Topological Correlators and Surface Defects from Equivariant Cohomology

Abstract

We find a one-dimensional protected subsector of N = 4 matter theories on a general class of three-dimensional manifolds. By means of equivariant localization we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on S 3 . Then, we apply it to the novel case of S 2 × S 1 and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models form a topological ring and that their correlation functions are naturally associated with a noncommutative star product. Finally, we couple the three-dimensional theory to general N = (2, 2) surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.

Schur correlation functions from q-deformed Yang-Mills

“…Recently, correlation functions of Schur operators on the torus are computed via supersymmetric localization [7,8] and are further mapped to the deformation quanzation of the Higgs branch of vacua [9][10][11][12][13][14] of the circlereduced theory in 3d. The aim of this letter is to propose a parallel approach to computing some of these correlators, by constructing the wave functions corresponding to the affine current operators for a class of 4d SCFTs and show that they are determined by the correlators of critical current algebras.…”