Topological Correlators and Surface Defects from Equivariant Cohomology

TL;DR

This paper develops a localization method to compute BPS correlators in 3D $ ext{N}=4$ theories, revealing a connection to one-dimensional quantum mechanics and topological structures, applicable to various manifolds and coupled defects.

Contribution

It introduces a novel equivariant localization approach that reduces 3D theories to 1D quantum mechanics, extending previous results and incorporating surface defects for comprehensive BPS correlator analysis.

Findings

Localization on $S^3$ reproduces known results.

New localization on $S^2 imes S^1$ yields disjoint quantum mechanics.

BPS operators form a noncommutative star product, with topological correlators.

Abstract

We find a one-dimensional protected subsector of $\mathcal{N}=4$ 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 $S^2 \times S^1$ and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models are naturally associated with a noncommutative star product, while their correlation functions are essentially topological. Finally, we couple the three-dimensional theory to general $\mathcal{N}=(2,2)$ surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.