Generalized complex geometry and T-duality

TL;DR

This paper explores how generalized complex geometry aligns with T-duality, providing a mathematical framework that extends known physical rules and offers new insights into the geometric structures involved.

Contribution

It demonstrates the compatibility of generalized complex geometry with T-duality through Courant algebroid isomorphisms and introduces a reinterpretation of T-duality as Courant reduction.

Findings

Transport of geometric structures via Courant algebroid isomorphisms

Extension of Buscher rules to generalized geometries

Viewing T-duality as a generalized complex submanifold (D-brane)

Abstract

We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists. T-duality relates topologically distinct torus bundles, and prescribes a method for transporting geometrical structures between them. We describe how this relation may be understood as a Courant algebroid isomorphism between the spaces in question. This then allows us to transport Dirac structures, generalized Riemannian metrics, generalized complex and generalized Kahler structures, extending the "Buscher rules" well-known to physicists. Finally, we re-interpret T-duality as a Courant reduction, and explain that T-duality between generalized complex manifolds may be viewed as a generalized complex submanifold (D-brane) of the product, in a way that establishes a direct analogy with the Fourier-Mukai transform.