# Comments on supersymmetric renormalization in two-dimensional curved spacetime

## TL;DR

This paper presents a gauge-invariant, supersymmetric regularization method for one-loop divergences in two-dimensional N=(2,2) theories on curved spacetime, demonstrating its application to supersymmetric localization.

## Contribution

It introduces a novel Pauli-Villars regularization procedure that preserves supersymmetry and gauge invariance in curved 2D theories, enabling consistent renormalization of partition functions.

## Key findings

- Partition functions are successfully renormalized using the new method.
- The approach maintains supersymmetry and gauge invariance during regularization.
- A Mathematica notebook is provided to implement supergravity transformations.

## Abstract

In this technical note we introduce a manifestly gauge-invariant and supersymmetric procedure to regularize and renormalize one-loop divergences of chiral multiplets in two-dimensional N=(2,2) theories in curved spacetime. We apply the method, a version of Pauli-Villars regularization, to known examples of supersymmetric localization and demonstrate that the partition functions are renormalized. An accompanying ancillary file, a Mathematica notebook, implements the relevant supergravity transformations.