# Quantum consistency in supersymmetric theories with $R$-symmetry in curved space

## TL;DR

This paper investigates the quantum consistency of four-dimensional rigid $ ext{N}=1$ supersymmetric theories with $U(1)_R$ symmetry in curved space, showing that anomaly cancellation is necessary for unbroken supersymmetry.

## Contribution

It demonstrates that the $U(1)_R$ anomaly coefficient must vanish for quantum consistency in supersymmetric curved space theories, based on correlation function analysis.

## Key findings

- $U(1)_R$ anomaly coefficient must be zero for consistency
- Supersymmetry remains unbroken only if the anomaly vanishes
- Correlation functions confirm the anomaly cancellation condition

## Abstract

We discuss consistency at the quantum level in the rigid $\mathcal N=1$ supersymmetric field theories with a $U(1)_R$ symmetry in four-dimensional curved space which are formulated via coupling to the new-minimal supergravity background fields. By analyzing correlation functions of the current operators in the $\mathcal{R}$-multiplet, we show that the quantum consistency with the (unbroken) supersymmetry requires the $U(1)_R$ anomaly coefficient, which depends only on the field content of the theory, to vanish. This consistency condition is obtained under the assumption that the supercurrent Ward identity is non-anomalous and that the vacuum is supersymmetric.