An introduction to supersymmetric field theories in curved space

TL;DR

This review presents a unified, supergravity-based framework for constructing and analyzing supersymmetric field theories on curved manifolds by treating the metric and other background fields as components of off-shell supergravity multiplets. It shows how rigid supersymmetry emerges from generalized Killing spinor equations and how $Z_\mathcal{M}$ and other BPS observables can be computed or constrained without full localization in many cases, through holomorphy and index-like structures. The authors apply the formalism to 4d $\mathcal{N}=1$ theories and 3d $\mathcal{N}=2$ theories, illustrating with examples such as $S^3 \times S^1$ and round/squashed $S^3$ to derive indices, $F$-maximization, and current correlator data. The work highlights the central role of background multiplets, $R$-symmetry, and geometric structures (complex structures, THFs) in determining holomorphic dependence and duality properties, providing a powerful toolkit for exact results in curved space QFTs.

Abstract

In this review, we give a pedagogical introduction to a systematic framework for constructing and analyzing supersymmetric field theories on curved spacetime manifolds. The framework is based on the use of off-shell supergravity background fields. We present the general principles, which broadly apply to theories with different amounts of supersymmetry in diverse dimensions, as well as specific applications to N=1 theories in four dimensions and their three-dimensional cousins with N=2 supersymmetry.