Natural curvature for manifest T-duality

TL;DR

This paper presents a reformulation of T-duality in string theory using a geometric approach that doubles coordinates and directly couples the Lorentz connection to the string, providing a covariant framework for T-dual curvature.

Contribution

It introduces a new geometric formulation of manifest T-duality that doubles coordinates for translations and Lorentz transformations, directly coupling the Lorentz connection to the string.

Findings

Reproduces the traditional T-dual torsion definition.

Provides a covariant, automatic definition of T-dual curvature.

Establishes a geometric framework for T-duality with doubled coordinates.

Abstract

We reformulate the manifestly T-dual description of the massless sector of the closed bosonic string, directly from the geometry associated with the (left and right) affine Lie algebra of the coset space Poincare/Lorentz. This construction initially doubles not only the (spacetime) coordinates for translations but also those for Lorentz transformations (and their dual). As a result, the Lorentz connection couples directly to the string (as does the vielbein), rather than being introduced ad hoc to the covariant derivative as previously. This not only reproduces the old definition of T-dual torsion, but automatically gives a general, covariant definition of T-dual curvature (but still with some undetermined connections).