Special Lagrangian and deformed Hermitian Yang-Mills on tropical manifold

TL;DR

This paper generalizes the correspondence between special Lagrangian submanifolds and deformed Hermitian Yang-Mills connections to tropical manifolds, removing previous simplifying assumptions and providing new data for the construction.

Contribution

It introduces data for gluing the construction on tropical manifolds and proves a generalized correspondence without assuming the Lagrangian is a section.

Findings

Generalization of the mirror symmetry correspondence to tropical manifolds.

Removal of previous assumptions about Lagrangian submanifolds.

New data for constructing the correspondence on tropical manifolds.

Abstract

From string theory, the notion of deformed Hermitian Yang-Mills connections has been introduced by Mari\~no, Minasian, Moore and Strominger. After that, Leung, Yau and Zaslow proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier-Mukai transform between dual torus fibrations. In their paper, some conditions are imposed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibration.