# Small gauge transformations and universal geometry in heterotic theories

## TL;DR

This paper explores the role of small gauge transformations in heterotic supergravity, establishing a gauge fixing method, deriving PDEs for moduli space metrics, and linking gauge choices to connections on the moduli space.

## Contribution

It introduces a holomorphic gauge fixing in heterotic theories, derives PDEs for moduli space metrics, and connects gauge transformations to moduli space connections, advancing understanding of heterotic moduli.

## Key findings

- Holomorphic gauge fixing simplifies heterotic supergravity analysis.
- Perturbative solutions for PDEs determine moduli space metrics.
- Connections on the moduli space relate to gauge choices and deformation theory.

## Abstract

The first part of this paper describes in detail the action of small gauge transformations in heterotic supergravity. We show a convenient gauge fixing is `holomorphic gauge' together with a condition on the holomorphic top form. This gauge fixing, combined with supersymmetry and the Bianchi identity, allows us to determine a set of non-linear PDEs for the terms in the Hodge decomposition. Although solving these in general is highly non-trivial, we give a prescription for their solution perturbatively in alpha' and apply this to the moduli space metric. The second part of this paper relates small gauge transformations to a choice of connection on the moduli space. We show holomorphic gauge is related to a~choice of holomorphic structure and Lee form on a `universal bundle'. Connections on the moduli space have field strengths that appear in the second order deformation theory and we point out it is generically the case that higher order deformations do not commute.