Localization for ${\cal N}=2$ Supersymmetric Gauge Theories in Four Dimensions

TL;DR

This paper reviews the application of supersymmetric localization techniques to four-dimensional  supersymmetric gauge theories on curved backgrounds, highlighting new geometries and detailed examples like the topological four-sphere.

Contribution

It introduces new geometric backgrounds for localization, including local T^2-bundle fibrations and detailed analysis of T^2-invariant metrics on the four-sphere.

Findings

Identification of new supersymmetric backgrounds on curved four-manifolds

Explicit construction of T^2-invariant metrics on the four-sphere

Demonstration of localization results in these novel geometries

Abstract

This is the 5th article in the collection of reviews "Exact results on N=2 supersymmetric gauge theories", ed. J. Teschner. We review the supersymmetric localization of $\mathcal{N}=2$ theories on curved backgrounds in four dimensions using $\mathcal{N}=2$ supergravity and generalised conformal Killing spinors. We review some known backgrounds and give examples of new geometries such as local $T^2$-bundle fibrations. We discuss in detail a topological four-sphere with generic $T^2$-invariant metric.