# Lie Algebroid Gauging of Non-linear Sigma Models

## TL;DR

This paper explores the conditions and methods for gauging non-linear sigma models using Lie algebroids, highlighting the non-uniqueness of gauging and limitations in extending T-duality.

## Contribution

It provides a comprehensive analysis of Lie algebroid gauging, including conditions, local-to-global issues, and the constraints preventing T-duality extension.

## Key findings

- Gauging with involutive vector fields is always locally possible.
- Gauging is not unique; flat connections can be chosen freely.
- Adding a field strength term to enforce equivalence is generally not feasible.

## Abstract

This paper examines a proposal for gauging non-linear sigma models with respect to a Lie algebroid action. The general conditions for gauging a non-linear sigma model with a set of involutive vector fields are given. We show that it is always possible to find a set of vector fields which will (locally) admit a Lie algebroid gauging. Furthermore, the gauging process is not unique; if the vector fields span the tangent space of the manifold, there is a free choice of a flat connection. Ensuring that the gauged action is equivalent to the ungauged action imposes the real constraint of the Lie algebroid gauging proposal. It does not appear possible (in general) to find a field strength term which can be added to the action via a Lagrange multiplier to impose the equivalence of the gauged and ungauged actions. This prevents the proposal from being used to extend T-duality. Integrability of local Lie algebroid actions to global Lie groupoid actions is discussed.