Towards mirror symmetry \`a la SYZ for generalized Calabi-Yau manifolds

TL;DR

This paper explores mirror symmetry for generalized Calabi-Yau manifolds by analyzing fibrations of flux backgrounds and demonstrating how T-duality on supersymmetric three-tori fibers realizes mirror symmetry, aligning flux properties with previous conjectures.

Contribution

It introduces a framework where mirror symmetry is realized via T-dualizing supersymmetric three-tori fibers in flux backgrounds with static SU(2) structures, extending previous conjectures.

Findings

Product M x t is doubly fibered by supersymmetric three-tori.

Mirror map corresponds to T-dualizing the fibers.

Flux properties match previous mirror symmetry conjectures.

Abstract

Fibrations of flux backgrounds by supersymmetric cycles are investigated. For an internal six-manifold M with static SU(2) structure and mirror \hat{M}, it is argued that the product M x \hat{M} is doubly fibered by supersymmetric three-tori, with both sets of fibers transverse to M and \hat{M}. The mirror map is then realized by T-dualizing the fibers. Mirror-symmetric properties of the fluxes, both geometric and non-geometric, are shown to agree with previous conjectures based on the requirement of mirror symmetry for Killing prepotentials. The fibers are conjectured to be destabilized by fluxes on generic SU(3)xSU(3) backgrounds, though they may survive at type-jumping points. T-dualizing the surviving fibers ensures the exchange of pure spinors under mirror symmetry.