T-dual branes on hyperk\"ahler manifolds

TL;DR

This dissertation explores the T-duality relationship between hyperk"ahler structures and branes on algebraic integrable systems, providing a geometric framework for mirror symmetry conjectures in Higgs bundle moduli spaces.

Contribution

It demonstrates that semi-flat hyperk"ahler structures are T-dual on algebraic integrable systems using generalized geometry techniques, and extends T-duality to generalized branes and their submanifolds.

Findings

T-duality relates hyperk"ahler structures on integrable systems.

T-duality of generalized branes is described and extended.

Specialization to BBB and BAA-branes confirms mirror symmetry conjectures.

Abstract

This submission is a PhD dissertation. Kapustin and Witten conjectured that there is a mirror symmetry relation between the hyperk\"ahler structures on certain Higgs bundle moduli spaces. As a consequence, they conjecture an equivalence between categories of BBB and BAA-branes. At the classical level, this mirror symmetry is given by T-duality between semi-flat hyperk\"ahler structures on algebraic integrable systems. In this thesis, we investigate the T-duality relation between hyperk\"ahler structures and the corresponding branes on affine torus bundles. We use the techniques of generalized geometry to show that semi-flat hyperk\"ahler structures are T-dual on algebraic integrable systems. We also describe T-duality for generalized branes. Motivated by Fourier-Mukai transform we upgrade the T-duality between generalized branes to T-duality of submanifolds endowed with U(1)-bundles and connections. This T-duality in the appropriate context specializes to T-duality between BBB and BAA-branes.