Non-Abelian localization for supersymmetric Yang-Mills-Chern-Simons theories on a Seifert manifold

Abstract

We derive non-Abelian localization formulae for supersymmetric Yang-Mills-Chern-Simons theory with matters on a Seifert manifold M , which is the threedimensional space of a circle bundle over a two-dimensional Riemann surface Σ, by using the cohomological approach introduced by Källén. We find that the partition function and the vev of the supersymmetric Wilson loop reduces to a finite dimensional integral and summation over classical flux configurations labeled by discrete integers. We also find the partition function reduces further to just a discrete sum over integers in some cases, and evaluate the supersymmetric index (Witten index) exactly on S 1 × Σ. The index completely agrees with the previous prediction from field theory and branes. We discuss a vacuum structure of the ABJM theory deduced from the

Seifert fibering operators in 3d $$ \mathcal{N}=2 $$ theories

“…Let us now consider the vector multiplet V for the gauge group G with Lie algebra g. The (q, p) fibering operator has contributions both from the 3d H = U (1) rk(G) vector multiplets V a in the maximal torus of G, and from the massive "W-bosons" on the Coulomb branch. 43 See Appendix D for details on the derivation of (4.74).…”

Exact results in quiver quantum mechanics and BPS bound state counting

“…A possible gauge invariant function made of Φ is a supersymmetric Wilson (Polyakov) loop operator along the Euclidean "time" direction: 24) where P stands for the path ordered product and the trace is taken over the representation R. We can evaluate the vev of W R (Φ) exactly by using the localization in principle. Secondly, another interesting physical observable is obtained from the dimensional reduction of the supersymmetric Chern-Simons (CS) action in three dimensions or BF action in two dimensions, which explains why lower dimensional gauge theories are exactly solvable [34][35][36][37]. In one dimensional model, dimensionally reduced CS type operator becomes…”

“…If we first take the limit of t → ∞, we find that the path integral becomes WKB exact and localizes at the fixed points QF I = QB I = 0, since the action with the coupling t is essentially Gaussian. The Gaussian integral also induces Jacobians (1-loop determinants) to the measure, which are given by super determinants (super Hessian) of the BRST transformations evaluated at the fixed points [37][38][39][40][41][42] ∆(Φ) = det…”

Coulomb branch localization in quiver quantum mechanics

“…Secondly, since the interaction term with the bulk gauge field is now pure imaginary and it is a phase in the path integral, we evaluate them at the fixed points just as a Q -closed operator, like the supersymmetric Chern-Simons term in the Euclidean three-dimensional space-time [17]. Using the gauge (3.18) of the minus sign, we find at the fixed point…”

“…From now on, we suppress the subscript 0 for the constant modes. The 1-loop determinant for the vector multiplet is given by 17) where Sdet denotes the superdeterminant. So we find…”