Supersymmetric localization in two dimensions

TL;DR

This paper reviews localization techniques in supersymmetric 2D gauge theories, detailing how to compute partition functions and correlators on various curved backgrounds, with applications to mirror symmetry and dualities.

Contribution

It provides an introductory overview of constructing Lagrangians and calculating exact partition functions and elliptic genera in 2D supersymmetric theories on curved spaces.

Findings

Computed partition functions on spheres and other backgrounds

Evaluated elliptic genus for N=(0,2) theories

Applied localization to study mirror symmetry and dualities

Abstract

This is an introductory review to localization techniques in supersymmetric two-dimensional gauge theories. In particular we describe how to construct Lagrangians of N=(2,2) theories on curved spaces, and how to compute their partition functions and certain correlators on the sphere, the hemisphere and other curved backgrounds. We also describe how to evaluate the partition function of N=(0,2) theories on the torus, known as the elliptic genus. Finally we summarize some of the applications, in particular to probe mirror symmetry and other non-perturbative dualities.