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Supersymmetric Wilson Loops in N=6 Super Chern-Simons-matter theory

We study supersymmetric Wilson loop operators in ABJM theory from both sides of the AdS_4/CFT_3 correspondence. We first construct some supersymmetric Wilson loops. The perturbative computations are performed in the field theory side at the first two orders. A fundamental string solution ending on a circular loop is also studied.


Introduction
Four-dimensional N = 4 super Yang-Mills theory is one of the most remarkable field theories. Among other things, this theory appears as low energy effective theory of multi D3-branes. This leads to the famous AdS 5 /CF T 4 correspondence [1,2,3,4]. Besides the local operators, supersymmetric Wilson loops play an important role in this correspondence. The vacuum expectation value (VEV) of half-BPS straight Wilson lines equals to unit due to the global supersymmetries they preserve. But the vacuum expectation value of half-BPS circular Wilson loop depends on the coupling constant in a nontrivial way. The computation of this VEV can be reduced to a computation via a Gaussian matrix model. Evidence of this reduction was found first by direct perturbative calculations [5]: in the Feynman gauge, the contributions from the diagrams which are not ladder/rainbow diagrams cancel each other. Later, this reduction was obtained from the conformal transformation linking the circular loop and the straight line [6]. Recently this reduction was obtained through localization technique [7]. For the Wilson loop in the fundamental representation of the gauge group, the VEV computed via this matrix model has the same large N, large λ behavior as the one from the string theory dual description in terms of fundamental strings in AdS 5 ending on this loop [8,9,10,11]. Similar conclusion was found for Wilson loops in symmetric (anti-symmetric) representation, in which case a better dual description is in terms of D3 (D5) branes [12,13,14,15,16]. Various aspects of Wilson loops with less supersymmetries were discussed in [17]- [24].
The low energy effective theory of multi M2-branes and M5-branes is one of the most important subjects in M-theory. For multi M2-branes in flat eleven dimensional spacetime, the low energy effective theory is the IR limit of N = 8 three dimensional super Yang-Mills theory. The construction of N = 8 three dimensional conformal field theory for multi-membranes has been a vexing question for more than ten years. Recently, based on early works [25], this question was addressed [26,27] with the help of a novel 3algebra. The theory based on seemingly the only unitary 3-algebra was found to be a theory for two M2-branes on an orbifold [28,29]. More interestingly it could be rewritten as a super-Chern-Simons theory of two gauge groups with opposite level [30]. This fact, combined with the earlier attempts of constructing three-dimensional super-conformal field theory as Chern-Simons theory [31,32], inspired people to search for the Chern-Simons-matter theory with higher supersymmetry. In [33], the one with N = 4 supersymmetry has been constructed. Very recently, a N = 6 super Chern-Simons-matter theory was proposed in [34] to describe arbitrary number of M2-branes on the orbifold C 4 /Z k (For some other discussions on three-dimension N = 4, 5, 6 super Chern-Simons-matter theories, see [35]- [39].) . In [34], the dual M/string description of large N limit of three-dimensional conformal theory was discussed. Depending on different values of the parameters in the field theory, this so-called ABJM theory can be dual to M-theory on AdS 4 ×S 7 /Z k or IIA theory on AdS 4 × CP 3 .
The subject of this paper is to study the supersymmetric Wilson loops in the ABJM theory. The pure Chern-Simons theory is a topological theory and the correlation functions of Wilson loops in this theory give various knot invariants [40]. The ABJM theory includes matters and is no more a topological theory. We construct 1/6-BPS Wilson loops which involve the matter fields. Both the dimensional considerations and the supersymmetries lead us to include the product of two scalar fields in these operators.
As a first step to study these BPS Wilson loops, we perform the perturbative computations up to the order of λ 2 in the planar limit. One the other hand, we find a fundamental string solution ending on a circular loop on the boundary. The dual fundamental string solution here is quite similar to the one in AdS 5 case. This leads us to expect that this F 1-string solution is half-BPS, and it should be dual to a half-BPS Wilson loop which is to be constructed.
The other part of this paper is organized as following: in the next section, we construct some supersymmetric Wilson loops in ABJM theory and discuss the supersymmetries they preserve. In section 3, we first perform the perturbative calculations at the first two orders. In section 4, we will discuss the string theory dual of the to-be-constructed half-BPS Wilson loops. The conclusion and discussions are put in the last section. We also include some needed Feynman rules in Appendix A.

Note added
When this project was in progress. we learnt that this subject was also studied by Nadav Drukker, Jan Plefka and Donovan Young, whose paper [41] has overlap with ours.

Supersymmetric Wilson loops in super Chern-Simons-matter theory
In this paper, we study supersymmetric Wilson loop operators in three dimensional N = 6 super Chern-Simons-matter theory proposed in [34]. The gauge group of this theory is U 1 (N) × U 2 (N). The matters include bifundamental superfields A 1 , A 2 and anti-bifundamental superfields B 1 , B 2 . The Chern-Simons part of the action is: Notice that the levels of these two group are opposite. The superpotential of this theory reads: We denote (A 1 , A 2 ,B 1 ,B 2 ) as Y A , A = 1, · · · , 4 which is in the fundamental representation of the R-symmetry group SU R (4). The supersymmetry transformation of the gauge fields and the scalar fields are 1 [42]: where ψ A is the supersymmetry partner of Y A and ω AB = −ω BA are the supersymmetry parameters and they are 6 Majorana spinors in three dimensional spacetime.
Here ω AB is defined as and we also have the following relation: We will search for supersymmetric spacelike Wilson loop operators in the following class: where s B A ands A B are Hermitian matrix, The indices i, j are the color indices of U 1 (N) and the indicesĩ,j are the color indices of U 2 (N). The normalization of the generators of the Lie algebra is For the above Wilson line operators to be supersymmetric, the supersymmetry transformation of each factor should vanish by itself. The transformation of the first factor is proportional to Then we get the following condition for the operator to be supersymmetric: In fact, eq. (12) is complex conjugation of eq. (11). Notice that ω AB is antisymmetric, we get from eq. (11): from which and the following equation from relabelling the indices in eq. (11) we have Since the general case is not easy to study, now we consider the special case in which s ands are diagonal matrices: In this case, from eq. (15), we get: this is to say that if s A = s B , ω AB = 0. Now we turn to consider the second factor: Similarly the supersymmetric condition for this factor gives: and its complex conjugation. As before, we can get From eqs. (11) and (19)(20), we get So if s A =s A , ω AB must be zero. Using all of the above results, and also taking the following condition into account: and the γ-matrices in [43,42], we find the 1/6−BPS Wilson loops which is defined by s 1,2 =s 1,2 = −s 3,4 = −s 3,4 = ± 2π k |ẋ|. Choose the positive sign for s 1 , the Wilson loop is Here for straight Wilson line, the preserved supercharges are global.
If we put the spacelike Wilson line along the x 2 direction, the above discussions give that the unbroken supersymmetries (Q AB ) α are (Q 12 ) 2 and (Q 34 ) 1 . They satisfy the following superalgebra: Notices that among three P µ 's, only P 2 appears in the right-hand side of superalgebra. This is as expected since only translation along x 2 direction is unbroken.
The Wilson loops preserve two supercharges in N = 2 and N = 3 Chern-Simons-matter theory were studied in [32]. Our 1/6−BPS operators are the generalization to ABJM theory of these operators. It is interesting to see if one can find other BPS Wilson loops within and/or out of the class eq. (9) we begin with.

Perturbative computations
In this section, we perform the perturbative computations of the vacuum expectation values of the above 1/6−BPS Wilson loops in the large N limit. Then the expansion parameter is λ = N/k. We will do the computation up to the order of λ 2 . We list some needed Feynman rules in Appendix A. And we choose the Landau gauge and use the regularization by dimensional reduction as done in [44]. In this regularization scheme, we first perform all tensor contractions in strict three dimensions: then we regular the divergent scalar integrals by the dimensional continuation with D = 3 − ǫ.
We will mainly focus on the circular Wilson loops. We parameterize the circle as with the x 3 direction being the time direction.

The calculations at the order of λ
We show in this subsection that the contribution at this order is zero. Due to the factor 2π k in front of the product of two scalars, we need only consider the contributions from the gauge field. There is only one contributing diagram as in Fig. 1. The contributions from this diagram reads: where the correlator of two gauge field is the free field one. The momentum space gluon propagator in the Landau gauge is proportional to 2 ǫ µνλ p λ p 2 .
By Fourier transformation, we get the position space propagator where x i = x(τ i ), i = 1, 2. Using this result, and eq. (26), one can easily see that eq. (27) is equal to zero.

The calculations at the order of λ 2
We first focus on the contribution due to the expansion of the first factor. The planar diagrams at the order of λ 2 are listed in Fig. 2 Since ǫ µνλ p ν p λ = 0, we need only consider in the last integration. From the Lorentz structure we know that this must proportional to p λ , so this will not contribute, either. We can check the above claim through direction computations using regularization by dimensional reduction in [44]: After the shift q → q + xp, we get the first term vanish and the second term is proportional to p λ , this confirms our claim.´ µ´ µ´µ The sum of the contributions from these diagrams is equal to: where the subscript 'ol' means one-loop. The contributions from the gluon loop and the ghost loop cancel each other [44], noticing that we are using the regularization by dimensional reduction. The contributions from onematter-loop are from Figs. 3.a-3.c. Both the one-fermion-loop and the onescalar-loop correction to the gluon self-energy are [44] (see also [32]) Taking the two gluon propagators into account, we get: In the ABJM theory, we have 4N scalars and 4N fermions, so totally we get: The second term of the above propagator will not contribute to the final answer due to the gauge invariance. In fact, the Fourier transformation of this term is proportional to where now we regularize the above integration as: where τ (ǫ) = τ 1 −ǫ andǫ is a regularization parameter. Then we get Using ∂ 1 µ log |x 1 − y| ∝ x µ (τ 1 ) and x(τ 1 ) ·ẋ(τ 1 ) = 0, we get that eq. (2) is zero even before we takeǫ → 0. So this term doesn't contribute as claimed.
By performing a Fourier transformation of the first term in eq. (39) to position space, we get So the contribution to at this order equals to where x ≡ x 1 , y ≡ x 2 . Now we turn to discuss the contribution due to the product of two scalar propagators which appears in Fig. 2.d. Using the position space propagator of (anti-)bifundamental scalars 1 4π we get that The sum of eqs. (46) and (48) When x(τ ) is among the circle in eq. (26), the above result equals to which is independent of the positions. From this we get that eq. (36) is equal to The second factor of the Wilson loop gives the same contribution. Then the total contribution is 2π 2 N 2 k 2 . Now at the order of λ 2 , we are left with the contribution from Fig. 2.e. This diagram comes from the pure Chern-Simons theory and was studied in details in [45]. The result from each factor in the Wilson line is −π 2 N 2 /(6k 2 ). So totally the contributions at the order of λ 2 is

Holographic description
Depending on the values of N and k, the gravity dual of the ABJM theory can be M-theory on AdS 4 × S 7 /Z k or type IIA theory on AdS 4 × CP 3 with two-form and four-form fluxes. In this section we will focus on the type IIA case which corresponds to the situation when k ≪ N ≪ k 5 . As the four dimensional super Yang-Mills case, the Wilson loop in the fundamental representation of the gauge group should be dual to a fundamental string in this background while the better description for the Wilson loop in higher dimensional representation should be higher dimensional probe D-branes or bubbling geometry.
In this section, we find a fundamental string solution ending on a circle on the boundary. The dual IIA string theory background is (We take α ′ = 1): where ǫ AdS 4 is the volume form of unit AdS 4 and J is the Kahler form of CP 3 , and R satisfies Using the Poincare coordinates in the AdS 4 part, the metric can be written as where y = 0 is the conformal boundary of AdS 4 . We put the circular Wilson loop at x 2 1 + x 2 2 = R 2 and x 3 = 0 on the boundary. As in the AdS 5 case [8,9], the worldsheet of this fundamental string should be completely embedded in AdS 4 ending on the Wilson loop [34]. The Nambu-Goto action of the fundamental string is whereg is the induced metric on the worldsheet. The fundamental string solution we are looking for is the same as the AdS 5 case [10,11]: The total action of this string is equal to Since this fundamental string solution is quite similar to the one in IIB theory on AdS 5 × S 5 , we expect that this solution is half-BPS. It should be very interesting to construct the dual half-BPS Wilson loop in the field theory side.

Conclusion and Discussions
In this paper, we studied the supersymmetric Wilson loops in ABJM theory from both sides of the AdS 4 /CF T 3 correspondence. In the field theory side, we constructed 1/6-BPS Wilson loop operators and perform the perturbative computations of their VEVs at the first two orders. In the string theory side, we found the fundamental string solution which could be dual to half-BPS circular Wilson loop.
There are a lot of interesting open questions we left for further studies. Here we list some of them. One natural question is to search for other BPS Wilson loops, especially the half-BPS one which seems to us not easy to construct. Even for the 1/6-BPS Wilson loops in this paper, our understanding is quite limited. Recently a class of Wilson loops operators are constructed in ABJM theory based on M-theory considerations [46] following the line in [9]. It is interesting to check if the 1/6-BPS Wilson loops belong to that class of operators and search for new supersymmetric Wilson loops in that class.
We also notice that Berkovits' pure string formalism [47,48] has been used in the string theory dual of ABJM theory [49,50,51]. Based on these works, maybe one can study the Wilson loops in ABJM theory using the ideas in [52] for the AdS 5 case.
The fundamental string dual to half-BPS operator is almost the same as the one in AdS 5 case. However the string solution dual to the 1/6-BPS operators may also be embedded in the CP 3 part of the background in a non-trivial manner as the string solution in AdS 5 × S 5 dual to 1/4-BPS Wilson loop in super Yang-Mills theory [18]. It is interesting to explicitly find this solution. The probe brane dual to Wilson loops in higher dimensional representations need further studies as well. Another interesting question is to find M2-brane in AdS 4 × S 7 /Z k dual to these Wilson loops.
Some membrane solutions in AdS 4 × S 7 is found in [53,54]. By generalizing them to membrane solutions AdS 4 × S 7 /Z k and using the reduction from M-theory to IIA theory, it is reasonable to get some fundamental string solution in AdS 4 × CP 3 . The field theory dual of these string solutions demands further studies.

Gluon propagator
in the Landau gauge, where the up/down sign is corresponding to the first/sector factor of the gauge group.
4. Three-gluon vertex: ± k 4π f abc ǫ µνλ . The up/down sign is corresponding to the first/sector factor of the gauge group.

Two scalars emitted the Wilson loop: ±i 2π
k for A's/B's. The last two vertices are from the Wilson loop operator (we omit the color factors in the expressions of these two vertices). We don't give the explicit express from the six-scalar vertex and scalar-scalar-fermion-fermion vertex. We also omit the ghost propagator and gluon-ghost-ghost vertex.