Supersymmetric Gauge Theories on the Five-Sphere

TL;DR

This work constructs Euclidean five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories on the five-sphere $S^5$ with vector and hypermultiplets. It derives the SUSY transformations and the action using both the Noether procedure and the rigid limit of off-shell 5d supergravity, showing that the preserved supercharge generates a circle fibration over $\mathbb{CP}^2$. Through localization, the authors show that the path integral localizes to a generalized instanton sector on $\mathbb{CP}^2$ together with covariantly constant Coulomb moduli, though the explicit integral remains to be evaluated. This framework clarifies the connection between 5d theories on $S^5$ and the 6d $\mathcal{N}=(2,0)$ CFT on a circle, and sets up a route to compute exact partition functions and their deformations, including squashed backgrounds. Key contributions include the explicit construction on a curved background and the reduction of the path integral to CP^2-instanton sectors plus Coulomb data.

Abstract

We construct Euclidean 5d supersymmetric gauge theories on the five-sphere with vector and hypermultiplets. The SUSY transformation and the action are explicitly determined from the standard Noether procedure as well as from off-shell supergravity. Using localization techniques, the path-integral is shown to be restricted to the integration over a generalization of instantons on CP^2 and the Coulomb moduli.