# Non-Abelian T-duality as a Transformation in Double Field Theory

## TL;DR

This paper demonstrates that Non-Abelian T-duality can be formulated as a coordinate-dependent O(d,d) transformation within Double Field Theory, simplifying calculations and clarifying conditions under which the transformed backgrounds solve supergravity or generalized supergravity equations.

## Contribution

It provides a natural embedding of NATD into Double Field Theory, showing how NATD transformations relate to O(d,d) covariant formulations and their solutions.

## Key findings

- NATD can be described as a coordinate-dependent O(d,d) transformation.
- Unimodular isometry algebras lead to supergravity solutions after NATD.
- Non-unimodular algebras result in solutions to generalized supergravity equations.

## Abstract

Non-Abelian T-duality (NATD) is a solution generating transformation for supergravity backgrounds with non-Abelian isometries. We show that NATD can be described as a coordinate dependent O(d,d) transformation, where the dependence on the coordinates is determined by the structure constants of the Lie algebra associated with the isometry group. Besides making calculations significantly easier, this approach gives a natural embedding of NATD in Double Field Theory (DFT), a framework which provides an O(d,d) covariant formulation for effective string actions. As a result of this embedding, it becomes easy to prove that the NATD transformed backgrounds solve supergravity equations, when the isometry algebra is unimodular. If the isometry algebra is non-unimodular, the generalized dilaton field is forced to have a linear dependence on the dual coordinates, which implies that the resulting background solves generalized supergravity equations.