Supersymmetry on curved spaces and superconformal anomalies

Understand

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.