Gauge Fields on Tori and T-duality

TL;DR

This paper explores gauge fields on tori across dimensions, constructs explicit solutions with topological charges, and demonstrates how T-duality can be understood through zero modes and dual gauge fields, linking topology and duality transformations.

Contribution

It introduces a gauge-theoretical approach to T-duality using explicit gauge fields and zero modes, providing new insights into duality transformations of D-brane systems.

Findings

Explicit gauge fields with topological charges are constructed.

Zero modes are used to derive dual gauge fields on the dual torus.

T-duality transformations are explained via index theorem and Heisenberg modules.

Abstract

We discuss gauge fields on tori in diverse dimensions, mainly in two and four dimensions. We construct various explicit gauge fields which have some topological charges and find the Dirac zero modes in the background of the gauge fields. By using the zero mode, we give new gauge fields on the dual torus, which is a gauge theoretical description of T-duality transformation of the corresponding D-brane systems including DDbar systems. From the transformation, we can easily see the duality expected from the index theorem. It is also mentioned that, for each topological charges, the corresponding constant curvature bundle can be constructed and their duality transformation can be performed in terms of Heisenberg modules.