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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
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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
4.2The 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 –
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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)
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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.
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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=1i<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)
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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
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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)
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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)
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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+112(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= ζ fixed.(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 –
Page 16
0.51.01.52.0 2.53.0
1
2
3
4
Figure 2: Plot of (4.52) as a function of λ, displaying clearly an interpolation between a quadratic
behavior near the origin and a square-root growth as λ becomes large.
and in this region the variables are related as
κ
8π= λ +π2λ3
3
−7π4λ5
60
+173π6λ7
1260
−37927π8λ9
181440
+ O?λ10?,(4.45)
which is obtained from the inversion of (4.40). By expanding (4.43) in power series around κ = 0,
and using the mirror map (4.45), we obtain
?W ? = eπiλ
?
1 +5π2λ2
6
−1
2iπ3λ3−29π4λ4
120
+1
12iπ5λ5+151π6λ6
1008
−1
10iπ7λ7−87449π8λ8
+ O?λ13??
362880
+2603iπ9λ9
15120
+3447391π10λ10
7983360
−1166161iπ11λ11
3628800
−5239372319π12λ12
6227020800
,
(4.46)
where we have extracted a framing factor to facilitate comparison with existing results like (2.4).
The first few terms agree with the calculations in [8, 9, 5, 11].
Of course, the main advantage of having analytic expressions is that one can perform a
weak-strong coupling interpolation easily. The strong coupling region is
κ ? 1,λ ? 1 (4.47)
and (4.40) leads to the asymptotic expansion
λ(κ) =log2(κ)
2π2
+1
24+ O
?1
κ2
?
, (4.48)
which is immediately inverted to
κ = eπ√2λ
?
1 + O
?
1
√λ,e−2π√2λ
??
.(4.49)
On the other hand, it is easy to check from (4.43) that
dI1
dκ= −1
2logκ +iπ
4+ O
?1
κ2
?
⇒ I1= −1
2κ logκ +
?1
2+iπ
4
?
κ + O
?1
κ
?
(4.50)
– 15 –
Page 17
It follows that
?W ? ∼
i
2π√2λeπ√2λ,λ ? 1.(4.51)
The leading exponential is in perfect agreement with the AdS prediction (2.5), and the exact
answer interpolates between the weak and the strong coupling behaviors. This is illustrated in
Fig. 2, which represents the function
f(λ) = log|?W ?|.(4.52)
This function interpolates between
f(λ) =
5π2λ2
6
,λ ? 1,
π
√
2λ,λ ? 1.
(4.53)
Let us now consider the vev of the 1/2 BPS Wilson loop of [16] in the fundamental repre-
sentation, which is given by (2.12). The expression for this vev follows from the specialization of
(4.16),
?SW ? =
κ
8πλ(κ). (4.54)
Since, in the ABJM slice, (2.8) and (2.9) are related by complex conjugation, this vev is manifestly
real. Notice also that, when adding the contributions of the two gauge groups as in (2.12), all
terms in the series expansion at small κ of the r.h.s. of (4.43) cancel except for the first constant
term. This confirms that the 1/2 BPS Wilson loop is much simpler than the 1/6 BPS Wilson
loop. At strong coupling we have,
?SW ? ∼
1
8πλeπ√2λ,λ ? 1,(4.55)
which displays the same leading exponential behavior predicted by the large N dual.
The computations at strong coupling can be easily extended to the case N1?= N2, but they
depend on the direction in which we take the limit in the space of ’t Hooft parameters. For ξ
fixed and ζ large (therefore λ1∼ λ2), we find the same exponential behavior
?W ? ∼ eπ√2λ1∼ eπ√λ1+λ2. (4.56)
5. Conclusions
In this paper we have related the matrix integral of [5], computing Wilson loop vevs in ABJM
theory, to a supergroup extension of the lens space matrix model introduced in [12] and solved in
[13, 14]. This has made possible to obtain exact expressions for the planar vev of 1/6 BPS Wilson
loops, and we have verified the strong coupling behavior predicted by the AdS dual [8, 10, 9].
There are various avenues for further research. It is clearly gratifying to see that the tech-
niques of mirror symmetry are relevant to the strong-weak interpolation problem in the AdS/CFT
correspondence, but it would be nice to have an a priori understanding of the relationship be-
tween topological string theory and the ABJM theories, maybe along the lines of [31]. This would
lead to further fruitful interactions between the two topics.
– 16 –
Page 18
We should also mention that the strong coupling analysis of Wilson loops in the context
of the gauge theory is different from what one does in topological string theory. There, the
interpolation from weak to strong coupling involves a different choice of flat coordinates and
a different choice of duality frame (see for example [32] and specially the analysis of Wilson
loop amplitudes in [18]). This means, in particular, that in topological string theory, analytic
continuation of the amplitudes is not enough. In the gauge theory analysis, in contrast, one
uses the same flat coordinates (the ’t Hooft parameters) and the same duality frame in the full
moduli space, and the weak and the strong coupling regions are simply related by an analytic
continuation.
One obvious problem is to find expressions for the Wilson loop vevs beyond the planar
approximation. As we have explained in this paper, this can be trivially done for the 1/2 BPS
Wilson loops constructed in [16], by simply applying the techniques of [17, 18]. For the 1/6 BPS
Wilson loop the calculation of 1/N corrections is more difficult, but in principle it can be done.
Finally, it would be interesting to compare the matrix model results with an AdS calculation
using D-branes, as it was done in [33] for the 1/2 BPS Wilson loop of N = 4 Yang–Mills theory.
Acknowledgments
We would like to thank Anton Kapustin for discussions. We are specially grateful to Nadav
Drukker for many conversations which helped us very much, and we want to thank him, as well
as Diego Trancanelli, for sharing their unpublished results with us. This work is supported in
part by the Fonds National Suisse.
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