A supermatrix model for
[3mm] super Chern-Simons-matter theory
The duality between string theory on asymptotically 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 /CFT duality was
proposed by Aharony, Bergman, Jafferis, and Maldacena for the maximally
supersymmetric gauge theory in three dimensions: supersymmetric
Chern-Simons-matter with gauge group
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 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 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  and do not get any supersymmetry enhancement due to the clever quiver construction of the 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 is a glaring gap in our understanding of this duality.
As another motivation, recall that the analog observable in super Yang-Mills theory in four dimensions has the remarkable property that its expectation value is a non-trivial function of the coupling and of 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 BPS Wilson loop a matrix model has been recently derived
in  and, despite its complexity, has been solved in
the planar approximation in .
Prompted by these considerations we construct here the 1/2 BPS Wilson loop for super Chern-Simons-matter. Furthermore we prove that the results of [25, 26] carry over to our case. The calculation of  uses localization with respect to a specific supercharge which is also shared by the BPS loop. We show that the BPS Wilson loop is cohomologically equivalent to a very specific choice of the BPS loop and is therefore also given by a matrix model. This matrix model has a supergroup structure and the 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 .
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 and is
defined in terms of the holonomy of a superconnection of this supergroup.
In Section 3 we relate this Wilson loop to the BPS one and show that it is indeed the most natural observable for the matrix model of . We interpret this matrix model as that of a supermatrix which represents the semiclassical expansion of pure Chern-Simons with supergroup on the lens space .
We conclude in Section 4 with a discussion of our results and some possible extensions. An appendix contains details about our notation.
2 The loop
We introduce now the construction of the Wilson loop in the Chern-Simons-matter theory. We denote the gauge field of the factor as and the gauge field of the factor as . These gauge fields are coupled to four scalar fields and their complex conjugates , and to four fermions and , with being an index and a spinor index. The scalars and the fermions are in the bi-fundamental representation of the gauge group. Our notation is such that and are in the adjoint of , whereas and are in the adjoint of . In the appendix we give more details about our conventions.
The central idea of this paper is to augment the connection of to a superconnection of the form
where parametrizes the curve along which the loop operator is supported and , , and are free parameters. A lot of the form of is dictated by dimensional analysis and by the index structure of the fields. In three dimensions the scalars have dimension , 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 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 and are Grassmann even, so that the off-diagonal blocks of are Grassmann odd and is a supermatrix.
Although has the structure of a superconnection,
the theory has only 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
2.1 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 .
The supercharges of this theory are parameterized by the two-component spinors (see the appendix). Motivated by the BPS string solution in , we want to find a loop operator invariant under the same six supercharges. They are in fact the same supercharges also annihilated by other brane solutions dual to the vortex loop operators of  and are parameterized by
As mentioned before, this loop should also preserve an subgroup of the -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
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 and , which appear anyhow in the
Wilson loop through the couplings to and .
Using the expressions in the appendix we find that for the particular choice
Turning to the off-diagonal entries in , the variation of the fermions and includes the covariant derivative . 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)
with exactly the choice (2.4) of and .
We finally find that
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 parameters
The last entry on the bottom right comes from the variation of and it has an extra minus sign since the supersymmetry parameter was permuted through the first fermion . We have also assumed that and 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 . A similar cancellation takes place for the supercharges.
preserves the six Poincaré supercharges (2.3) and is therefore BPS. We performed the same calculation to the next loop order by multiplying the diagonal part of with another 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 into the diagonal part and the off-diagonal entries . We leave the bosonic piece in the exponent and expand only
The supersymmetry variation can act on the exponent, bringing down an extra integral of or can act on one of the , giving a matrix with an off-diagonal entry of the form (or ). As mentioned before, this is the covariant derivative with the modified connection appearing in . This allows us to integrate by parts these terms in the presence of the path ordered . 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 coming from pulling out through the insertions. Reordering the terms we see that this exactly cancels the insertion of the variation into the term in (2.15) with integrals of .
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.
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 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 or in the Lorentzian theory in , as we do here. Indeed it is simple to check that the replacement gives the 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
The scalar couplings should not be affected by the conformal transformation, so for the diagonal part of the superconnection (2.1) we again use the shorthands
We still should couple only to the fermion fields and . The spinor index is chosen by taking and to be eigenstates of the projector
with an arbitrary function 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 label , should be related to the super-Poincaré transformation parametrized by . We take the ansatz
and using the explicit superconformal transformations  determine the supersymmetric circular loop. Note that the choice in (2.21) is consistent with the reality condition (A.13) on and the analog one for .
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 replacement . Using that , with our choice of 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
The extra terms in the variations of and are written explicitly in . We would like to write the last two expressions as total derivatives, which gives the equations
From this we deduce that the extra function in (2.20) is . The product of the two couplings is then , as in the case of the line.
The superconnection for the circular Wilson loop is therefore
with and defined in (2.18) and
Collecting all the pieces, we find that the variation is
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 is antiperiodic on the circle,
. This calculation in fact determines that the Wilson
loop is supersymmetric only when taking the trace of the holonomy,
and not the supertrace.
We can repeat the all-order proof outlined in (2.1). Expanding the exponential in 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 from integrating over the first variation and at from the last variation . 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.
3 Localization to a matrix model
Recently, in a very nice paper , the evaluation of
supersymmetric Wilson loop operators in Chern-Simons-matter theories
with 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.
The Wilson loop studied in  is the one constructed in . The Chern-Simons-matter 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 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 and
3.1 Relation between the different Wilson loops
The calculation of  uses localization with respect to a single supercharge, which is also shared by the BPS Wilson loop. We will show now that the BPS Wilson loop is related to the BPS loop — they are in the same cohomology class under this supercharge. Hence the localization calculation immediately applies also to the BPS Wilson loop.
We start by analyzing the case of the infinite straight line. We notice that the BPS Wilson loop shares all four supercharges preserved by the BPS one. These are the ones parameterized by and and their superconformal counterparts. For the BPS Wilson loop the couplings of the scalars is given by the matrices and there is no coupling to the fermions. One can therefore write the difference between the superconnection for the BPS loop and the connection of the BPS one as
The off-diagonal term in is the same as defined above in (2.15). The diagonal piece comes from the difference in the scalar couplings and between the two loops.
We want now to show that the 1/2 BPS loop, , and the 1/6 BPS one, , 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 of the supersymmetries with parameters and , namely that there exists a such that
To find it is useful to rewrite the difference between the loops as
is such that . Acting with on and recalling that , one finds the following two terms
To evaluate the second contribution we can use a similar logic to the all-order proof (2.1) and Taylor expand the in the exponent, the difference being that the integral in the exponent is now between and infinity rather than between minus infinity and infinity. The cancellation between bosons and fermions is therefore incomplete and when acts on the first the integration by parts introduces an extra boundary term
This is nothing else than the diagonal part of . Combining it with the term in 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 in a power series of terms with the connection
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 , then and , next and , etc.
Before finding we should choose one of the supercharges annihilating
the BPS Wilson loop. We take
It is easy to check that
The covariant derivative acting on in has the generalized connection with and in (2.18), but its action on is the same as a covariant derivative in the connection, since the difference between the two, involving and , cancels when acting on . We can therefore integrate the total derivative inside a Wilson loop with either the or connection.
We now solve for in a power series. We take , where the term has field insertions into the Wilson loop with connection . The first few are