1 Introduction

SNUST 080903

arXiv:0809.3786[hep-th]


Wilson Loops in Superconformal Chern-Simons Theory

and

Fundamental Strings in Anti-de Sitter Supergravity Dual

Soo-Jong Rey,     Takao Suyama,     Satoshi Yamaguchi

School of Physics & Astronomy, Seoul National University, Seoul 151-747 KOREA

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

ABSTRACT

We study Wilson loop operators in three-dimensional, superconformal Chern-Simons theory dual to IIA superstring theory on AdS. Novelty of Wilson loop operators in this theory is that, for a given contour, there are two linear combinations of Wilson loop transforming oppositely under time-reversal transformation. We show that one combination is holographically dual to IIA fundamental string, while orthogonal combination is set to zero. We gather supporting evidences from detailed comparative study of generalized time-reversal transformations in both D2-brane worldvolume and ABJM theories. We then classify supersymmetric Wilson loops and find at most supersymmetry. We next study Wilson loop expectation value in planar perturbation theory. For circular Wilson loop, we find features remarkably parallel to circular Wilson loop in super Yang-Mills theory in four dimensions. First, all odd loop diagrams vanish identically and even loops contribute nontrivial contributions. Second, quantum corrected gauge and scalar propagators take the same form as those of super Yang-Mills theory. Combining these results, we propose that expectation value of circular Wilson loop is given by Wilson loop expectation value in pure Chern-Simons theory times zero-dimensional Gaussian matrix model whose variance is specified by an interpolating function of ‘t Hooft coupling. We suggest the function interpolates smoothly between weak and strong coupling regime, offering new test ground of the AdS/CFT correspondence.

1 Introduction

The proposal of holographic principle put forward by Maldacena [1] has changed fundamentally the way we understand quantum field theory and quantum gravity. In particular, the AdS-CFT correspondence between super Yang-Mills theory and Type IIB superstring on AdS, followed by diverse variant setups thereafter, enormously enriched our understanding of nonperturbative aspects of gauge and string theories. In exploring holographic correspondence between gauge and string theory sides, an important class of physical observable is provided by semiclassical fundamental strings and D-branes in string theory side and by topological defects in gauge theory side. In particular, the Wilson loop operator [2] extended to super Yang-Mills theory was proposed and identified with macroscopic fundamental string on AdS [3, 4]. During the ensuing development of holographic correspondence between gauge and string theories, the proposal of [3, 4] became an essential toolkit for extracting physics from diverse variants of gauge-gravity correspondence. Among those further developments, one important step was the observation that the exact expectation value of the -supersymmetric circular Wilson loop is computable by a Gaussian matrix model [5, 6, 7].

Recently, Aharony, Bergman, Jafferis and Maldacena (ABJM) [8] put forward a new account of the AdS-CFT correspondence: three-dimensional superconformal Chern-Simons theory dual to Type IIA string theory on AdS. Both sides of the correspondence are characterized by two integer-valued coupling parameters and . On the superconformal Chern-Simons theory side, they are the rank of product gauge group U( and Chern-Simons levels , respectively. On the Type IIA string theory side, they are related to spacetime curvature and Ramond-Ramond fluxes, all measured in string unit. Much the same way as the counterpart between super Yang-Mills theory and Type IIB string theory on AdS, we can put the new correspondence into precision tests in the planar limit:

(1.1)

by interpolating ‘t Hooft coupling parameter between superconformal Chern-Simons theory regime at and semiclassical AdS string theory regime at .

The purpose of this paper is to identify Wilson loop operators in the ABJM theory which corresponds to a macroscopic Type IIA fundamental string on AdS and put them to a test by studying their quantum-mechanical properties. The proposed Wilson loop operators involve both gauge potential and a pair of bi-fundamental scalar fields, a feature already noted in four-dimensional super Yang-Mills theory. Typically, functional form of the Wilson loop operator is constrained severely by the requirement of affine symmetry along the contour , by superconformal symmetry on , and by gauge and SU(4) symmetries. We shall find that, in the ABJM theory, there are two elementary Wilson loop operators determined by these symmetry requirement:

(1.2)

We first determine conditions on in order for the Wilson loop to keep unbroken supersymmetry. We shall find that there is a unique Wilson loop preserving of superconformal symmetry. We shall then study vacuum expectation value of these Wilson loops both in planar perturbation theory of the ABJM theory and in minimal surface of the string worldsheet in AdS. We also study determine functional form of from various symmetry considerations. We shall then propose that the linear combination of Wilson loops:

(1.3)

is identifiable with appropriate Type IIA fundamental string configuration and that the opposite linear combination is mapped to zero. We gather evidences for these proposal from detailed study for relation between the ABJM theory and the worldvolume gauge theory of D2-branes, from identification of time-reversal invariance in these theories, and from explicit computation of Wilson loop expectation values in planar perturbation theory.

Out of these elementary Wilson loops, we can also construct composite Wilson loop operators encompassing the two product gauge groups, for example, or , etc. As in four-dimensional super Yang-Mills theory, we expect that these Wilson loop operators constitute an important class of gauge invariant observables, providing an order parameter for various phases of the ABJM theory. In fact, even in pure Chern-Simons theory (obtainable from ABJM theory by truncating all matter fields), it was known that expectation value of Wilson loop operators yields nontrivial topological invariants [9, 10] 111See [11] for an earlier discussion on Wilson loops in ABJM theory..

We organized this paper as follows. In section 2, we collect relevant results on macroscopic IIA fundamental string in AdS, adapted from those obtained in AdS previously. We discuss two possible configurations with different stabilizer subgroup and number of supersymmetries preserved. In section 3, we formulate Wilson loop operators in ABJM theory. In subsection 3.2, we propose Wilson loop operators and constrain their structures by various symmetry considerations. We find from these that, up to SU(4) rotation, functional form of the Wilson loop operator is determined uniquely. Still, this leaves separate Wilson loops for U() and gauge groups, respectively. To identify relation between the two, in subsection 3.3, we first recall the argument of [12, 13, 14, 15, 16] relating three-dimensional super Yang-Mills theory and ABJM superconformal Chern-Simons theory 222This procedure is first proposed by Mukhi and Papageorgakis for relating (variants of) Bagger-Lambert-Gustavsson (BLG) theory[17, 18] to 3-dimensional super Yang-Mills theory.. We then identify that fundamental IIA string ending on D2-brane couples to diagonal linear combination of U() and . In section 4, we study supersymmetry condition of the Wilson loop operator and deduce that tangent field along the contour should be constant. From this, we find that unique supersymmetric Wilson loop operator is the one preserving of the superconformal symmetry. In section 5, we revisit the time-reversal symmetry in ABJM theory. Based on the results of sections 3 and 4, we find that one combination of the elementary Wilson loops with a definite time-reversal transformation is dual to a fundamental IIA string on AdS, while orthogonal combination is mapped to zero. In section 6, we study expectation value of the Wilson loop operator to all orders in planar perturbation theory. For straight Wilson loop operator, we find that Feynman diagrams vanish identically at each loop order. For circular Wilson loop operator, we find that Feynman diagrams vanish at one loop order, nonzero at two loop order and zero again at three loop order. Remarkably, the two loop contribution consists of a part exactly the same as one-loop part of Wilson loop in super Yang-Mills theory and another part exactly the same as unknotted Wilson loop in pure Chern-Simons theory. Up to three-loop orders, all Feynman diagrams involve gauge and matter kinetic terms only. Features of full-fledged superconformal ABJM theory, in particular Yukawa and sextet scalar potential, begin to enter at four loops and beyond. Nevertheless, we show that the Feynman diagrams vanish identically for all odd number of loops. In other words, expectation value of the ABJM Wilson loop operator is a function of . In section 7, based on the results of section 6 and under suitable assumptions, we make a conjecture on the exact expression of circular Wilson loop expectation value in terms of a Gaussian matrix model and of unknot Wilson loop of the pure Chern-Simons theory. To match with weak and strong coupling limit results, variance of the matrix model ought to be a transcendental interpolating function of the ‘t Hooft coupling. Since this is different from super Yang-Mills theory, we discuss issues associated with the interpolating function. Section 8 is devoted to discussions for future investigation. In appendix A, we collect conventions, notations and Feynman rules. In appendix B, we give details of analysis for Wilson loops of generic contour. In appendix C, we recapitulate the one-loop vacuum polarization in ABJM theory, obtained first in [19]. In appendix D, we give details for the analysis of three-loop contributions.

While writing up this paper, we noted the papers [20, 21] posted on the arXiv archive, which have some overlap with ours. We also found [22] discuss some closely related issue.

2 Macroscopic IIA Fundamental String in AdS

We begin with strong ‘t Hooft coupling regime, . In this regime, by the AdS/CFT correspondence, IIA string theory on AdS is weakly coupled and provides dual description to strongly coupled ABJM theory. As shown in [3, 4], correlation function of the Wilson loop operators is calculated by the on-shell action of fundamental string whose worldsheet boundaries at the boundary of AdS space are attached to each Wilson loop operators. Following this, we shall consider a macroscopic IIA fundamental string in AdS and compute expectation value of the Wilson loop operator for a straight or a circular path.

The radius of the AdS is as measured in unit of the IIA string tension. IIA string worldsheet configurations corresponding to straight and circular Wilson loops are exactly the same as the corresponding IIB string worldsheet configurations in AdS background. The results are 333Our convention for the relation between the IIA string coupling and rank of ABJM theory is .

(2.1)

for timelike straight path  [3, 4] and spacelike circular path  [23], respectively. Extended to multiply stacked strings of same orientation, the ratio between the two Wilson loops is given by

(2.2)

In IIB string theory, both string configurations are known to be supersymmetric. In section 7, we shall try to relate these string theory results with perturbative computations in superconformal Chern-Simons theory side.

We briefly recapitulate how to get the above result. In the limit , the string becomes semiclassical and sweeps out a macroscopic minimal surface in AdS-space. The metric of AdS is expressed in Poincaré coordinates as

(2.3)

In this coordinate system, the boundary is located at . We choose a macroscopic string configuration in the static gauge and it corresponds to a timelike straight Wilson loop sitting at . Here, following the prescription of [3, 4], we regularize the AdS-space to , remove divergence (corresponding to self-energy) and finally lift off the regularization  444Alternatively, we can prescribe renormalization scheme by adding a boundary counter-term, as in [24]. The result is the same.. The renormalized string worldsheet action is and the result (2.1) follows.

After Wick rotation, timelike straight Wilson loop can be conformally transformed to spacelike circular Wilson loop. Let us examine this string configuration in Euclidean AdS. The metric of Euclidean AdS is written as

(2.4)

We choose the fundamental string configuration in the static gauge and , and we also take an ansatz . It corresponds to a circular Wilson loop whose center sits at . The string worldsheet action is given by

(2.5)

where . The solution with circular boundary is , and its on-shell action is written as

(2.6)

Here again, we regularized the AdS-space to . After removing the divergent part, we obtain the renormalized on-shell action as . Expectation value of the Wilson loop is and the result (2.1) follows.

We now would like to identify spacetime symmetries preserved by these classical string solutions. Each classical string configuration wraps a suitably foliated AdS submanifold in AdS, so it preserves SLSO(2) symmetry of the isometry SO(2,3) of AdS. If the string were sitting at a point in , the isometry group SU(4) of is broken to stabilizer subgroup U(1) SU(3). If the string were distributed over in , the isometry group SU(4) is broken further to stabilizer subgroup U(1)SU(2)SU(2). Variety of other configurations are also possible, but we shall primarily focus on these two configurations. In the background AdS, there are 24 supercharges. They form a multiplet of the SO(2,3)Sp(4,) and the SU(4) isometry groups. We can see that these two strings are supersymmetric by identifying supercharges that annihilate each configurations.

The first configuration turns out supersymmetric. Unbroken supersymmetries ought to be organized in multiplets of the stabilizer subgroup SL SU(3). Branching rules of SO(2,3)SU(4) into SL SU(3) follows from

(2.7)

Therefore, the minimal possibility is of SL SU(3). Noting that of SU(3) is a complex representation, we deduce that the number of unbroken supercharges is either or . There is no possibility that all the supercharges are preserved since the configuration does not preserve the SU(4) symmetry. So, we conclude that the string sitting at a point on preserves 12 of the 24 supercharges.

The second configuration is supersymmetric. Branching rules of SO(2,3)SU(4) into SLSU(2)SU(2) follow from

(2.8)

The minimum possibility is . Since each pair are charged oppositely under U(1), we deduce that possible number of unbroken supercharges are 4, or 16 (apart from 12 or 24 we have already analyzed). We see that a supersymmetric string distributed over preserves at least 4 of the 24 supercharges.

In summary, for both straight and circular string, we identified two representative supersymmetric configurations. A configuration localized in preserve supercharges (corresponding to -BPS) and SLSO(2) U(1) SU(3) isometries. A configuration distributed over in preserves at least supercharges (corresponding to -BPS) and SLSO(2)U(1)SU(2)SU(2) isometries.

3 Wilson Loop: Proposal and Simple Picture

3.1 Wilson Loop in Super Yang-Mills Theory

We first recapitulate a few salient features of Wilson loop operator in four-dimensional super Yang-Mills theory and its holographic dual, macroscopic Type IIB superstring in AdS. On , the Wilson loop operator for defining representation was proposed [3, 4] to be

(3.1)

Here, is a vector specifying in , is a vector in SO(6) internal space, and where s are a set of Lie algebra generators, and Tr is trace in fundamental representation. It is motivated by ten-dimensional Wilson loop operator Tr over a path specified by on D9-brane worldvolume. T-dualizing to D3-brane, the gauge potential and the path are split to and , and , respectively. We then obtain (3.1), where the vector is described in terms of internal coordinates as:

(3.2)

We can also motivate that this Wilson loop operator is related to Type IIB fundamental string in AdS by noting that that the gauge potential lives in is conformally equivalent to AdS:

(3.3)

In this situation, the Wilson loop sweeps out a path in or its conformal equivalent in AdS.

Depending on the choice of the velocity vector , the Wilson loop preserves different subgroup of the SO(6) R-symmetry. If , the Wilson loop preserves SO(6). If is -independent, the Wilson loop preserves SO(5) subgroup of SO(6) since can be rotated by a rigid SO(6) rotation to, say, . Moreover, may also develop a discontinuity at some . In holographic dual, the Wilson loop expectation value is given by a saddle-point of the string worldsheet whose boundary at AdS infinity is prescribed by the vectors of the Wilson loop. In general, there can be a continuous family of string worldsheets satisfying the same boundary condition, parametrized by zero-modes. In that case, each worldsheet preserves a subgroup smaller than the subgroup preserved by the corresponding Wilson loop. In order to restore the subgroup preserved by the Wilson loop, one then needs to integrate over a parameter space of the zero-modes for the string worldsheet.

One can also study the Wilson loop operators averaged over the boundary condition . For example,

(3.4)

is an averaged Wilson loop operator in which the vector is averaged to over a domain . Each configuration of preserves different subgroup of SO(6) symmetry, so the above average Wilson loop operator would retain a stabilizer subgroup common to each of in .

3.2 Wilson Loops in Superconformal Chern-Simons Theory

In this subsection, paving steps parallel to the four-dimensional super Yang-Mills theory, we shall construct a Wilson loop operator in the ABJM theory and find an interpretation from holographic dual side. In particular, we pay attention to features that contrast the ABJM Wilson loop operators against the Wilson loop operators in super Yang-Mills theory.

Our proposal for the Wilson loop operators in the ABJM theory is as follows. Denote coordinates of as and of SU(4) internal space as . With two gauge fields and of U() and gauge groups, respectively, we can construct two types of Wilson loop operators associated with each gauge fields. Consider the U() gauge group. Our proposal of the U() Wilson loop operator is

(3.5)

Here, and , where ’s are Lie algebra generators of U() gauge group. Again, the vector field specifies the path in and is a tensor in SU(4) internal space. A choice that is a direct counterpart of (3.1) is

(3.6)

Since , eigenvalues of are .

We also motivate functional form of the Wilson loop from the following symmetry considerations:

  • Wilson loop describes a trajectory of a heavy particle probe. Charge of the particle is characterized by a representations under U() and gauge groups. Mass of the particle is set by scalar fields and should carry scaling dimension 1. In (2+1) dimensions, the scalar fields have scaling dimension 1/2. It also should transform in adjoint representation of U. These requirements fix uniquely the requisite combination as .

  • Functional form of the tensor given in (3.6) is largely determined by spacetime translational symmetry and by affine reparametrization and parity symmetries along the path . Transitive motion on embedding space is described by for a constant . The tensor is manifestly invariant under such motion since it depends only on .

  • Affine reparametrization is induced by . The tensor is manifestly invariant under such motion since it transforms with Jacobian . This cancels against the Jacobian induced by the measure .

Likewise, our proposal for the Wilson loop operator of gauge group is

(3.7)

where .

From ABJM theory viewpoint, various composites of these Wilson loop operators are possible (in addition to the choice of and ). Taking the above Wilson loop operators as building blocks, composite Wilson loops involving both gauge groups are constructible. For example, one can construct

(3.8)
(3.9)

etc. However, under suitable conditions, they turn out not independent one another. For example, at large limit, expectation values of these composite Wilson loop operators are all equal because of large factorization property. One might have expected that the composites are further restricted if the Wilson loops are to preserve part of the supersymmetry. This is not so, since supersymmetry acts on and independently.

In comparison with super Yang-Mills theory, one distinguishing feature of the ABJM theory is that there are two sets of Wilson loops, one for U() gauge group and another for gauge group. From holographic perspectives, this raises a puzzle. We expect that these Wilson loops are mapped to a string. While there are two variety of Wilson loops in the ABJM theory, there is one and only one fundamental string in AdS. We first resolve this puzzle by analyzing the way a fundamental string is coupled to a stack of D2-branes, whose worldvolume gauge theory is in turn related to the ABJM theory by moving away appropriately from conformal point.

3.3 Fundamental String Ending on D2-Brane

Consider a D2-brane and a macroscopic IIA fundamental string ending on it. From IIA supergravity field equations in the presence of the string and the D2-brane, we see that the string endpoint on the D2-brane carries an electric charge of the worldvolume gauge field of the D2-brane. How is the electric charge related to charges in the ABJM theory?

Answer to this question is obtainable simply by identifying relation between the D2-brane worldvolume gauge field and the two gauge fields in the ABJM theory. The identification is in fact already made in [12]. By giving a nonzero vacuum expectation value to one of the bi-fundamental scalar fields in ABJM theory, one linear combination of the gauge fields becomes massive. Integrating out the massive gauge field, we are left with orthogonal linear combination of the gauge fields. This is identified with the D2-brane worldvolume gauge field . Relevant part of the ABJM Lagrangian is

(3.10)

The last line is to indicate how an external source with gauge currents couples to the two ABJM gauge potentials.

Turn on vacuum expectation value of one of the scalar fields, say, the real part of :

(3.11)

We also decompose the two gauge potentials as

(3.12)

The corresponding field strengths are

(3.13)

We then find that the Chern-Simons terms are reduced to

(3.14)

while the kinetic terms are reduced to

(3.15)

The equations of motion for

(3.16)

can be solved perturbatively at large . Collecting terms in increasing power of derivatives and redefining , we find that the Lagrangian is reduced to

(3.17)

To retain nontrivial gauge dynamics at quadratic order and suppress all higher order terms, we take the scaling limit:

(3.18)

We see that, around the vacuum given by the above expectation value, the ABJM theory is reduced to maximally supersymmetric U() gauge theory of the gauge potential below the energy scale set by , viz. it describes worldvolume dynamics of the D2-brane.

From the Lagrangian, we derive equations of motion for the gauge potential as

(3.19)

If a fundamental string ends on the D2-brane, it acts as a source to the worldvolume gauge field . In the scaling limit that reduces ABJM theory to (2+1)-dimensional super Yang-Mills theory, all but the first term drop out. This in turn implies that the string endpoint creates one unit (in unit of ) of from ABJM currents. We also note that the non-minimal coupling of to the current is suppressed in the above scaling limit.

In this section, we identified that is the gauge field for the D2-brane worldvolume dynamics, while is decoupled from the dynamics. Therefore, a fundamental string ending on D2-brane is described by the Wilson loop operator composed solely of (plus an appropriate combination of eight scalar fields). We emphasize that, under time-reversal, this Wilson loop operator transforms in the standard way. For timelike , the representation of the Wilson loop is mapped to conjugate representation but the internal tensor remains intact. For spacelike , representation remains intact but the internal tensor is mapped to conjugate tensor .

4 Supersymmetric Wilson loop

We now would like to understand under what choices of and the proposed Wilson loop preserves some of the superconformal symmetry. The same question was addressed previously for super Yang-Mills theory [25] and for the holographic dual [26]. There, assuming that the Wilson loop sweeps a calibrated surface in , it was found that the Wilson loop preserving of the superconformal symmetry ought to lie in on either a timelike straight path or a spacelike circular path. Here, we shall check if the same choice of of the ABJM Wilson loop operators is supersymmetric. More general choice of the contour will be discussed later in this section.

Begin with the ABJM Wilson loop over a timelike straight path. By a Lorentz boost, we can always bring the path to , so . We first focus on the U() Wilson loop operator:

(4.1)

As in [25, 26], we take the ansatz that is a -independent, constant tensor.

The Poincaré supersymmetry transformations for the gauge and scalar fields are [30, 31, 32]

(4.2)
(4.3)

where are supersymmetry parameters satisfying the following relations:

(4.4)

Consider a point along the contour . The supersymmetry variation of the integrand in the exponent of (4.1) becomes

(4.5)

In order to be supersymmetric, the following two equations must be satisfied for some of the supersymmetry parameters:

(4.6)

By unitary transformation, diagonalize the constant Hermitian matrix as

(4.7)

In this frame, the supersymmetry condition (4.6) reads

(4.8)

We see that each eigenvalues must take values in order to satisfy the conditions (4.8). If one of the eigenvalues, say , is not , since the eigenvalues of are , (4.8) implies . In this case, the second relation of (4.4) reads for as well and no supersymmetry is preserved.

Modulo overall sign and permutations of the eigenvalues, there are three possible combinations. We examine each of them separately.

  • : This configuration preserves full SU(4) symmetry. The supersymmetry conditions (4.8) now read

    (4.9)

    These two equations cannot be satisfied simultaneously because of the reality condition (4.4). So, there is no supersymmetric Wilson loop with unbroken SU(4) symmetry. The same conclusion holds for .

  • : This configuration breaks SU(4) to SU(3)U(1). From the supersymmetry condition (4.8) for and and the first relation of (4.4), it follows that . This and the second relation of (4.4) imply that for all . Again, there is no supersymmetric Wilson loop with unbroken SU(3)U(1) symmetry. The same conclusion holds for .

  • : This configuration breaks SU(4) to SU(2)SU(2)U(1). In this case, supersymmetry parameters and satisfying the projection conditions:

    (4.10)

    exists. Other components of should vanish. We thus find that this Wilson loop preserves real supercharges. Since conformal supersymmetry transformations of are obtainable from Poincaré supersymmetry by the substitution , we also find that this Wilson loop preserves real conformal supercharges. We conclude that this Wilson loop preserves of the superconformal symmetry.

In summary, the supersymmetric Wilson loop in ABJM theory is unique: it has the tensor which has maximal rank , preserves SU(2)SU(2)U(1) symmetry of SU(4), and corresponds to a -BPS configuration of the superconformal symmetry 555There are other supersymmetric configurations. For example, a -BPS configuration is obtainable by and . However, since , this configuration is actually a generating functional of all -BPS local operators. A direct counterpart in super Yang-Mills theory is the and configuration. Again, with , this Wilson loop is a generating functional of -BPS local operators [27] (see also [28, 29]). .

Actually, the Wilson loop operator (3.5) is closely related to the Wilson loop considered in [33] in superconformal Chern-Simons theory. The -BPS configuration we found above is the same as the -BPS configuration of the superconformal symmetry: for a straight timelike path, both preserves two Poincaré supersymmetries and two conformal supersymmetries. So, features we find in this paper ought to hold to various superconformal Chern-Simons theories.

Notice that the tensor of the -BPS configuration has the properties ( positive integer)

(4.11)

Though trivial looking, these properties will play a crucial role when we evaluate in the next section the Wilson loop expectation value explicitly in planar perturbation theory.

We can also generalize the supersymmetric Wilson loops to a general contour specified by tangent vector . The supersymmetry condition now reads

(4.12)

We assume that is smooth, implying that is a smooth function of . We also set using the reparametrization invariance. The important point is that (4.12) ought to satisfy the supersymmetry conditions at each . Without loss of generality, we assume at that and the only non-zero components of are and : these are the eigenstates of with eigenvalue and , respectively. It is then possible to show that (4.12) allows only a constant and . The details of the proof of this statement is given in Appendix B. In plain words, tangent vector along the contour should remain constant. We conclude that the Wilson loop is supersymmetric only if is a straight line. The circular Wilson loop, which is a conformal transformation of this supersymmetric Wilson loop, is annihilated not by the Poincaré supercharges, but by linear combinations of the Poincaré supercharges and the conformal supercharges. The conformal transformation on cannot affect . So, is also the tensor relevant for the circular supersymmetric Wilson loops.

Still, the above result poses a puzzle. We argued that the Wilson loops proposed are unique in the sense that the supersymmetry considerations fix its structure completely. We also found that these Wilson loops preserve of the supersymmetry, but no more. On the other hand, the macroscopic IIA fundamental string preserves of the supersymmetry. At present, we do not have a satisfactory resolution. We expect that the supersymmetric Wilson loop corresponds to a string worldsheet whose location on is averaged over, perhaps, in a manner similar to the prescription (3.4). An encouraging observation is that the R-symmetry preserved by the Wilson loop is the same as the isometry preserved by the string smeared over in , and the number of preserved supercharges also match. This also fits to the observation that above cannot be written as (3.6) for any choice of since the trace of (3.6) does not vanish.

5 Consideration of Time-Reversal Symmetry

Though it involves Chern-Simons interactions, the ABJM theory is invariant under (suitably generalized) time-reversal transformations. This also fits well with the observation in section 3.2 that, by vacuum expectation value of scalar fields, the ABJM theory is continuously connected to the worldvolume gauge theory of multiple D2-branes. The latter theory is invariant under parity and time-reversal transformations. In section 3.2, we also identified as the right combination of the ABJM gauge potentials that couples to the current of the string endpoint on D2-brane. We shall now combine this observation and time-reversal transformation properties to identify , where

(5.1)

as the timelike Wilson loop dual to the fundamental IIA string. We shall now show that (5.1) transforms under the time-reversal precisely the same as the D2-brane worldvolume gauge potential that couple to the fundamental string. Moreover, since the other orthogonal combination is not present in the worldvolume gauge theory of multiple D2-branes, we are led to identify that expectation value of Wilson loops for the other combination vanishes identically:

(5.2)

Consider a timelike Wilson loop in . We take its path along the time direction, . By definition,

(5.3)

where denotes exponent of the Wilson loop:

(5.4)

Under the time-reversal transformation, . In the ABJM theory, this is adjoined with involution that exchanges the two gauge groups U() and . The resulting generalized time-reversal transformation then acts on relevant fields as

(5.5)

Being anti-linear, also acts as

(5.6)

Moreover, since the path is timelike, also reverses ordering of the path. To bring the path ordering back, we take transpose of products of s inside trace. Together with minus sign from time reversal, the generators are mapped to . These are the generators for the complex conjugate representation. Thus, the exponent of the timelike Wilson loop transforms as

(5.7)

where

(5.8)

We see that the time-reversal acts on the Wilson loop as

(5.9)

Notice, however, that does not change the path and the internal tensor .

With (5.9), we identify that (5.1) is the linear combination of elementary Wilson loops that transform under the generalized time-reversal transformation :