Supersymmetry on curved spaces and superconformal anomalies

TL;DR

This paper explores how unbroken supersymmetry constrains anomalies and background fields in four-dimensional curved space theories, revealing simplifications and topological features in the anomalies, with implications for holography.

Contribution

It demonstrates how supersymmetry fixes background vector fields and simplifies anomalies in curved space field theories, especially relating to the Weyl tensor and topological terms.

Findings

Trace anomaly becomes purely topological.

Supersymmetry imposes relations between background fields and curvature.

Vanishing of certain anomaly terms under specific supersymmetric conditions.

Abstract

We study the consequences of unbroken rigid supersymmetry of four-dimensional field theories placed on curved manifolds. We show that in Lorentzian signature the background vector field coupling to the R-current is determined by the Weyl tensor of the background metric. In Euclidean signature, the same holds if two supercharges of opposite R-charge are preserved, otherwise the (anti-)self-dual part of the vector field-strength is fixed by the Weyl tensor. As a result of this relation, the trace and R-current anomalies of superconformal field theories simplify, with the trace anomaly becoming purely topological. In particular, in Lorentzian signature, or in the presence of two Euclidean supercharges of opposite R-charge, supersymmetry of the background implies that the term proportional to the central charge c vanishes, both in the trace and R-current anomalies. This is equivalent to the vanishing of a superspace Weyl invariant. We comment on the implications of our results for holography.