Novel BPS Wilson loops in three-dimensional quiver Chern-Simons-matter theories

We show that generic three-dimensional $\mathcal N=2$ quiver super Chern-Simons-matter theories admit Bogomol'nyi-Prasad-Sommerfield (BPS) Drukker-Trancanelli (DT) type Wilson loops. We investigate both Wilson loops along timelike infinite straight lines in Minkowski spacetime and circular Wilson loops in Euclidean space. In Aharnoy-Bergman-Jafferis-Maldacena theory, we find that generic BPS DT type Wilson loops preserve the same number of supersymmetries as Gaiotto-Yin type Wilson loops. There are several free parameters for generic BPS DT type Wilson loops in the construction, and supersymmetry enhancement for Wilson loops happens for special values of the parameters.


Introduction
Construction and classification of Bogomol'nyi-Prasad-Sommerfield (BPS) Wilson loops are certainly important subjects in the study of supersymmetric gauge theories. The situation in three dimensions is more complicated than the four-dimensional case. Gaiotto and Yin constructed BPS Wilson loops (along a straight line or a circle) in N = 2 and N = 3 super Chern-Simons-matter (CSM) theories by including scalar fields [1]. This construction is quite similar to 1/2 BPS Wilson loops in fourdimensional N = 4 super Yang-Mills theory [2,3]. The idea of Gaiotto and Yin was adopted in [4][5][6] to construct BPS Wilson loops in Aharnoy-Bergman-Jafferis-Maldacena (ABJM) theory [7]. With some surprise, people only found 1/6 BPS Wilson loops within the class of Gaiotto-Yin (GY) type. However, the study of dual fundamental string solutions in AdS 4 ×CP 3 [4,6] indicates that there should be 1/2 BPS Wilson loops in ABJM theory. About one year later, such Wilson loops were finally constructed by Drukker and Trancanelli [8] via including fermions in a clever way. This construction was explained elegantly through the Brout-Englert-Higgs (BEH) mechanism in [9], and 2/5 BPS Drukker-Trancanelli (DT) type Wilson loops in N = 5 CSM theories [10,11] were also constructed in this paper. Later DT type 1/2 BPS Wilson loops in N = 4 CSM theories were constructed in [12,13]. On the other hand, GY type Wilson loops generically preserve two Poincaré supercharges. The only known exceptional case is the Wilson loops associated with certain end of N = 4 linear quiver theories, where supersymmetry (SUSY) enhancement appears [13].
The previous results may tend to let people assume that DT type Wilson loops are very rare and their existence requires that the theory have a quite large number of supersymmetries. DT type Wilson loops also seem to preserve more supersymmetries than the GY type Wilson loops when they are along the same curve. The result in this letter will show that it is not the case. For generic N = 2 quiver CSM theories, for each bifundamental (or adjoint) matter chiral multiplet we can construct 1/2 BPS DT type Wilson loops. The BPS Wilson loops include the ones along timelike straight lines in Minkowski spacetime and the circular ones in Euclidean space. This construction can be generalized to CSM theories with more supersymmetries. In ABJM theory, we find that generic DT type BPS Wilson loops are 1/6 BPS, and they preserve the same number of supersymmetries as GY type Wilson loops. There are several free parameters in the generic 1/6 BPS DT type Wilson loops, and for special values of the parameters the preserved supersymmetries are enhanced to 1/2 BPS. The generic Wilson loops constructed here are not invariant under local SU (3) transformations which form a subgroup of the SU (4) R-symmetry. This is a big difference from the previously constructed less BPS Wilson loops in ABJM theory [14][15][16][17][18]. In this short letter, we only give the main results and the details of the derivations and some generalizations are presented in [19].

N = quiver CSM theories
We consider generic N = 2 quiver CSM theories with bifundamental matters. We pick two adjacent nodes in the quiver diagram and assume that the corresponding gauge groups are U (N ) and U (M ).
The vector multiplet for gauge group U (N ) includes A µ , σ, χ, D with the last three ones being auxiliary fields. Similarly, for the gauge group U (M ), we have the vector multiplet with fieldsÂ µ ,χ,σ,D. The chiral multiplet in the bifundamental representation of U (N ) × U (M ) consists of the scalar φ, spinor ψ and auxiliary field F . There could be additional matters couple to these two gauge fields. They will enter into the on-shell values of σ andσ in the Wilson loops which we will construct. However the structure of these Wilson loops will not be affected.
As shown in [21] For BPS Wilson loops along straight lines, Poincaré SUSY and conformal SUSY are separately preserved and similar, and so it is enough to consider only Poincaré SUSY. The preserved Poincaré supersymmetries are given by We now construct the DT type Wilson loop along To make it preserve the supersymmetries (2.5), it is enough to require that [9] δL for some Grassmann odd matrix Concretely, one needs We find that the necessary and sufficient conditions for the existence of suchḡ 1 and g 2 arē and the DT type Wilson loop is The 1/2 BPS GY and DT type Wilson loops preserve the same supersymmetries We would like to point out that this construction can also be applied to the case when U (N )×U (M ) is replaced by SO(N ) × U Sp(2M ), and the case when there are matter fields in adjoint representation.

N = 2 quiver CSM theories with multiple matters
Now we turn to the case when there are multiple matter fields in the bifundamental and antibifundamental representations in the N = 2 theory. These multiplets include fields φ i , ψ i , F i and φî, ψî, Fî, respectively. The off-shell SUSY transformations of the gauge fields part (2.1) do not change, and those of the matters part include Here we have i = 1, 2, · · · , N f , andî =1,2, · · · ,Nf . The definitions of the covariant derivatives are We want it to preserve the supersymmetries (2.5). Starting from (2.9), we have the parameterizations withᾱ i , γî, β i , and δî being complex constants, and we also have the following conditions These lead to four classes of solutions.
• Class I γî =δî = 0. • Class III In Euclidean space we still have the GY type 1/2 BPS Wilson loop (2.11), and the preserved supersymmetries are (2.13). We construct the DT type Wilson loop along x µ = (cos τ, sin τ, 0) The on-shell SUSY transformations of ABJM theory are [10,[22][23][24] with the SUSY parameters ǫ IJ = θ IJ + x µ γ µ ϑ IJ ,ǭ IJ =θ IJ −θ IJ x µ γ µ . In Minkowski spacetime θ IJ , θ IJ , ϑ IJ ,θ IJ are Dirac spinors with constraints Symbol ǫ IJKL is totally antisymmetric with ǫ 1234 = 1. In Euclidean space the constraints become We have definitions of covariant derivatives In Minkowski spacetime, a general GY type Wilson loop along x µ = τ δ µ 0 takes the form We find that, up to some SU (4) R-symmetry transformation, the only BPS GY type Wilson loop is the one with R I J = S I J = diag(−1, −1, 1, 1). It is 1/6 BPS and preserves the supersymmetries γ 0 θ 12 = iθ 12 , γ 0 θ 34 = −iθ 34 , This is just the Wilson loop that was constructed in [4][5][6]. Especially, we find that we do not need to require that R I J or S J I is a hermitian matrix a priori, and we can show that it is the result of SUSY invariance.
We turn to constructing a DT type Wilson loop that preserves at least the supersymmetries (4.6).
In Minkowski spacetime, a general DT type Wilson loop is [8]
In Euclidean space, one can construct the circular 1/6 BPS GY type Wilson loop along x µ = (cos τ, sin τ, 0) The preserved supersymmetries are We also construct the DT type Wilson loop along x µ = (cos τ, sin τ, 0) and (4.14) that make this DT type Wilson 1/6 BPS, and the preserved supersymmetries are (4.22).

Conclusion and Discussion
The Our results inspire quite a few interesting problems for further exploration. It would be nice to figure out the holographic duals of these novel BPS Wilson loops. It is also worth searching for the origin of these loop operators, beginning with the BEH mechanism as in [9]. One can also try to construct and study some new BPS cusped Wilson loops [14,18] using these new DT type BPS Wilson loops as building blocks. We hope such cusped BPS Wilson loops still play the role as one of the bridges between the localization computations and integrable structure as first proposed in [26]. It is also quite interesting to investigate whether DT type BPS Wilson loops exist in four-dimensional N = 4 or N = 2 quiver gauge theories.

Acknowledgement
We would like to thank Nan Bai, Bin Chen, Song He, Yu Jia, Wei Li, Jian-Xin Lu, Gary Shiu and