A Killing spinor on \mathbb{S}^{4}

Exact Results and Holography of Wilson Loops


Superconformal (Quiver) Gauge Theories

Soo-Jong Rey ,     Takao Suyama

School of Physics and Astronomy & Center for Theoretical Physics

Seoul National University, Seoul 141-747 KOREA

School of Natural Sciences, Institute for Advanced Study, Princeton NJ 08540 USA

sjrey@snu.ac.kr     suyama@phya.snu.ac.kr


Using localization, matrix model and saddle-point techniques, we determine exact behavior of circular Wilson loop in superconformal (quiver) gauge theories in the large number limit of colors. Focusing at planar and large ’t Hooft couling limits, we compare its asymptotic behavior with well-known exponential growth of Wilson loop in super Yang-Mills theory with respect to ’t Hooft coupling. For theory with gauge group SU coupled to fundamental hypermultiplets, we find that Wilson loop exhibits non-exponential growth – at most, it can grow as a power of ’t Hooft coupling. For theory with gauge group SU( and bifundamental hypermultiplets, there are two Wilson loops associated with two gauge groups. We find Wilson loop in untwisted sector grows exponentially large as in super Yang-Mills theory. We then find Wilson loop in twisted sector exhibits non-analytic behavior with respect to difference of the two ’t Hooft coupling constants. By letting one gauge coupling constant hierarchically larger/smaller than the other, we show that Wilson loops in the second type theory interpolate to Wilson loops in the first type theory. We infer implications of these findings from holographic dual description in terms of minimal surface of dual string worldsheet. We suggest intuitive interpretation that in both classes of theory holographic dual background must involve string scale geometry even at planar and large ’t Hooft coupling limit and that new results found in the gauge theory side are attributable to worldsheet instantons and infinite resummation therein. Our interpretation also indicates that holographic dual of these gauge theories is provided by certain non-critical string theories.

1 Introduction

AdS/CFT correspondence [1] between super Yang-Mills theory and Type IIB string theory on has been studied extensively during the last decade. One remarkable result obtained from the study is exact computation for expectation value of Wilson loop operators at strong coupling [2][3]. For a half-BPS circular Wilson loop, based on perturbative calculations at weak ’t Hooft coupling [4], exact form of the expectation value was conjectured in [5], precisely reproducing the result expected from the string theory computation [2], [3] and conformal anomaly therein. Their conjecture was confirmed later in [6] using a localization technique.

In this paper, we study aspects of half-BPS circular Wilson loops in supersymmetric gauge theories. We focus on a class of superconformal gauge theories — the (quiver) gauge theory of gauge group SU and fundamental hypermultiplets and quiver gauge theory of gauge group SU(SU and bifundamental hypermultiplets — and compute the Wilson loop expectation value by adapting the localization technique of [6]. We then compare the results with the super Yang-Mills theory, which is a special limit of the quiver gauge theory of gauge group SU() and an adjoint hypermultiplet. Their quiver diagrams are depicted in Fig. 1.

Figure 1: Quiver diagram of superconformal gauge theories under study: (a) theory with = SU and one adjoint hypermultiplet, (b) theory with =SU() and fundamental hypermultiplets, (c) theory with SU( SU() and bifundamental hypermultiplets. The theory is obtainable from theory by tuning ratio of coupling constants to 0 or . See sections 3 and 4 for explanations.

We show that, on general grounds, path integral of these superconformal gauge theories on is reducible to a finite-dimensional matrix integral. The resulting matrix model turns out very complicated mainly because the one-loop determinant around the localization fixed point is non-trivial. This is in shartp contrast to the super Yang-Mills theory, where the one-loop determinant is absent and further evaluation of Wilson loops or correlation functions is straightforward manipulation in Gaussian matrix integral.

Nevertheless, in the planar limit, we show that expectation value of the half-BPS circular Wilson loop is determinable provided the ’t Hooft coupling is large. In the large limit, the one-loop determinant evaluated by the zeta-function regularization admits a suitable asymptotic expansion. Using this expansion, we can solve the saddle-point equation of the matrix model and obtain large behavior of the Wilson loop expectation value. In super Yang-Mills theory, it is known that the Wilson loop grows exponentially large as becomes infinitely strong.

In gauge theory, we find that the Wilson loop expectation value grows exponentially, exactly the same as the super Yang-Mills theory. The result for gauge theory is surprising. We find that the Wilson loop is finite at large . This means that the Wilson loop exhibits non-exponential growth. The quiver gauge theory is also interesting. There are two Wilson loops associated with each gauge groups, equivalently, one in untwisted sector and another in twisted sector. We find that the Wilson loop in untwisted sector scales exponentially large, coinciding with the behavior of the Wilson loop super Yang-Mills theory and the gauge theory. On the other hand, the Wilson loop in twisted sector exhibits non-analytic behavior with respect to difference of two ’t Hooft coupling constants. We also find that we can interpolate the two surprising results in and gauge theories by tuning the two ’t Hooft couplings in theory hierarchically different. In all these, we ignored possible non-perturbative corrections to the Wilson loops. This is because, recalling the fishnet picture for the stringy interpretation of Wilson loops, the perturbative contributions would be the most relevant part for exploring the AdS/CFT correspondence and the holography therein.

We also studied how holographic dual descriptions may explain the exact results. Expectation value of the Wilson loop is described by worldsheet path integral of Type IIB string in dual geometry and that, in case the dual geometry is macroscopically large such as AdS, it is evaluated by saddle-points of the path integral – worldsheet configurations of extremal area surface. We first suggest that non-exponential growth of the Wilson loop arise from delicate cancelation among multiple — possibly infinitely many — saddle-points. This implies that holographic dual geometry of the gauge theory ought to be (AdS where the internal space necessarily involves a geometry of string scale. The string worldsheet sweeps on average an extremal area surface inside AdS, but many nearby saddle-point configurations whose worldsheet sweep two cycles over cancel among the leading, exponential contributions of each. We next suggest that Wilson loop in untwisted sector is given by a macroscopic string in AdS and hence grows exponentially with average of the two ’t Hooft coupling constants. In twisted sector, however, it is negligibly small and scales with difference of the two ’t Hooft coupling constants. This is again due to delicate cancelation among multiple worldsheet instantons that sweep around collapsed two cycles at the orbifold fixed point. We also demonstrate that Wilson loop expectation values are interpolatable between and behaviors (or vice versa) by tuning NS-NS 2-form potential on the collapsed two cycle from to or vice versa.

This paper is organized as follows. In section 2, we show that evaluation of the expectation value of the half-BPS circular Wilson loop in a generic superconformal gauge theory reduces to a related problem in a one-matrix model. The reduction procedure is based on localization technique and is parallel to [6]. Compared to [6], our derivations are more direct and elementary and hence makes foregoing analysis in the planar limit far clearer physicswise. In section 3, we evaluate the Wilson loop at large ’t Hooft coupling limit. Based on general analysis for one-matrix model (subsection 3.1), we evaluate the matrix model action which is induced by the one-loop determinant (subsection 3.2). As a result, we obtain a saddle-point equation whose solution provides the large ’t Hooft coupling behavior of the Wilson loop (subsection 3.3). In section 5, we discuss interpretation of these results in holographic dual string theory. For both and types, we argue contribution of worldsheet instanton effects can explain non-analytic behavior of the exact gauge theory results. Section 7 is devoted to discussion, including a possible implication of the present results to our previous work [7] (see also [8][9]) on ABJM theory [10]. We relegated several technical points in the appendices. In appendix A, we summarize Killing spinors on . In appendix B, we work out off-shell closure of supersymmetry algebra. In appendix C, we present asymptotic expansion of the Wilson loop. In appendix D, we present detailed computation of that arise in the evaluation of one-loop determinant.

Results of this work were previously reported at KEK workshop and at Strings 2009 conference. For online proceedings, see [11] and [12], respectively.

2 Reduction to One-Matrix Model

The work [6] provided a proof for the conjecture [4, 5] that the evaluation of the half-BPS Wilson loop in super Yang-Mills theory [2, 3] is reduced to a related problem in a Gaussian Hermitian one-matrix model. In this section, we show that the similar reduction also works for superconformal gauge theories of general quiver type. The resulting matrix model is, however, not Gaussian but includes non-trivial vertices due to nontrivial one-loop determinant.

2.1 From to

A shortcut route to an gauge theory of general quiver type — with matters in various different representations and coupling constants in different values — is to start with super Yang-Mills theory. In this section, for completeness of our treatment, we elaborate on this route. Let be the gauge group. The latter theory consists of a gauge field with , scalar fields and an Majorana-Weyl spinor , all in the adjoint representation of . The action can be written compactly as


where and


Note that the metric of the base manifold is taken in the Euclidean signature, while the ten-dimensional ’metric’ is taken Lorentzian with . As usual in the dimensional reduction, the derivatives other than are set to zero.

The action (2.1) is invariant under the supersymmetry transformations


where is a constant Majorana-Weyl spinor-valued supersymmetry parameter satisfying the chirality condition . In what follows, we rewrite the action (2.1) so that the resulting action provides a useful guide to deduce the action of an gauge theory with hypermultiplet fields of arbitrary representations.

We first choose which half of the supercharges of the supersymmetry is to be preserved. This choice corresponds to the choice of embedding the SU(2) R-symmetry of theory into the SU(4) R-symmetry of the theory. Consider one such embedding defined by the matrix


Its determinant is


so it is obvious that any transformation of the form


belongs to the SO(4) transformation acting on . Note that this transformation preserves the embedding (2.7). In the ten-dimensional language, SU(4) R-symmetry of the theory is realized as the rotational symmetry SO(6) of . Therefore, one embedding of SU(2) R-symmetry into SU(4) is chosen by selecting SU or SU. We choose the latter as the R-symmetry of the theories.

There is a U(1) subgroup of SU generated by . Let be an element of this U(1). This is -rotation in 67-plane and -rotation in 89-plane. In the following, we require that the supercharges preserved in theory should be invariant under the . For an infinitesimal , acts on the supersymmetry transformation parameter as


Therefore, should satisfy


selecting eight components out of the original sixteen ones.

The scalar fields with can be combined into the doublet () of SU as


and their conjugates . Gamma matrices are defined similarly in terms of . They satisfy


Note that, for arbitrary vectors and , one has


The Majorana-Weyl spinor is split into the chirality eigenstates with respect to as follows:


Both fermions are Majorana-Weyl. We further split into , which are eigenstates of


Note that is the generator for and hence satisfies


Now, are not Majorana-Weyl. In fact, they are related by charge conjugation


where is the index for the adjoint representation of and is the complex conjugation matrix. So, we shall denote by . Then, modulo a phase factor, is .

In terms of , , and , the action (2.1) can be written as


with the understanding that the dimensional reduction sets for . The supersymmetry transformations (2.5),(2.6) can be written as


Again, if obeys the projection condition (2.11), the action (2.19) has supersymmetry.

At this stage, we shall be explicit of representation contents of fields and their conjugates. Let be the generators of Lie in the adjoint representation. We also impose on the projection condition (2.11). In terms of them, the action (2.19) can be written as


where the gauge covariant derivatives are


The supersymmetry transformation rules are


The above action (2.25) is equivalent to the original action (2.1): we have just rewritten the original action in terms of renamed component fields. The supersymmetry transformations (2.29)-(2.33) are also equivalent to (2.5) - (2.6) in so far as is projected to supersymmetry as (2.11).

It turns out that the action (2.25) is invariant under supersymmetry transformations (2.29)-(2.33) even for in a generic representation of the gauge group , which can also be reducible. Therefore, (2.25) defines an gauge theory with matter fields in the representation and their conjugates.

It is also possible to treat quiver gauge theories on the same footing. We embed the orbifold action into SU. In this paper, we shall focus on quiver gauge theory. In this case, we should substitute


into (2.19). Note that the supersymmetry (2.29)-(2.33) is preserved even when the gauge coupling constant is replaced with the matrix-valued one:


In general, and can be extended to complex domain. Extension to is straightforward.

2.2 Superconformal symmetry on

Following [6], we now define the superconformal gauge theory on of radius . For definiteness, we consider the round-sphere with the metric induced through the standard stereographic projection. Details are summarized in Appendix A.

For this purpose, it also turns out convenient to start with super Yang-Mills theory defined on . To maintain conformal invariance, the scalars ought to have the conformal coupling to the curvature scalar of . The action thus reads


where . The action is invariant under the supersymmetry transformations


provided that and satisfy the conformal Killing equations:


Explicit form of the solution to these equations are given in Appendix A.

The action of an gauge theory on with a hypermultiplet of representation can be deduced easily as in the previous subsection. One obtains


where . The action is invariant under the superconformal symmetry

where satisfies the conformal Killing equations (2.39) in addition to the projection condition (2.11). We emphasize that this is the transformation of the superconformal symmetry, not just the Poincaré part of it. This can be checked explicitly, for example, by examining the commutator of two transformations on the fields.

We find it convenient to define a fermionic transformation corresponding to the above superconformal transformation . It is obtained easily by the replacement and with a real Grassmann parameter. The resulting transformation is


where now and are bosonic SO(9,1) Majorana-Weyl spinors satisfying projection (2.11) and conformal Killing equation (2.39).

2.3 Localization

By extending the localization technique of [6], we now show that computation of Wilson loop expectation value in superconformal gauge theory of quiver type can be reduced to computation of a one-matrix integral.

Let be a fermionic transformation. Suppose that an action under consideration is invariant under . Then, the following modification


does not change the partition function provided that


Likewise, correlation functions remain unchanged if operators under consideration are -invariant. We shall choose such that the bosonic part of is positive semi-definite. For this choice, since can be chosen to be an arbitrary value, we can take the limit so that the path-integral is localized to configurations where the bosonic part of vanishes. It will turn out later that the vanishing locus of is parametrized by a constant matrix. This is why the evaluation of the expectation value of a -invariant operator reduces to a matrix integral. The action of the resulting matrix model is the sum of evaluated at the vanishing locus and the one-loop determinant obtained from the quadratic terms of when expanded around the vanishing locus.

One might think that the fermionic transformation defined in the previous section can be used as above. In fact, is a sum of bosonic transformations, and therefore, (2.43) appears to hold as long as is invariant under the transformations. The problem of this choice is that is such a sum only on-shell. According to [13],[14] and [15], has to be modified so that the resulting closes to a sum of bosonic transformations for off-shell.

To this end, we introduce auxiliary fields , and . They transform in the adjoint, and representations of the gauge group , respectively. Utilizing them, we modify the action (2.40) in a trivial manner:


Evidently, this action is physically equivalent to the original one. The modified action (2.44) is now invariant under the following transformations:


To make close to a sum of bosonic transformations off-shell, the spinors , , should be chosen appropriately out of . Details on them are summarized in Appendix B. With the correct choice, closes, for example, on as follows:


This shows that is a sum of a diffeomorphism on , a gauge transformation and a global SU transformation. In particular, notice that turns out to be independent of . The action of on the auxiliary fields is slightly different. For example, on , one obtains


Here, the index does not transform as a part of the four-vector on . This is not a problem since is contracted with in defined below, and not with some other four-vectors. The defined above is the right transformation available for the localization procedure.

We are at the position to choose . We take




Note that is a scalar with respect to a particular combination of the diffeomorphism on , the gauge transformation and the global SU transformation. This follows from the identities for the spinors, for example,


and similar ones for and which are summarized in Appendix A and B. Therefore, (2.43) is satisfied with this choice, as required.

After straightforward but tedious algebra, one obtains the bosonic part of expressed as


where is the polar angle on , and


Here, the hatted indices are the Lorentz ones. The above expression shows that, after a suitable Wick rotation for and the auxiliary fields, the bosonic part of is positive semi-definite. Therefore, by taking the limit , the path-integral is localized at the vanishing locus of . It turns out that, as in [6], non-zero fields at the vanishing locus are


where is a constant Hermitian matrix. The coefficients are chosen for later convenience.

Now, the path-integral is reduced to an integral over the Hermitian matrix . The action of the corresponding matrix model is a sum of the action (2.44) evaluated at the vanishing locus and the one-loop determinant for the quadratic terms in . Note that higher-loop contributions vanish in the large limit since plays the role of the loop-counting parameter. At the vanishing locus, the action (2.44) takes the value


An important difference from the super Yang-Mills theory is that the one-loop determinant around the vanishing locus does not cancel and has a complicated functional structure. In the next section, we show that the presence of the non-trivial one-loop determinant is crucial for determining the large ’t Hooft coupling behavior of the half-BPS Wilson loop.

The half-BPS Wilson loop of gauge theory has the following form:


The functions , are chosen appropriately to preserve a half of the superconformal symmetry. We shall choose to be the great circle at the equator of (i.e. ) specified by


and as


For this choice, one can show that


where . See Appendix A for the explicit expressions of . This implies that is invariant under due to the identity


Thus, we have shown that is calculable by a finite-dimensional matrix integral. The operator whose expectation value in the matrix model is equal to is


Notice that it is solely governed by the constant-valued, Hermitian matrix . This enables us to compute the Wilson loops in terms of a matrix integral. This observation will also play a role in identifying holographic dual geometry later.

3 Wilson loops at Large ’t Hooft Coupling

We have shown that evaluation of the Wilson loop is reduced to a related problem in a one-Hermitian matrix model. Still, the matrix model is too complicated to solve exactly. In the following, we focus our attention to either the superconformal gauge theory of type with U coupled to fundamental hypermultiplets and of type with UU, both at large limit. For these theories, we show that the large ’t Hooft coupling behavior is determinable by a few quantities extracted from the one-loop determinant. This allows us to exactly evaluate the Wilson loop in the large and large ’t Hooft coupling limit.

3.1 General results in one matrix model

Consider a matrix model for an Hermitian matrix . In the large limit, expectation value of any operator in this model is determinable in terms of eigenvalue density function of the matrix . By definition, is normalized by


Let denote the support of . We assume that1


Expectation value of the operator