# Supersymmetry anomalies in $\mathcal{N}=1$ conformal supergravity

## TL;DR

This paper determines the superconformal anomalies in four-dimensional $
=1$ conformal supergravity, including fermionic anomalies, and explores their implications for supersymmetry on curved backgrounds, extending previous holographic results.

## Contribution

It explicitly solves the Wess-Zumino consistency conditions for $
=1$ supergravity, generalizing anomaly results to arbitrary $a$ and $c$ coefficients and including fermionic anomalies.

## Key findings

- $
$-supersymmetry is anomalous if and only if R-symmetry is anomalous.
- The anomalies deform the supersymmetry algebra on curved backgrounds with Killing spinors.
- Results extend holographic anomaly calculations to arbitrary $a$ and $c$.

## Abstract

We solve the Wess-Zumino consistency conditions of $\mathcal{N}=1$ off-shell conformal supergravity in four dimensions and determine the general form of the superconformal anomalies for arbitrary $a$ and $c$ anomaly coefficients to leading non trivial order in the gravitino. Besides the well known Weyl and $R$-symmetry anomalies, we compute explicitly the fermionic $\mathcal{Q}$- and $\mathcal{S}$-supersymmetry anomalies. In particular, we show that $\mathcal{Q}$-supersymmetry is anomalous if and only if $R$-symmetry is anomalous. The $\mathcal{Q}$- and $\mathcal{S}$-supersymmetry anomalies give rise to an anomalous supersymmetry transformation for the supercurrent on curved backgrounds admitting Killing spinors, resulting in a deformed rigid supersymmetry algebra. Our results may have implications for supersymmetric localization and supersymmetry phenomenology. Analogous results are expected to hold in dimensions two and six and for other supergravity theories. The present analysis of the Wess-Zumino consistency conditions reproduces the holographic result of arxiv:1703.04299 and generalizes it to arbitrary $a$ and $c$ anomaly coefficients.