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

TL;DR

The paper develops a unified framework for exact results in four-dimensional ${\rm N}=2$ supersymmetric gauge theories on curved spaces by coupling to ${\rm N}=2$ supergravity and solving generalized conformal Killing spinor equations. It surveys background geometries—most notably ${S^4_{\epsilon_1,\epsilon_2}}$ and local ${T^2}$-bundles—that preserve supersymmetry and enable localization, including topological twists, Omega-backgrounds, and conformal Killing spinor setups. Localization reduces the partition function to fixed-point data computed via equivariant indices, yielding a factorization into north/south pole contributions tied to Nekrasov-like instanton sums and one-loop determinants. The work connects the four-dimensional theory to rich geometric and algebraic structures (e.g., the Atiyah-Singer index theorem, fixed-point localization, and the AGT-like relations via sphere partition functions) and outlines open directions in classifying admissible backgrounds beyond the analyzed ${T^2}$-bundle 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.