Rigid Supersymmetric Theories in Curved Superspace

TL;DR

This paper provides a simplified, uniform framework for constructing rigid supersymmetric theories in curved four-dimensional spacetimes, analyzing specific examples like AdS4, S4, and S3×R, and exploring their partition functions and symmetries.

Contribution

It introduces a streamlined method using classical auxiliary fields for formulating supersymmetric theories in curved backgrounds, simplifying previous approaches and clarifying conditions for various geometries.

Findings

Reproduces known results for AdS4.

Constructs supersymmetric Lagrangian on S4, noting non-reflection positivity for non-conformal theories.

Shows partition function on S3×S1 is parameter-independent and holomorphic in background gauge fields.

Abstract

We present a uniform treatment of rigid supersymmetric field theories in a curved spacetime $\mathcal{M}$, focusing on four-dimensional theories with four supercharges. Our discussion is significantly simpler than earlier treatments, because we use classical background values of the auxiliary fields in the supergravity multiplet. We demonstrate our procedure using several examples. For $\mathcal{M}=AdS_4$ we reproduce the known results in the literature. A supersymmetric Lagrangian for $\mathcal{M}=\mathbb{S}^4$ exists, but unless the field theory is conformal, it is not reflection positive. We derive the Lagrangian for $\mathcal{M}=\mathbb{S}^3\times \mathbb{R}$ and note that the time direction $\mathbb{R}$ can be rotated to Euclidean signature and be compactified to $\mathbb{S}^1$ only when the theory has a continuous R-symmetry. The partition function on $\mathcal{M}=\mathbb{S}^3\times \mathbb{S}^1$ is independent of the parameters of the flat space theory and depends holomorphically on some complex background gauge fields. We also consider R-invariant $\mathcal{N}=2$ theories on $\mathbb{S}^3$ and clarify a few points about them.