Construction and classification of novel BPS Wilson loops in quiver Chern-Simons-matter theories

In this paper we construct and classify novel Drukker-Trancanelli (DT) type BPS Wilson loops along infinite straight lines and circles in $\mathcal N=2,3$ quiver superconformal Chern-Simons-matter theories, Aharony-Bergman-Jafferis-Maldacena (ABJM) theory, and $\mathcal N=4$ orbifold ABJM theory. Generally we have four classes of Wilson loops, and all of them preserve the same supersymmetries as the BPS Gaiotto-Yin (GY) type Wilson loops. There are several free complex parameters in the DT type BPS Wilson loops, and for two classes of Wilson loops in ABJM theory and $\mathcal N=4$ orbifold ABJM theory there are supersymmetry enhancements at special values of the parameters. We check that the differences of the DT type and GY type Wilson loops are $Q$-exact with $Q$ being some supercharges preserved by both the DT type and GY type Wilson loops. The results would be useful to calculate vacuum expectation values of the DT type Wilson loops in matrix models if they are still BPS quantum mechanically.


BPS Wilson loop in ABJM theory was constructed in
To provide a matrix model calculation for the VEV of this N = 2 quiver CSM theories -vector multiplets Let us consider generic N = 2 quiver SCSM theories with bifundamental matters.
Let us pick two adjacent nodes in the quiver diagram and the corresponding gauge group are U (N 1 ) and U (N 2 ). The Chern-Simons levels are k 1 and k 2 , respectively.
The vector multiplet for gauge group U (N 1 ) include A µ , σ, χ, D and the last three fields are the auxiliary fields. Similarly for gauge group SU (N 2 ) we have the vector multipletÂ µ ,χ,σ,D. N = 2 quiver CSM theories -chiral multiplets The chiral multiplet in the bifunamental representation of U (N 1 ) × U (N 2 ) includes the scalar φ, the spinor ψ and the auxiliary field F .

Supersymmetry transformation
For the vector multiplet part, we only need the off-shell supersymmetry transformation of A µ , σ,Â µ ,σ is,

Supersymmetry transformation
For the vector multiplet part, we only need the off-shell supersymmetry transformation of A µ , σ,Â µ ,σ is, For the matter part we only need the off-shell supersymmetry transformation of φ and ψ δφ = iθψ, δφ = iψθ,

GY type BPS Wilson loops
In Minkowski spacetime, one can construct a GY type 1/2 BPS Wilson loop along an infinite straight line x µ = τ δ µ 0 as Jun-Bao Wu IHEP-CAS 31/65

GY type BPS Wilson loops
In Minkowski spacetime, one can construct a GY type 1/2 BPS Wilson loop along an infinite straight line x µ = τ δ µ 0 as The preserved SUSY can be denoted as Jun-Bao Wu IHEP-CAS 32/65
To make it preserve the SUSY in (4) at least classically, it is enough to require that [K. Lee, S. Lee, 2010] for some Grassmann odd matrix

DT-type Wilson loops -II
We find that the necessary and sufficient conditions for the existence of suchḡ 1 and g 2 arē Such DT type Wilson loop is 1/2 BPS, and the preserved SUSY is the same as (4). Note that there are two free complex parametersᾱ and β in the Wilson loop, and they can be any complex constants. Whenᾱ = β = 0, it becomes the GY type Wilson loop.

Generalizations
There could be other matters couple to these two gauge fields. They will change the on-shell values of σ andσ in the Wilson loops we will construct. The structure of these Wilson loops will not be changed.
We also constructed half-BPS circular Wilson loops for N = 2 superconformal quiver Chern-Simons theory in Euclidean space.
Jun-Bao Wu IHEP-CAS 37/65 Generalizations There could be other matters couple to these two gauge fields. They will change the on-shell values of σ andσ in the Wilson loops we will construct. The structure of these Wilson loops will not be changed.
We also constructed half-BPS circular Wilson loops for N = 2 superconformal quiver Chern-Simons theory in Euclidean space.
This construction can be also applied to the case when U (N ) is replaced by SO(N ) or U Sp(N ), and the case when there are matter fields in the adjoint representation. For the last case, one just simply letÂ µ ≡ A µ andσ ≡ σ. Generalizations There could be other matters couple to these two gauge fields. They will change the on-shell values of σ andσ in the Wilson loops we will construct. The structure of these Wilson loops will not be changed.
We also constructed half-BPS circular Wilson loops for N = 2 superconformal quiver Chern-Simons theory in Euclidean space.
This construction can be also applied to the case when U (N ) is replaced by SO(N ) or U Sp(N ), and the case when there are matter fields in the adjoint representation. For the last case, one just simply letÂ µ ≡ A µ andσ ≡ σ.
The case with multi matter fields in the bifundamental and anti-bifundamental representations were also considered. The DT type Wilson loops can be divided into four classes.

GY-type Wilson loops in ABJM theory -I
In ABJM theory, there are four scalars φ I and four fermions ψ I in the bifundamental representation.
A general GY type Wilson loop along the timelike infinite straight line x µ = τ δ µ 0 takes the form Novel DT-type Wilson loops in ABJM theory -I We turn to constructing a DT type Wilson loop along a straight line that preserves at least the supersymmetries (10). A general DT type Wilson loop is f 1 = 2π kζ I ψ I |ẋ|, f 2 = 2π kψ I η I |ẋ|.