Central charges of T-dual branes for toric varieties

TL;DR

This paper establishes a precise relationship between the SYZ T-dual of equivariant sheaves on toric orbifolds and genus-zero Gromov-Witten invariants, linking mirror symmetry and enumerative geometry.

Contribution

It proves that the oscillatory integral over the T-dual Lagrangian equals the genus-zero Gromov-Witten potential with a Gamma class insertion, connecting mirror symmetry to enumerative invariants.

Findings

Oscillatory integral matches Gromov-Witten potential

Establishes a link between T-duality and enumerative geometry

Provides explicit formulas for toric orbifolds

Abstract

Given any equivariant coherent sheaf $\mathcal L$ on a compact semi-positive toric orbifold $\mathcal X$, its SYZ T-dual mirror dual is a Lagrangian brane in the Landau-Ginzburg mirror. We prove the oscillatory integral of the equivariant superpotential in the Landau Ginzburg mirror over this Lagrangian brane is the genus-zero $1$-descendant Gromov-Witten potential with a Gamma-type class of $\mathcal L$ inserted.