Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

T-dual branes on hyperkähler manifolds

Maria Anna Sisak

Abstract

This submission is a PhD dissertation. Kapustin and Witten conjectured that there is a mirror symmetry relation between the hyperkähler 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 hyperkähler structures on algebraic integrable systems. In this thesis, we investigate the T-duality relation between hyperkähler structures and the corresponding branes on affine torus bundles. We use the techniques of generalized geometry to show that semi-flat hyperkähler 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.

T-dual branes on hyperkähler manifolds

Abstract

This submission is a PhD dissertation. Kapustin and Witten conjectured that there is a mirror symmetry relation between the hyperkähler 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 hyperkähler structures on algebraic integrable systems. In this thesis, we investigate the T-duality relation between hyperkähler structures and the corresponding branes on affine torus bundles. We use the techniques of generalized geometry to show that semi-flat hyperkähler 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.

Paper Structure

This paper contains 81 sections, 82 theorems, 848 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment:
  3. Affine torus bundles and integrable systems
  4. Tori
  5. Complex tori.
  6. Dual tori.
  7. Affine torus bundles
  8. Invariant forms.
  9. Connections.
  10. Leray spectral sequence.
  11. Algebraic integrable systems
  12. Special Kähler manifolds.
  13. Special Kähler geometry on the base of an algebraic integrable system.
  14. Semi-flat hyperkähler structure on an algebraic integrable system
  15. Generalized geometry
  16. ...and 66 more sections

Key Result

Theorem 1

Let ${\mathcal{L}}=(S,F)$ be a locally T-dualizable brane in $M$ with $\pi(S)$ simply connected. Then, there exists a foliation of $S\times_{\pi(S)}\hat{M}$ by affine torus subbundles denoted by $Z$. For each such $Z$ we denote by $\hat{S}_Z\subset \hat{M}$ the image of $Z$ under the projection $S\t where $P\in \Omega^2(M\times_B\hat{M})$ is a closed invariant two-form and $p_Z:Z\rightarrow S$ and

Theorems & Definitions (205)

  • Definition : Definition \ref{['locally T-dualizable brane']}
  • Theorem : Theorem \ref{['local gg thm']}
  • Theorem : Theorem \ref{['global thm1']}
  • Theorem : Theorem \ref{['tdual U(1) bundles trivial base']}
  • Theorem : Theorem \ref{['tdual u(1) bundle general base']}
  • Theorem : Theorem \ref{['last global']}
  • Proposition : Proposition \ref{['leaf space of coisotropic 1']} and Proposition \ref{['leaf space of coisotropic 2']}
  • Theorem : Theorem \ref{['coisotropic in integrable system']}
  • Theorem : Theorem \ref{['semiflat tdual']}
  • Definition 2.1.1
  • ...and 195 more