## Abstract

We study the consequences of unbroken rigid supersymmetry of fourdimensional 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 fieldstrength 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.

Original language | English |
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Article number | 25 |

Number of pages | 30 |

Journal | Journal of High Energy Physics |

Volume | 2013 |

Issue number | 10 |

DOIs | |

Publication status | Published - 3 Oct 2013 |

## Keywords

- Supersymmetric gauge theory
- Anomalies in Field and String Theories
- AdS-CFT Correspondence
- Differential and Algebraic Geometry
- HOLOGRAPHIC RENORMALIZATION
- GAUGED SUPERGRAVITY
- CONFORMAL SUPERGRAVITY
- COEFFICIENTS
- MANIFOLDS
- SYMMETRY
- FIELDS