Fourier-Mukai Transform and Mirror Symmetry for D-Branes on Elliptic Calabi-Yau

TL;DR

This paper explores the Fourier-Mukai transform and mirror symmetry for D-branes on elliptic Calabi-Yau three-folds, revealing how twisted operations relate to monodromies and moduli spaces, with explicit examples and spectral cover connections.

Contribution

It establishes the adiabatic form of fibrewise T-duality for D-branes on elliptic Calabi-Yau three-folds using twisted cohomology operations and links Fourier-Mukai transforms to monodromies in mirror symmetry.

Findings

Twisted cohomology operations are necessary for the expected T-duality form.

Explicit connection between Fourier-Mukai transforms and monodromies is demonstrated.

Analysis of the moduli space of D4-branes under Fourier-Mukai transform with spectral cover application.

Abstract

Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau three-fold $X$ is seen to have an expected adiabatic form for its induced cohomology operation only when an appropriately twisted operation resp. twisted charge is defined. Some differences with the case of $K3$ as well as connections with the spectral cover construction for bundles on $X$ are pointed out. In the context of mirror symmetry Kontsevich's association of line bundle twists (resp. a certain 'diagonal' operation) with monodromies (esp. the conifold monodromy) is made explicit and checked for two example models. Interpreting this association as a relation between FM transforms and monodromies, we express the fibrewise FM transform through known monodromies. The operation of fibrewise duality as well as the notion of a certain index relevant to the computation of the moduli space of the bundle is transported to the sLag side. Finally the moduli space for D4-branes and its behaviour under the FM transform is considered with an application to the spectral cover.