Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds

TL;DR

This paper computes exact partition functions of supersymmetric gauge theories on non-compact hyperbolic three-manifolds, specifically Euclidean AdS$_3$ quotients, using localization and multiple determinant evaluation techniques.

Contribution

It provides a detailed localization framework for supersymmetric theories on non-compact hyperbolic manifolds, including boundary condition analysis and multiple determinant computation methods.

Findings

Exact partition functions computed for theories on hyperbolic three-manifolds.

Demonstrated equivalence of different one-loop determinant calculation methods.

Analyzed supersymmetry preservation and boundary conditions in non-compact settings.

Abstract

We study supersymmetric gauge theories with an R-symmetry, defined on non-compact, hyperbolic, Riemannian three-manifolds, focusing on the case of a supersymmetry-preserving quotient of Euclidean AdS$_3$. We compute the exact partition function in these theories, using the method of localization, thus reducing the problem to the computation of one-loop determinants around a supersymmetric locus. We evaluate the one-loop determinants employing three different techniques: an index theorem, the method of pairing of eigenvalues, and the heat kernel method. Along the way, we discuss aspects of supersymmetry in manifolds with a conformal boundary, including supersymmetric actions and boundary conditions.