From weak to strong coupling in ABJM theory

TL;DR: In this paper, the authors show that the planar free energy of ABJM theory matches the classical IIA supergravity action on a zero-dimensional super-matrix model and gives the correct N 3/2 scaling for the number of degrees of freedom of M2 brane theory.

Abstract: The partition function of $${\mathcal{N}=6}$$ supersymmetric Chern–Simons-matter theory (known as ABJM theory) on $${\mathbb{S}^3}$$ , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on $${{\rm AdS}_4\times\mathbb{C}\mathbb{P}^3}$$ and gives the correct N 3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in $${\mathbb{C}\mathbb{P}^3}$$ . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.

Summary

1. Introduction and Summary

• The authors will mostly concern ourselves with the 1/2 BPS Wilson loop in the fundamental representation of U (N1|N2) and the 1/6 BPS Wilson loop in the fundamental representation of U (N1).
• Section 5 addresses the strong coupling limit of the theory, where the matrix model should reproduce the semiclassical expansion of these observables in type IIA string theory on AdS4 ×CP3.
• The brave souls that will make it to Sects.
• All the quantities captured by the matrix model should exist also in a topologically twisted version of ABJM theory, possibly along the lines of [36].

3. Moduli Space, Picard–Fuchs Equations and Periods

• In this section the authors present the tools for solving the lens space matrix model using special geometry.
• As it is well-known, these periods are annihilated by a pair of differential operators called Picard–Fuchs operators.
• 4. The moduli space of the ABJM theory.
• Some of the explicit relations needed for this identification will be presented only in the following sections, but the authors would still like to present here the main points on the moduli space.
• The orbifold point (weakly coupled ABJM theory) maps to the monopole point, while the large radius point (strongly coupled ABJM theory) corresponds to the semi-classical region (see Fig. 3).

4. Weak Coupling

• In principle, to study the matrix model at weak coupling one does not need the sophisticated tools presented in the previous section.
• One can do perturbative calculations directly in the integral expressions (2.1) or (2.3) for the matrix model.
• This agrees with the weak coupling expansion of the inverse of the exact mirror map (5.5), obtained in [8].
• This expression agrees with the 2-loop calculations in [15–17].
• Here the authors notice that the period in (2.15) gives only the derivative of the free energy.

5. Strong Coupling Expansion and the AdS Dual

• The authors turn now to the strong coupling limit of the matrix model, where they have to find the analytic continuation of the ’t Hooft parameters to the strong coupling region, as functions of the global parameters of moduli space.
• The authors are now ready to discuss the calculation of the planar free energy at strong coupling.
• The authors will show this now, and in particular they will match the numerical coefficient in (5.37).
• The counterterm action includes higher order corrections which are not relevant for the case of AdS4 and will not be considered here [52].
• These instanton corrections are also present in the Wilson loop result (5.16), again with an infinite series of corrections.

6. Conifold Expansion

• The expansion around the conifold locus corresponds to a region in the moduli space of the ABJM model where one of the gauge groups has finite coupling, while the other one is weakly coupled.
• When g = 0, the expansion (6.10) can be computed from the exact planar solution in various ways.
• This procedure gives a method to compute the expansion (6.10) directly in the matrix model.
• There are extra graphs which are not drawn, with fermionic tadpoles on the scalar lines, or vice–versa.
• The essential part of the expression for the Wilson loop at order O(λ2), (6.28) is the connected 8 Wilson loops arise out of dressing propagators of matter fields also in [57].

7. Modular Properties and the Genus Expansion

• This is based on the modular properties of the solution and the technique of direct integration of the holomorphic anomaly equations.
• This is then used to estimate non-perturbative effects in the large N expansion.
• As noted in [41], the authors can use the relation between the local F0 theory and Seiberg– Witten theory to write all the quantities in the model in terms of modular forms.
• The first one concerns its functional dependence.
• Since (7.23) is a power series in Q with no constant term, the authors obtain one extra condition, which, together with the g − 1 conditions from the orbifold point and the 2g − 2 conditions of the conifold point, completely fixes the 3g − 2 unknowns in the holomorphic ambiguity.

A. Normalization of the ABJM Matrix Model

• Here the authors shall fix the overall normalization of the matrix model.
• This term appears as a consequence of integrating out the matter hypermultiplets at one-loop.

Figures

Fig. 4. Comparison of the exact result for ∂λF0(λ) given in (5.29), plotted as a solid blue line, and the weakly coupled and strongly coupled results. In the figure on the left, the red dashed line is the supergravity result (5.45), while in the figure on the right, the black dashed line is the Gaussian result (5.46)

Fig. 2. The moduli space of the ABJM theory, describing the possible values of the ’t Hooft couplings λ1,2, can be parametrized by a real submanifold of the moduli space of local F0, here depicted as a sphere. The orbifold point maps to the origin, while the conifold locus (which is represented by a dashed line) maps to the two axes

Fig. 1. Cuts in the z-plane and in the Z -plane

Fig. 5. Several Feynman graphs which may contribute at order λ2 to the 1/6 BPS Wilson loop with gluons stripped. The big circle is the Wilson loop, dashed lines are bosons and the solid line a fermion, all presented in double-line, double-color notation

Fig. 3. The moduli space of the ABJM theory for B = 1/2 can be mapped to the line [1,∞) in the u plane of Seiberg–Witten theory, which is here shown in red. The monopole point corresponds to the weakly coupled ABJM theory, while the semiclassical limit corresponds to the strongly coupled theory

Full text

Article
Reference
From Weak to Strong Coupling in ABJM Theory
DRUKKER, Nadav, MARINO BEIRAS, Marcos, PUTROV, Pavel
DRUKKER, Nadav, MARINO BEIRAS, Marcos, PUTROV, Pavel. From Weak to Strong Coupling
in ABJM Theory. Communications in Mathematical Physics, 2011, vol. 306, no. 2, p.
511-563
DOI : 10.1007/s00220-011-1253-6
Available at:
http://archive-ouverte.unige.ch/unige:17556
Disclaimer: layout of this document may differ from the published version.
1 / 1

Digital Object Identifier (DOI) 10.1007/s00220-011-1253-6
Commun. Math. Phys. 306, 511–563 (2011)
Communications in
Mathematical
Physics
From Weak to Strong Coupling in ABJM Theory
Nadav Drukker
1
, Marcos Mariño
2,3
, Pavel Putrov
3
1
Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany.
E-mail: drukker@physik.hu-berlin.de
2
Département de Physique Théorique, Université de Genève, Genève CH-1211, Switzerland.
E-mail: marcos.marino@unige.ch
3
Section de Mathématiques, Université de Genève, Genève CH-1211, Switzerland.
E-mail: pavel.putrov@unige.ch
Received: 26 August 2010 / Accepted: 14 October 2010
Published online: 18 May 2011 – © Springer-Verlag 2011
Abstract: The partition function of N = 6 supersymmetric Chern–Simons-matter
theory (known as ABJM theory) on S
3
, as well as certain Wilson loop observables, are
captured by a zero dimensional super-matrix model. This super–matrix model is closely
related to a matrix model describing topological Chern–Simons theory on a lens space.
We explore further these recent observations and extract more exact results in ABJM
theory from the matrix model. In particular we calculate the planar free energy, which
matches at strong coupling the classical IIA supergravity action on AdS
4
× CP
3
and
gives the correct N
3/2
scaling for the number of degrees of freedom of the M2 brane
theory. Furthermore we find contributions coming from world-sheet instanton correc-
tions in CP
3
. We also calculate non-planar corrections, both to the free energy and to
the Wilson loop expectation values. This matrix model appears also in the study of
topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises
between the space of couplings of the planar ABJM theory and the moduli space of
this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and
strong coupling (AdS) expansions, a third natural expansion locus is the line where one
of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus
of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory.
We present some explicit results for the partition function and Wilson loop observables
around this locus.
Contents
1. Introduction and Summary .......................... 512
2. The ABJM Matrix Model and Wilson Loops ................ 515
2.1 The matrix model and its planar limit .................. 515
2.2 Wilson loops .............................. 519
3. Moduli Space, Picard–Fuchs Equations and Periods ............ 520
3.1 Orbifold point, or weak coupling .................... 522

512 N. Drukker, M. Mariño, P. Putrov
3.2 Large radius, or strong coupling .................... 523
3.3 Conifold locus .............................. 525
3.4 The moduli space of the ABJM theory ................. 527
4. Weak Coupling ................................ 529
5. Strong Coupling Expansion and the AdS Dual ............... 530
5.1 Analytic continuation and shifted charges ............... 530
5.2 Wilson loops at strong coupling and semi–classical strings ...... 533
5.3 The planar free energy and a derivation of the N
3/2
behaviour .... 534
5.4 Calculation of the free energy in the gravity dual ............ 536
6. Conifold Expansion ............................. 539
6.1 Expansion from the exact planar solution ................ 539
6.2 Conifold expansion from the matrix model ............... 541
6.3 On the near Chern–Simons expansion of ABJM theory ........ 544
7. Modular Properties and the Genus Expansion ................ 546
8. More Exact Results on Wilson Loops .................... 551
8.1 1/N corrections ............................. 551
8.2 Giant Wilson loops ........................... 554
A. Normalization of the ABJM Matrix Model ................. 558
B. Giant Wilson Loops in Chern–Simons Theory ............... 559
1. Introduction and Summary
The discovery of Aharony, Bergman, Jafferis and Maldacena (ABJM) of the world-
volume theory of coincident M2-branes [1] (following Bagger-Lambert and Gustavsson
[2,3]) provides a new interacting field theory with well defined weak and strong cou-
pling expansions. A great deal of effort has been given to studying these two limits of the
theory: three dimensional N = 6 supersymmetric Chern–Simons-matter and type IIA
string theory on AdS
4
×CP
3
(or M-theory on AdS
4
×S
7
/Z
k
). For better or worse, both
descriptions of the theory are much harder than the D3-brane analog: 4d N = 4SYM
and type IIB string theory on AdS
5
× S
5
. At weak coupling perturbative calculations
in ABJM theory are rather subtle and for many quantities are in even powers of the
coupling, while at strong coupling the geometry of CP
3
is more complicated than S
5
and has, for example, non-trivial 2-cycles.
An important breakthrough, which is the underpinning of the present study, was the
work of Kapustin, Willett and Yaakov [4], who use the localization techniques of [5]to
reduce the calculation of certain quantities in the gauge theory on S
3
to finite dimensional
matrix integrals.
1
These matrix integrals can be evaluated in a systematic expansion in
1/N. Indeed, they have a natural supergroup structure, i.e., they are super-matrix mod-
els [7,8], and are related to some previously studied bosonic matrix models [9,10]by
analytical continuation [8].
The solution of this matrix model allowed for the evaluation of the first exact inter-
polating function in this theory [8] giving a closed form expression for the expectation
value of the 1/2 BPS Wilson loop operator of [7] at all values of the coupling. This
expression derived from the matrix-model reduction of the gauge theory reproduces
exactly the known leading strong coupling result, the classical action of a macroscopic
string in AdS
4
.
1
Similar results apply also to other 3d theories with N = 2 supersymmetry [4,6].

From Weak to Strong Coupling in ABJM Theory 513
The purpose of this paper is to explore further what can be learnt from the matrix
model and its solution to the understanding of the physical 3d gauge theory and its
string/M-theory dual.
This is a broad subject, connected through the matrix model to special geometry,
Chern–Simons (CS) theory, topological strings and more. One of the avenues we explore
is the relation between the moduli space of the matrix model and the space of couplings
of the gauge theory. It is very useful to consider the generalization of the gauge theory
where the rank of the two gauge groups are not equal [11].
2
The space of couplings is
two dimensional and upon complexification, it matches the moduli space of the Riemann
surface solving the planar matrix model. This surface is also the mirror to a well studied
toric Calabi–Yau manifold known as local F
0
, where F
0
= P
1
× P
1
is a Hirzebruch
surface. As we review in Sect. 3, this moduli space has three special loci: the orbifold
point, the large radius limit and the conifold locus.
These can be identified in the gauge theory respectively as the weakly coupled gauge
theory, the strongly coupled theory described by string theory on AdS, and lastly the
conifold locus is where the rank of one of the gauge group vanishes, so ABJM theory
reduces to topological CS theory [12]. The first two are known duality frames with the
AdS/CFT rules on how to evaluate observables on both sides. The simplicity of the con-
ifold locus suggests that there should be another duality frame where ABJM theory is
considered as a deformation of topological CS theory. We explore this in Sect. 6, where
we calculate the partition function and Wilson loop observables around this point. It
would be very interesting to learn how to calculate other quantities in this regime.
We present the matrix model for the ABJM theory and that for CS theory on the lens
space L(2, 1) = S
3
/Z
2
in the next section. The matrix model of ABJM has an underly-
ing U(N
1
|N
2
) symmetry while that of the lens space has U(N
1
+ N
2
) symmetry, which
in both cases are broken to U(N
1
)×U(N
2
). It is easy to see that the expressions for them
are related by analytical continuation of N
2
→−N
2
, or analogously a continuation of
the ’t Hooft coupling N
2
/k →−N
2
/k (which may be attributed to the negative level
of the CS coupling of this group in the ABJM theory). We can then go on to study the
lens space model and analytically continue to ABJM at the end.
Conveniently, the lens space matrix model has been studied in the past [8,10,13,14].
The planar resolvent is known in closed form and the expressions for its periods are
given as power series at special points in moduli space. We review the details of this
matrix model and its solution in Sects. 2 and 3.
The matrix model of ABJM theory was derived by localization: it captures in a finite
dimensional integral all observables of the full theory which preserve certain super-
charges. At the time it was derived in [4], the only such observable (apart for the vacuum)
was the 1/6 BPS Wilson loop constructed in [15–17] and 1/2 BPS vortex loop operators
[18]. Indeed, the expectation value of the 1/6 BPS Wilson loop can be expressed as an
observable in the ABJM matrix model, and by analytical continuation in the lens space
model.
Another class of Wilson loop operators, which preserve 1/2 of the supercharges,
was constructed in [7] and studied further in [19]. It is the dual of the most symmetric
classical string solution in AdS
4
×CP
3
. This Wilson loop is based on a super-connection
in space-time and reduces upon localization to the trace of a supermatrix in the ABJM
matrix model [7]. The different 1/2 BPS Wilson loops are classified by arbitrary repre-
sentations of the supergroup U (N
1
|N
2
), and the 1/6 BPS ones are classified by a pair
2
Though commonly known as ABJ theory, for simplicity we still call the theory with this extra parameter
ABJM theory. When specializing to the case of equal rank we refer to it as the “ABJM slice”.

514 N. Drukker, M. Mariño, P. Putrov
of representations
3
of U(N
1
) and U(N
2
). We will mostly concern ourselves with the
1/2 BPS Wilson loop in the fundamental representation of U(N
1
|N
2
) and the 1/6BPS
Wilson loop in the fundamental representation of U(N
1
). The exception is Sect. 8.2 and
Appendix B, where we study the 1/2 BPS Wilson loop in large symmetric and antisym-
metric representations. There we also make contact with the vortex loop operators of
[18].
Of course, the natural observables in CS theory are the partition function and Wilson
loops, so these quantities were also studied earlier in the matrix models of CS (see, for
example, [9,10,13,20–22]). This information is encoded in different period integrals
on the surface solving the matrix model, as we explain in Sect. 2.2. It turns out that
the 1/6 BPS loop is captured by a period integral around one of the two cuts in the
planar solution and the 1/2 BPS Wilson loop by a period integral around both cuts, or
alternatively, around the point at infinity, and is much easier to calculate [8].
With all this machinery presented in Sects. 2 and 3 in hand, we are ready to calculate,
and in Sects. 4, 5 and 6 we study the partition function and Wilson loop observables in
the three natural limits of the matrix model. First, in Sects. 4 we look at the orbifold
point, which is the weak coupling point of the matrix model and likewise of the physical
ABJM theory. The calculations there are straightforward and we present the answers
to these quantities. A single term (1/6 BPS loop at 2-loops) was calculated indepen-
dently directly in the field theory. All other terms are predictions for the higher order
perturbative corrections.
Section 5 addresses the strong coupling limit of the theory, where the matrix model
should reproduce the semiclassical expansion of these observables in type IIA string
theory on AdS
4
×CP
3
. The expectation value of the Wilson loop was already derived in
[8] and matched with a classical string in AdS. We first generalize the strong coupling
expansion for the case of N
1
= N
2
, which corresponds to turning on a B-field in the
AdS dual. This version of the theory was studied in [11] and a more precise analysis of
the dictionary, capturing shifts in the charges, was presented in [23,24]. Interestingly,
it turns out that the matrix model knows about these shifted charges, and the strong
coupling parameter turns out to be exactly the one calculated in [24], rather than the
naive coupling.
In the same section we present also the calculation of the free energy in the matrix
model. The result is proportional to N
2
/
√
λ (or a slight generalization for N
1
= N
2
).
This scales at large N like N
3/2
, which is indeed the M-theory prediction for the number
of degrees of freedom on N coincident M2-branes [25]. Comparing with a supergravity
calculation, we find precise agreement with the classical action of AdS
4
×CP
3
.Thisis
the first derivation of this large N scaling in the field theory side. The matrix model also
provides an infinite series of instanton/anti–instanton corrections to both the partition
function and to the Wilson loop expectation value, which we interpret as fundamental
strings wrapping the CP
1
inside CP
3
.
We then turn to a third limit of the theory, when one of the gauge couplings is pertur-
bative and the other one not. In the strict limit the ABJM theory reduces to topological
CS and in the matrix model one cut is removed. We show how to perform explicit calcu-
lations in this regime both from the planar solution of the matrix model and directly by
performing matrix integrals. In both approaches one can see the full lens space matrix
model arising as a (rather complicated) observable in topological CS theory on S
3
.We
3
Special combinations of representations of U(N
1
) ×U(N
2
) are also representations of U(N
1
|N
2
),and
in this case the 1/6BPSand1/2 BPS loops will have the same expression in the matrix model and the same
VEVs. The proof of localization for the 1/2 BPS loop [7] relied on this equivalence.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "From weak to strong coupling in abjm theory" ?

The authors explore further these recent observations and extract more exact results in ABJM theory from the matrix model. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. The authors present some explicit results for the partition function and Wilson loop observables around this locus. Furthermore the authors find contributions coming from world-sheet instanton corrections in CP3. In particular it suggests that, in addition to the usual perturbative and strong coupling ( AdS ) expansions, a third natural expansion locus is the line where one of the two ’ t Hooft couplings vanishes and the other is finite. 

Q2. What is the common topic covered in the article?

Another topic covered there is that of “giant Wilson loops” [33–35], where in the supergravity dual (at least in AdS5 × S5) a fundamental string is replaced by a D-brane. 

Q3. How can the expectation value of the 1/6 BPS Wilson loop be expressed?

the expectation value of the 1/6 BPS Wilson loop can be expressed as an observable in the ABJM matrix model, and by analytical continuation in the lens space model. 

Q4. What is the nature of the matrix integrals?

they have a natural supergroup structure, i.e., they are super-matrix models [7,8], and are related to some previously studied bosonic matrix models [9,10] by analytical continuation [8]. 

Q5. What is the surface of a well studied toric Calabi–Yau manifold?

This surface is also the mirror to a well studied toric Calabi–Yau manifold known as local F0, where F0 = P1 × P1 is a Hirzebruch surface. 

Q6. What is the significance of the study?

An important breakthrough, which is the underpinning of the present study, was the work of Kapustin, Willett and Yaakov [4], who use the localization techniques of [5] to reduce the calculation of certain quantities in the gauge theory on S3 to finite dimensional matrix integrals. 

Q7. What is the value of the Wilson loop operator?

The solution of this matrix model allowed for the evaluation of the first exact interpolating function in this theory [8] giving a closed form expression for the expectation value of the 1/2 BPS Wilson loop operator of [7] at all values of the coupling. 

Q8. What is the connection between ABJM and a topological string theory?

Since CS and the matrix model are related to topological strings, the authors expect there to be a direct connection between ABJM theory and a topological string theory. 

Q9. What are the natural observables in CS theory?

Of course, the natural observables in CS theory are the partition function and Wilson loops, so these quantities were also studied earlier in the matrix models of CS (see, for example, [9,10,13,20–22]). 

Q10. What is the full expansion of the free energy on S3?

In Sect. 7 the authors show that the full 1/N expansion of the free energy on S3 is completely determined by a recursive procedure based on direct integration [28,29] of the holomorphic anomaly equations [30].