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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 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. 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. We study the consequences of unbroken rigid supersymmetry of four-dimensional field theories placed on curved manifolds.

Curved space supersymmetry in 2d