A supermatrix model for N=6 super Chern-Simons-matter theory

We construct the Wilson loop operator of N=6 super Chern-Simons-matter which is invariant under half of the supercharges of the theory and is dual to the simplest macroscopic open string in AdS_4 x CP^3. The Wilson loop couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bi-fundamental representation of the U(N) x U(M) gauge group. These ingredients are naturally combined into a superconnection whose holonomy gives the Wilson loop, which can be defined for any representation of the supergroup U(N|M). Explicit expressions for loops supported along an infinite straight line and along a circle are presented. Using the localization calculation of Kapustin et al. we show that the circular loop is computed by a supermatrix model and discuss the connection to pure Chern-Simons theory with supergroup U(N|M).


Introduction
The duality between string theory on asymptotically AdS spaces and conformal field theories has been an exciting area of research for over ten years now, with string theory providing answers to strong coupling questions in the gauge theory and vice-versa.
A year and a half ago, a new example of an AdS/CFT duality was proposed by Aharony, Bergman, Jafferis, and Maldacena for the maximally supersymmetric gauge theory in three dimensions: N = 6 supersymmetric Chern-Simons-matter with gauge group U(N) × U(N) [1]. 1 The proposal was inspired by a construction of the gauge theory with even more supersymmetry, N = 8, but which applied only to the gauge group SU(2) × SU (2) [3,4]. The gravity dual of this theory is M-theory on AdS 4 × S 7 /Z k , where k is the level of the Chern-Simons term, or, for large enough k, type IIA string theory on AdS 4 × CP 3 .
This gauge theory and the dual string theory have been studied extensively, but so far one of the most interesting observables in the gauge theory has not been constructed. Like in all gauge theories, one can define Wilson loop operators, which in the dual string theory are given by semi-classical string surfaces [5,6]. The most symmetric string of this type preserves half of the supercharges of the vacuum (as well as an U(1) × SL(2, R) × SU(3) bosonic symmetry) but its dual operator in the field theory has not been identified yet.
So far the most symmetric Wilson loop operators in this theory, constructed in [7][8][9], preserve only 1/6 of the supercharges and are therefore not viable candidates to be the dual of this classical string. In fact, these operators exist also in Chern-Simons theories with less supersymmetry [10] and do not get any supersymmetry enhancement due to the clever quiver construction of the N = 6 theory.
One reason to look for these operators is that Wilson loops are interesting observables in all gauge theories but in particular in Chern-Simons theories. In Chern-Simons without matter they are in fact the main observables. Beyond that, the lack of the gauge theory dual of the simplest string solution in AdS 4 × CP 3 is a glaring gap in our understanding of this duality.
As another motivation, recall that the analog observable in N = 4 super Yang-Mills theory in four dimensions has the remarkable property that its expectation value is a nontrivial function of the coupling and of N which can be calculated exactly by a Gaussian matrix model and interpolates from weak to strong coupling [11][12][13].
Another exact interpolating function which exists in the 4-dimensional theory is the cusp anomalous dimension, also known as the universal scaling function [14][15][16][17][18][19][20][21] which captures the scaling dimension of twist two operators. Trying to compute similar quantities in the 3-dimensional theory does not go through as nicely. In the calculation of the spectrum of local operators there is a matching with the square root structure of the dispersion relation of giant magnons, but this involves one extra function of the coupling [22][23][24], whose value is only known at weak and at strong coupling but not in the intermediate regime.
It is therefore interesting to revisit the question of Wilson loop operators in the hope that there are exact interpolating functions for them. For the 1/6 BPS Wilson loop a matrix model has been recently derived in [25] and, despite its complexity, has been solved in the planar approximation in [26]. 2 Their results indeed match the string theory calculation and provide a first non-trivial interpolation function for this theory.
Prompted by these considerations we construct here the 1/2 BPS Wilson loop for N = 6 super Chern-Simons-matter. Furthermore we prove that the results of [25,26] carry over to our case. The calculation of [25] uses localization with respect to a specific supercharge which is also shared by the 1/2 BPS loop. We show that the 1/2 BPS Wilson loop is cohomologically equivalent to a very specific choice of the 1/6 BPS loop and is therefore also given by a matrix model. This matrix model has a supergroup structure and the 1/2 BPS loop is the most natural observable within this model. Indeed it can be calculated for all values of the coupling also beyond the planar approximation [26].
In the coming section we present the loop and verify its symmetry. Our derivation uses in an essential way the quiver structure of the theory. In addition to the gauge fields, the Wilson loop couples to bilinears of the scalar fields and, crucially, also to the fermionic fields transforming in the bi-fundamental representation of the two gauge groups. Our loop is classified by representations of the supergroup U(N|M) and is defined in terms of the holonomy of a superconnection of this supergroup. 3 In our analysis we consider both a loop supported along an infinite straight line and one supported along a circle.
In Section 3 we relate this Wilson loop to the 1/6 BPS one and show that it is indeed the most natural observable for the matrix model of [25]. We interpret this matrix model as that of a supermatrix which represents the semiclassical expansion of pure Chern-Simons with supergroup U(N|M) on the lens space S 3 /Z 2 .

The loop
We introduce now the construction of the Wilson loop in the U(N) k × U(M) −k Chern-Simons-matter theory. We denote the gauge field of the U(N) factor as A µ and the gauge field of the U(M) factor as A µ . These gauge fields are coupled to four scalar fields C I and their complex conjugatesC I , and to four fermions ψ α I andψ I α , with I = 1, 2, 3, 4 being an SU(4) R index and α = +, − a spinor index. The scalars and the fermions are in the bifundamental representation of the gauge group. Our notation is such that CC andψψ are in the adjoint of U(N), whereasCC and ψψ are in the adjoint of U(M). In the appendix we give more details about our conventions.
The central idea of this paper is to augment the connection of U(N) × U(M) to a superconnection of the form where x µ parametrizes the curve along which the loop operator is supported and M I J ,M I J , η α I andη I α are free parameters. A lot of the form of L is dictated by dimensional analysis and by the index structure of the fields. In three dimensions the scalars have dimension 1/2, so they should appear as bi-linears, which are in the adjoint and therefore enter in the diagonal blocks together with the gauge fields. The fermions have dimension 1 and should appear linearly. Since they transform in the bi-fundamental, they are naturally placed in the off-diagonal entries of the matrix. Note that η I andη I are Grassmann even, so that the off-diagonal blocks of L are Grassmann odd and L is a supermatrix.
Although L has the structure of a U(N|M) superconnection, the theory has only U(N) × U(M) gauge symmetry. It is nevertheless possible, given a path and the extra parameters, to calculate the holonomy of this superconnection and end up with a supermatrix. For a closed curve one can then take the trace 4 in any representation R of the supergroup U(N|M). This gives the Wilson loop (2.2)

Infinite straight line
In order to find the maximally supersymmetric Wilson loop, we consider an operator defined along an infinite straight line in the temporal direction, parameterized by x µ = (τ, 0, 0). As mentioned before, this loop should also preserve an SU(3) subgroup of the R-symmetry group. Given that and the chirality of the supercharges, this suggests the ansatz We define the modified connections which appear in the diagonal blocks of L One can easily verify [7][8][9] that the supersymmetry variation of these terms does not vanish. Instead we demand that their variation contains only ψ + 1 andψ 1 + , which appear anyhow in the Wilson loop through the couplings to η α I andη I α . Using the expressions in the appendix we find that for the particular choice 5 of m 1 = m 2 = 1 the variation is (noticing that ψ + = ψ − and ψ + = −ψ − ) (2.6) Turning to the off-diagonal entries in L, the variation of the fermionsψ 1 and ψ 1 includes the covariant derivative γ µ D µ . Since the fermions appearing in the loop have specific chiralities, as do the supercharges (2.3), the covariant derivative gets projected to be along the direction of the loop by Furthermore, all the non-linear terms appearing in the variation of the fermions can be repackaged into a covariant derivative with the modified connection (2.5) We finally find that Combining (2.6) and (2.9) the variation of L for the time-like line is given by (2.10) The proof of supersymmetry-invariance of the Wilson loop requires one additional step, namely integration by parts. Expanding to second order, the Wilson loop is The off-diagonal pieces of the linear term are total derivatives, as can be seen in (2.10) and integrate away. The diagonal part of the linear term does not vanish on its own, but it is canceled by the variation of the fermions in the quadratic term. To see that, we write the relevant terms for the variations with parametersθ 1I The last entry on the bottom right comes from the variation of L(τ 2 ) and it has an extra minus sign since the supersymmetry parameterθ was permuted through the first fermion ψ + 1 (τ 1 ). We have also assumed that η andη are constant, so we have pulled them out of the integrals.
Integrating by parts and ignoring any possible contributions from infinity, one obtains The two integrals clearly cancel each other for ηη = 2i. A similar cancellation takes place for theθ IJ+ supercharges.
To summarize, we have shown at leading order in the expansion (2.11) that the Wilson loop (2.1), (2.2) with preserves the six Poincaré supercharges (2.3) and is therefore 1/2 BPS. We performed the same calculation to the next loop order by multiplying the diagonal part of δL with another L and the off-diagonal pieces with two more and integrating one of the three integral by parts. After including all the terms, the final result vanishes again.
This analysis can be carried over to all orders. To do that we separate L into the diagonal part L B and the off-diagonal entries L F . We leave the bosonic piece in the exponent and expand only L F (2.15) The supersymmetry variation can act on the exponent, bringing down an extra integral of δL B or can act on one of the L F , giving a matrix with an off-diagonal entry of the form D 0 C I (or D 0C K ). As mentioned before, this D 0 is the covariant derivative with the modified connection appearing in L B . This allows us to integrate by parts these terms in the presence of the path ordered exp(i L B ). As in the case considered explicitly above, these will give non-zero contributions at the limits of integration, where in general we have with the factor (−1) n−1 coming from pulling outθ 1I + through the L F insertions. Reordering the terms we see that this exactly cancels the insertion of the variation δL B into the term in (2.15) with p − 2 integrals of L F . This calculation proves that this Wilson loop preserves six of the twelve Poincaré supercharges. Similarly, one can show that six conformal supercharges are also preserved.

Circle
Under a conformal transformation a line is transformed into a circle. While conformal transformations are a symmetry of the theory, they change the topology of the curve and, as it turns out, also the expectation value of the loop. In the case of the 1/2 BPS Wilson loops of N = 4 super Yang-Mills one finds that, whereas the straight line has trivial expectation value, the circular loop depends in an interesting way on the coupling constant of the theory. It is therefore of great interest to consider circular Wilson loops also in this 3-dimensional theory.
First we consider the Wick rotation of the time-like line to a space-like line. The latter can be defined either for the theory in Euclidean R 3 or in the Lorentzian theory in R 1,2 , as we do here. Indeed it is simple to check that the replacement |ẋ| → −i|ẋ| gives the 1/2 BPS Wilson loop for a space-like line. This replacement affects the scalar bi-linear term and the fermionic terms.
To get the circle one should perform a conformal transformation. The path is now given by 6 x 1 = cos τ , The scalar couplings should not be affected by the conformal transformation, so for the diagonal part of the superconnection L (2.1) we again use the shorthands We still should couple only to the fermion fields ψ α 1 andψ 1 α . The spinor index is chosen by taking η α I (τ ) andη I α (τ ) to be eigenstates of the projector with an arbitrary function η(τ ) which is determined by checking the supersymmetry variation of the loop.
The loop along the line preserved six super-Poincaré symmetries and six superconformal ones, for the circle we expect to find twelve which are linear combinations of the two. The parameters of the superconformal transformations, which we labelθ IJ , should be related to the super-Poincaré transformation parametrized byθ IJα . We take the ansatz and using the explicit superconformal transformations [31] determine the supersymmetric circular loop. Note that the choice in (2.21) is consistent with the reality condition (A.8) on θ IJ and the analog one forθ IJ .
To do the calculation we note that, apart for one extra term in the variation of the spinors, the superconformal transformations of the fields are the same as the super-Poincaré transformations, modulo the replacementθ IJ →θ IJ x µ γ µ . Using that x µ γ µẋ ν γ ν = iσ 3 , with our choice of ϑ IJ we find Another useful relation involves the change of spinor indices on the projector As mentioned in the appendix, unless we write it explicitly, we always use the indices as in the left-hand side of this equation. Lastly, we note that Using these relations we get the variations under Poincaré and superconformal transformations of the fields in L The extra terms in the variations of ψ andψ are written explicitly in [29]. We would like to write the last two expressions as total derivatives, which gives the equations with A and A defined in (2.18) and Collecting all the pieces, we find that the variation is (2.29) It is instructive to repeat the supersymmetry analysis at leading order also for the circle. Expanding the loop as in (2.11) and varying it as in (2.29), one finds (we consider just one kind of supercharges and write explicitly only the terms in the diagonal blocks) As done for the line, it is easy to integrate by parts and verify the cancellation of the bulk terms between the first and the second lines of this expression. From the integration by parts one has now also the following boundary terms which cancel once taking the trace, since η 1 is antiperiodic on the circle, η 1 (2π) = −η 1 (0). This calculation in fact determines that the Wilson loop is supersymmetric only when taking the trace of the holonomy, and not the supertrace. 7 We can repeat the all-order proof outlined in (2.16). Expanding the exponential in L F one can see again cancellations between bosons and fermions similarly to what happened for the line. The only difference from that case are the new boundary terms arising at τ = 0 from integrating over the first variation δL F (τ 1 ) and at τ = 2π from the last variation δL F (τ p ). As in the leading order case studied above, upon taking the trace these two contributions cancel.
The same analysis carried out above applies also to the six other supercharges. We have shown then that the circle operator is invariant under the twelve supercharges in (2.22) and is therefore 1/2 BPS. 7 The trace of a supermatrix in an arbitrary representation is defined on diagonal matrices by the supertrace as Tr

Localization to a matrix model
Recently, in a very nice paper [25], the evaluation of supersymmetric Wilson loop operators in Chern-Simons-matter theories with N = 2 supersymmetry was reduced to a 0-dimensional matrix model. In this section we show how to apply the same result to the circular Wilson loop constructed in the preceding section. 8 We can then use the solution of this matrix model [26] to evaluate the Wilson loop at arbitrary values of the coupling constants.
The Wilson loop studied in [25] is the one constructed in [10]. The N = 2 Chern-Simonsmatter theories have supersymmetric Wilson loops with the gauge connection and an extra coupling to the scalar in the vector multiplet. This scalar has an algebraic equation of motion and after integrating it out we find the Wilson loop with a coupling to some of the other scalar fields of the theory.
Specializing to the case of the theory with N = 6 supersymmetry, one ends up with the Wilson loops of the type constructed in [7][8][9], where the connection is given by (2.1) with η α I =η I α = 0 and M I J = M I J = diag(−1, −1, 1, 1). 9 In the following we will denote the resulting connection matrix by L 1/6 and that for the loop constructed in Section 2 by L 1/2 . The reason for this notation is that while these Wilson loops preserve half of the supercharges of the the N = 2 theories, they do not see the supersymmetry enhancement of the gauge theory from N = 2 to N = 6, so they are 1/6 BPS. The loops constructed in Section 2 preserve instead half of the supercharges of the N = 6 theory.

Relation between the different Wilson loops
The calculation of [25] uses localization with respect to a single supercharge, which is also shared by the 1/2 BPS Wilson loop. We will show now that the 1/2 BPS Wilson loop is related to the 1/6 BPS loop -they are in the same cohomology class under this supercharge. Hence the localization calculation immediately applies also to the 1/2 BPS Wilson loop.
We start by analyzing the case of the infinite straight line. We notice that the 1/2 BPS Wilson loop shares all four supercharges preserved by the 1/6 BPS one. These are the ones parameterized byθ 12 + andθ 34+ and their superconformal counterparts. For the 1/6 BPS Wilson loop the couplings of the scalars is given by the matrices M I J = M I J = diag(−1, −1, 1, 1) and there is no coupling to the fermions. One can therefore write the 8 The calculation of [25] was done for a Wilson loop on the equator of S 3 , while here we discuss a circle in R 3 or R 1,2 . These operators should have the same expectation value when normalizing by the partition function. 9 The choice of 1/6 BPS Wilson loop is not unique. The constructions in [7][8][9] all preserve the same supercharges but have slight differences. We use this definition, since it will turn out to be related to the loop constructed in Section 2, as we show below. difference between the superconnection for the 1/2 BPS loop and the connection of the 1/6 BPS one asL The off-diagonal term inL is the same as L F defined above in (2.15). The diagonal piece comes from the difference in the scalar couplings M I J and M I J between the two loops. We want now to show that the 1/2 BPS loop, W 1/2 , and the 1/6 BPS one, W 1/6 , are cohomologically equivalent with respect to the aforementioned supercharges. This means that the difference between the two loops is exact with respect to a linear combination Q of the supersymmetries with parametersθ 12 + andθ 34+ , namely that there exists a V such that To find V it is useful to rewrite the difference between the loops as We take is such that Q Λ = L F . Acting with Q on V and recalling that Q L 1/6 = 0, one finds the following two terms (3.6) To evaluate the second contribution we can use a similar logic to the all-order proof (2.16) and Taylor expand the L F in the exponent, the difference being that the integral in the exponent is now between τ and infinity rather than between minus infinity and infinity. The cancellation between bosons and fermions is therefore incomplete and when Q acts on the first L F the integration by parts introduces an extra boundary term . (3.7) This is nothing else than the diagonal part ofL. Combining it with the term in L F in (3.6), one finds which, upon Taylor expansion, can be seen to be exactly equal to (3.3).
We analyze now the circular loop. The difference between the connections is now So we can write W 1/2 − W 1/6 in a power series of terms with the W 1/6 connection (3.10) As we saw in the supersymmetry analysis, terms with different numbers of integrals mix. It will be therefore useful to separate this sum into terms with different numbers of field insertions. First L F , thenL B and L 2 F , next L FLB and L 3 F , etc. Before finding V we should choose one of the supercharges annihilating the 1/6 BPS Wilson loop. We take 10 and define It is easy to check that The covariant derivative acting on Λ in Q L F has the generalized connection with A andÂ in L 1/2 (2.18), but its action on Λ is the same as a covariant derivative in the L 1/6 connection, since the difference between the two, involving C 2C 2 andC 2 C 2 , cancels when acting on Λ. We can therefore integrate the total derivative inside a Wilson loop with either the L 1/2 or L 1/6 connection.
We now solve for V in a power series. We take V = ∞ p=1 V p , where the term V p has p field insertions into the Wilson loop with connection L 1/6 . The first few are 14) Using (3.13) it is easy to check that These indeed are the terms in the expansion of W 1/2 − W 1/6 around the L 1/6 connection with one, two and three fields insertions. We have also checked the next term in the expansion and expect this pattern to extend to all orders.
This comparison with the 1/6 BPS loop allows for an alternative, immediate verification that the loop constructed in Section 2 is indeed 1/2 BPS. This derivation shows that the Wilson loop is invariant under the single supercharge used above. Then we note that from inspecting L 1/2 , the Wilson loop preserves an SU(3) subgroup of the SU(4) R-symmetry group of the theory. Note though that the supercharge is not invariant under this SU(3), so acting with this symmetry we automatically generate more supercharges preserved by this loop. In a similar fashion one can generate the full supergroup with twelve supercharges preserved by the loop as the minimal one containing the SU(3) generators and the four preserved by the 1/6 BPS loop.

Supermatrix model and supergroup Chern-Simons
Since the 1/2 BPS loop and the 1/6 BPS one are in the same cohomology class with respect to Q, we can immediately conclude that the localization argument used in [25] for the 1/6 BPS circular loop will also apply unaltered to our operator. Generalizing the matrix model derived in [25] to the case of M = N gives the following expression for the partition function a a<b sinh 2 (π(λ a − λ b )) â<b sinh 2 (π(λâ −λb)) a,â cosh 2 (π(λ a −λâ)) . (3.16) Here λ a (a = 1, . . . , N) andλâ (â = 1, . . . , M) are two sets of eigenvalues corresponding to the two gauge groups of the theory. Our 1/2 BPS Wilson loop in the fundamental representation is evaluated by inserting into the integral above (3.17) For a general representation the insertion is (see footnote 7) Examining these expressions one sees that if the cosh functions in (3.16) were in the numerator rather than in the denominator, this would be the matrix model for pure Chern-Simons theory with gauge group SU(N + M) on a lens space S 3 /Z 2 , where an SU(N) subgroup is expanded around the trivial vacuum and an SU(M) subgroup around the other flat connection [32][33][34]. 11 Compared to the trivial saddle point with unbroken SU(N + M), the non-trivial connection is represented by the shift of the eigenvalue λ → λ + i/2. This replaces some of the sinh functions with cosh functions and also gives the (−) factor in the lower-right block on the right-hand side of (3.18).
In (3.16) the cosh functions are in the denominator rather than in the numerator. This arises naturally when considering instead the matrix model and Chern-Simons theory (and the same saddle point) for the gauge supergroup SU(N|M). For a fuller discussion see [26].

Discussion
In this paper we have found the so far elusive 1/2 BPS Wilson loop operator of N = 6 super Chern-Simons-matter. We have considered both a loop supported along an infinite straight line and one supported along a circle. The former preserves separately six Poincaré supercharges and six conformal supercharges, whereas the latter preserves twelve linear combinations of the two. The proof of the invariance of our operator under these supercharges is quite novel, for it requires to Taylor expand the exponential and to integrate by parts some of the variations in order to have cancellations between terms of different order. While the theory has only U(N) × U(M) gauge symmetry, our loop can be defined for arbitrary representations of the supergroup U(N|M).
We have shown that this loop is related to another one, which is only 1/6 BPS, by the addition of a term exact under a supercharge Q. This in turn implies that the expectation values of the two loops are equal and allows us to use the matrix model (3.16), derived using localization in [25]. Beyond this formal derivation it would be interesting to check this expression by an explicit perturbative computation. It is straightforward to do the matrix model calculation perturbatively, plugging (3.17) into (3.16). The first few orders are This is a prediction for a corresponding computation to be performed directly in the gauge theory (with framing one) by summing Feynman diagrams [35].
The matrix model gives, in principle, an exact expression for the expectation value of the Wilson loop valid for all values of N, M and k. Unfortunately, unlike the 4-dimensional analog [11][12][13], this matrix model is quite complicated (for similar models see e.g. [32-34, 36,37]). Still, it can be solved [26] and gives the correct expression for the expectation value of the Wilson loop at strong coupling as evaluated by a macroscopic fundamental string extending in AdS 4 × CP 3 and ending along the circular loop on the boundary of AdS 4 (or by an M2 brane wrapping the orbifolded direction of S 7 /Z k ) Note that while the matrix model is related to the supergroup Chern-Simons theory, it is not exactly the same. The matrix model calculates the contribution of a single saddle point in a perturbative expansion of the Chern-Simons theory. This is reminiscent of the situation for the Wilson loops on S 2 in N = 4 supersymmetric Yang-Mills in four dimensions and their relation to two-dimensional Yang-Mills [38,39]. They are not given by the full answer in the 2-dimensional theory, but rather by a semiclassical expansion around the zero-instanton sector [40][41][42]. Recently an interpretation was given for the other saddle points, as the correlation function of Wilson and 't Hooft loops [43]. It would be interesting to understand if there are any observables in N = 6 Chern-Simons-matter theory which give the other saddle points in the perturbative expansion of the U(N|M) pure Chern-Simons theory.
Another direction worth investigating is related to the construction of Wilson loops via the higgsing of membranes, as done in four dimensions in [44]. In [30,9] the coupling of the Wilson loop to the scalar fields was found by separating membranes and computing the mass of the resulting off-diagonal modes stretching between them. This indeed gives the scalar couplings in (2.4), but did not include the fermions, that, as we have seen, are crucial to enhance the supersymmetry of the loop operator. It would be therefore interesting to repeat the calculation considering also fermionic off-diagonal modes and to reproduce the couplings η α I andη I α in this way. There are other objects in this theory which are very closely related to the Wilson loops constructed here. These are the vortex loop operators of [29], which have a semiclassical description in the gauge theory. Along the loop the gauge symmetry is broken to some subgroup and different U(1) factors have vortices. In addition, the scalar fields can have square-root branch cuts. So, parameterizing the transverse plane to the line by complex coordinates z andz, the field configuration (in one U(1) factor) is with α and β being two real parameters. The vortex loops carry k unit of electric flux and should be an alternative description for k coincident Wilson loops. In fact, while one can construct the Wilson loop in any anti-symmetric representation, in Chern-Simons theory the dimension of the symmetric representation should be smaller than k. We expect the vortex loops to take over as the description of the object carrying k units of electric flux. In the M-theory picture the 1/2 BPS Wilson loop is an M2-brane wrapping the orbifolded direction of S 7 /Z k . The k-th symmetric loop is the brane wrapping the circle in the covering space. This brane then develops extra allowed deformations, including opening up in AdS 4 , which corresponds to the β parameter above, and rotating on S 7 , thus leading to the 1/3 BPS vortex of [29].
Still, a fuller classification of all 1-dimensional defects in this theory is in order. In N = 4 super Yang-Mills in four dimensions the classification of Wilson and 't Hooft loops gives rise to a rich structure of objects in the dual string theory, including probe branes wrapping various cycles (see e.g. [45,46,44,47]) and fully backreacted geometries, the socalled "bubbling" solutions (see e.g. [48][49][50][51]). So far this classification has only been partially undertaken in the M-theory dual of N = 6 super Chern-Simons-matter. We plan to complete this in a future publication [35].
Apart for the theory with N = 6 supersymmetry, there are closely related 3-dimensional Chern-Simons-matter theories with N = 4 and N = 5 supersymmetry [52][53][54]. These theories are based on more complicated quivers and have a richer structure of allowed gauge groups and matter representations. We expect that constructions similar to ours will give the 1/2 BPS loops of these theories. These should probably be related to Wilson loop observables in the topological theories discussed in [28].
It would be also of some importance to find other loop operators of N = 6 super Chern-Simons-matter preserving reduced amounts of supersymmetry, following the spirit of [55,56]. In particular, note that the couplings satisfy the relationsη I α η β I = i(1 +ẋ µ γ µ ) β α and M I J = δ I J + iη α Jη I α . Such relations may play a role in a more comprehensive analysis as do the pure spinors in the treatment of [57].
where θ and ψ are arbitrary spinors.
Regarding the gauge index structure, if we denote by a,â the gauge indices in the fundamental of the first and the second gauge group, respectively, we have (C I )â a , (C I ) â a , (ψ I ) â a , (ψ I )â a . (A.5) Under supersymmetry the fields transform as (for clarity we indicate here all the spinor indices explicitly) where we have written the transformations only in terms of the parametersθ and not θ, by using the following relation The supersymmetry parameters are antisymmetric,θ IJ = −θ JI , and obey the reality conditionθ IJ = (θ IJ ) * . (A.8)