

supermatrix model, and the physical supermatrix model [23]. The ordinary supermatrix model is

obtained by requiring A, C to be real Hermitian matrices, while the physical model is obtained by

requiring that, after diagonalizing Φ by a superunitary transformation, the resulting eigenvalues

are real. Here we will be interested in the physical supermatrix model. Its partition function

reads, in terms of eigenvalues [23, 24]

Zs(N1|N2) =

?

N1

?

i=1

dµi

N2

?

j=1

dνj

?

i<j(µi− µj)2(νi− νj)2

?

i,j(µi− νj)2

e−1

gs(?

iV (µi)−?

jV (νj)). (4.4)

– 9 –

Page 11

When the two groups of eigenvalues µi, νjare expanded around two different critical points, the

partition function (4.4) is well-defined as an asymptotic expansion in gs. It is easy to show that

(4.4) is related to the partition function of the corresponding bosonic, two-cut matrix model

?

i=1j=1i<j

Zb(N1,N2) =

N1

?

dµi

N2

?

dνj

?

(µi− µj)2(νi− νj)2?

i,j

(µi− νj)2e−1

gs(?

iV (µi)+?

jV (νj))

(4.5)

after changing N2→ −N2:

Zs(N1|N2) = Zb(N1,−N2).(4.6)

Such a flip of sign is trivially performed if one knows the exact solution of the model in the

1/N expansion. The relation (4.6) can be proved diagramatically by introducing Faddeev–Popov

ghosts as in [24, 25].

We now see that the relationship between the ABJM matrix model and the lens space matrix

model is identical to the one we have between supergroup matrix models and multi-cut bosonic

matrix models, with the only difference that the interaction between the eigenvalues has been

promoted to the sinh interaction typical of Chern–Simons matrix models. Indeed, the lens space

matrix model is a two-cut matrix model where the µ, ν eigenvalues are expanded around two

different saddle points, z = 0 and z = πi. The ABJM matrix model is just its supergroup version.

We then conclude that

ZABJM(N1,N2,g) = ZL(2,1)(N1,−N2,g).

Notice that the change N2→ −N2is equivalent to setting

t1= 2πiλ1,

(4.7)

t2= −2πiλ2,(4.8)

where

λi=Ni

k,

i = 1,2, (4.9)

are the ’t Hooft parameters of the ABJM model.

The appearance of a hidden supergroup structure in the matrix model of [5] is not surprising,

since N = 4 Chern–Simons–matter theories are classified by supergroups [26]. In fact, the ABJM

theory can be constructed as an N = 4 theory with supergroup U(N1|N2) and containing both

hypermultiplets and twisted hypermultiplets [27]. This hidden supergroup structure in the ABJM

theory is explicitly used in the construction of half-BPS Wilson loops in [16].

Let us now discuss Wilson loops. The most natural correlator in the standard lens space

matrix model is

?

where R is a representation of U(N1+ N2). In Chern–Simons gauge theory on L(2,1), this

computes the vev of a trivial knot, expanded around a generic, fixed flat connection. In the

topological string large N dual, it corresponds to an open string amplitude for a toric D-brane

(see for example [18] for more details). These vevs can be computed for any R and to all orders

in the 1/N expansion [17, 18].

In order to consider its supergroup extension, notice that a representation of U(N1+ N2)

induces a super-representation of U(N1|N2), defined by the same Young tableau R (see for

example [28]). Therefore, the supergroup generalization of (4.10) is simply

?

TrR

?eµi

0

0 −eνj

??

,(4.10)

StrR

?eµi

0

0 −eνj

??

(N1|N2)

.(4.11)

– 10 –

Page 12

This can be also written as [28]

StrR

?eµi

0

0 −eνj

?

=

?

?

?k

χR(?k)

z?k

?

?

?

?

Str

?eµi

?

0

0 −eνj

e?µi?

???k?

=

?k

χR(?k)

z?k

?

?

Tr

− (−1)?Tr

?

e?νj??k?.

(4.12)

In this equation, which is the supergroup generalization of Frobenius formula,?k = (k?) is a vector

of non-negative, integer entries, which can be regarded as a conjugacy class of the symmetric

group, χR(?k) is the character of this conjugacy class in the representation R, and

?

We then have

?

(N1|N2)

z?k=

?

?k?k?! (4.13)

StrR

?eµi

0

0 −eνj

??

=

?

TrR

?eµi

0

0 −eνj

??

(N1,−N2,g) (4.14)

which extends (4.7) to correlation functions. In view of (2.10), we conclude that the vevs of the

1/2 BPS Wilson loops constructed in [16] can be computed to all orders in the 1/N expansion

by calculating the correlator (4.10) in the lens space matrix model and changing N2→ −N2

1

sR

0 −eνj

?SWR? =

?

TrR

?eµi

0

??

(N1,−N2,g). (4.15)

Let us give an example of how to calculate the vev (4.15) when R =

we have, for the lens space matrix model correlator (4.10), the exact answer

?

. In the planar limit

1

N1− N2

Tr

?eµi

0

0 −eνj

??

=

1

t1− t2

ζ

2, (4.16)

where ζ is defined in (3.21). After setting (4.8) we obtain the weak coupling expansion

?SW ? = 1+iπ(λ1−λ2)−1

3π2?2λ2

1− 5λ2λ1+ 2λ2

2

?−1

3iπ3?λ3

1− 4λ2λ2

1+ 4λ2

2λ1− λ3

2

?+··· (4.17)

This is of course (up to normalizations) the first term in equation (6.54) of [18]. Higher genus

corrections can be extracted from the higher genus resolvents ωg(z).

computed by using standard matrix model techniques applied to the spectral curve (3.28), as in

[17, 18].

These in turn can be

4.2 The 1/6 BPS Wilson loop

If the densities of eigenvalues ρ1(µ) and ρ2(ν) given in (3.17) are known, it is possible to calculate

the exact planar limit of the correlator

?

1

N?Treµi? =

C1

ρ1(µ)eµdµ =

?

C1

ρ1(X)XdX,(4.18)

– 11 –

Page 13

as well as of multiple-winding correlators

1

N?Trenµi? =

?

C1

ρ1(µ)enµdµ.(4.19)

We then conclude, in view of (2.8), that the planar limit of the vev of a 1/6 BPS Wilson loop is

given by

?

after changing variables as in (4.8).

The densities ρ1(µ) and ρ2(ν) can be explicitly calculated from (3.17) and (3.18). We find,

??

?W ? =

C1

ρ1(µ)eµdµ (4.20)

ρ1(X)dX =

1

πt1tan−1

αX − 1 − X2

βX + 1 + X2

??

?

dX

X,

?

ρ2(Y )dY = −1

πt2tan−1

βY + 1 + Y2

αY − 1 − Y2

dY

Y

.

(4.21)

In terms of the variable x = logX we have

ρ1(x) =

1

πt1tan−1

??

α − 2coshx

β + 2coshx

?

, (4.22)

and a similar expression for ρ2(y). Notice that, if t2= 0, one has β = 2, α = 4et− 2, and ρ1(x)

becomes the density of eigenvalues for the matrix model of Chern–Simons theory on S3[29]

ρ1(x) =

1

πttan−1

?

et− cosh2?x

2

?

cosh?x

2

?

. (4.23)

Remark 4.1. We can write the density of eigenvalues (4.22) as

ρ1(x) = f(x)

?

A2− x2, (4.24)

where

f(x) =

1

πt1

√A2− x2tan−1

??

α − 2coshx

β + 2coshx

?

=

∞

?

k=0

βkx2k. (4.25)

We find for example

β0=

1

πt1Atan−1

??α − 2

β + 2

?

. (4.26)

It is easy to check that ρ1(x) agrees, in the special case t1= −t2= t, with the perturbative expan-

sion obtained in [11] up to order 10 in t. Our expression (4.22) gives then the full resummation

of the expression obtained in [11], and extends it to any t1,t2.

The integral (4.20) is then given by

?W ? =

1

πt1I1,I1=

?a

1/a

tan−1

??

αX − 1 − X2

βX + 1 + X2

?

dX.(4.27)

– 12 –

Page 14

This integral is not easy to calculate in closed form, but its derivatives w.r.t. ζ and ξ can be

expressed in terms of elliptic integrals. We find

∂I1

∂ζ

=1

2

a

?

1/a

XdX

?(αX − 1 − X2)(βX + 1 + X2)

?(βX + 1 + X2) − (αX − 1 − X2)?dX

= −

1

√ab(1 + ab)(aK(k) − (a + b)Π(n|k)),

∂I1

∂ξ

=1

2

a

?

1/a

?(αX − 1 − X2)(βX + 1 + X2)

=

√ab

a + bE(k),

(4.28)

where Π(n|k) is the complete elliptic integral of the third kind, K(k),E(k) are elliptic integrals

of the first and second kind, respectively, the modulus is given by (3.27), and finally

n =b

a

a2− 1

1 + ab.

(4.29)

This determines the planar limit of the 1/6 BPS Wilson loop exactly. As we mentioned in the

introduction, the Wilson loop is naturally expressed in terms of the “bare” coordinates ζ,ξ, and

we have to use the mirror map (3.33) in order to write it in terms of the ’t Hooft parameters.

As an application of these formulae, we will present the first few terms of the weak coupling

expansion in t1,t2. To do this, we simply calculate

∂I1

∂ti

=∂I1

∂ζ

∂ζ

∂ti

+ 2∂I1

∂ξet, (4.30)

we use the expressions (4.28), and we integrate w.r.t. t1, t2. This produces the expansion

?W ? = 1+t1

2+1 12(2t2

1+3t1t2)+1

48(2t3

1+6t2

1t2+4t1t2

2)+

1

960(8t4

1+35t3

1t2+30t2

1t2

2+10t1t3

2)+···

(4.31)

The result for the Wilson loop in the ABJ theory is obtained by simply changing variables to

(4.8). The result agrees at the first few orders with a perturbative matrix model calculation of

[16]. Finally, the vev of the other Wilson loop (2.9) is obtained by exchanging t1↔ t2, or, in the

ABJ theory, by exchanging λ1↔ λ2and complex conjugating the result.

Interestingly, the strong coupling limit of the above expressions depends on the direction in

which one goes to infinity. We will discuss one such direction in detail in the next subsection,

where we consider the restriction to the original ABJM model N1= N2.

4.3 The ABJM slice

In the original ABJM theory with N1= N2= N (the case N1?= N2was considered in [7]) we

should look at the slice

t1= −t2= 2πiλ,

in the moduli space of the dual topological string. From the point of view of the periods σ1, σ2

in (3.33) this means that we should set

σ1= 0,

λ =N

k

(4.32)

(4.33)

therefore x1= 0. In order to have a nontrivial σ2, we must consider the double-scaling limit

x1→ 0,x1x2= ζfixed.(4.34)

– 13 –

Page 15

The one-dimensional subspace (4.32) corresponds, in terms of the variables ζ,ξ, to ξ = 2. As in

[30], we can find simplified expressions for the periods in this subspace. It is easy to see from the

structure of σ2that, in the limit (4.34), one has

σ2=

∞

?

m=0

amζ2m+1,am= c2m+1,2m+1, (4.35)

and from the recursion relation (3.34) we find

am=

2−4mΓ?m +1

2

?2

π(2m + 1)Γ(m + 1)2. (4.36)

We then obtain

dσ2

dζ

=2

πK

?ζ

4

?

, (4.37)

which is in fact a particular case of (3.26), as it can be easily seen by using the transformation

properties of the elliptic integral K(k). The period t1 itself can be written as a generalized

hypergeometric function:

t1(ζ) =ζ

4

3F2

?1

2,1

2,1

2;1,3

2;ζ2

16

?

. (4.38)

In the physical ABJM theory, t1is purely imaginary. This means that ζ is purely imaginary as

well, so we set

ζ = iκ (4.39)

and we finally obtain

λ(κ) =

κ

8π

3F2

?1

2,1

2,1

2;1,3

2;−κ2

16

?

. (4.40)

Let us now calculate the planar limit of the vev of the 1/6 BPS Wilson loop in the ABJM

slice N1= N2. Since

α = 2 + iκ,β = 2 − iκ,(4.41)

the endpoints of the cuts are given by

a(κ) =1

2

?

?

2 + iκ +

?

?

κ(4i − κ)

?

,

?

b(κ) =1

2

2 − iκ +

−κ(4i + κ).

(4.42)

The planar vev of the Wilson loop is then determined, as a function of the ’t Hooft coupling λ,

by the single equation

d

dκ(λ(κ)?W ?) = −

1

2π2√ab(1 + ab)(aK(k) − (a + b)Π(n|k)), (4.43)

together with the explicit relation between λ and κ in (4.40) –yet another example of mirror

map.

As a check, we can perform a weak coupling expansion. The weakly coupled region corre-

sponds to

κ ? 1,λ ? 1, (4.44)

– 14 –

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