Moment maps, nonlinear PDE, and stability in mirror symmetry

TL;DR

This paper investigates the deformed Hermitian-Yang-Mills equation through a variational lens, establishing existence results for geodesics, linking algebraic invariants to stability, and exploring mirror symmetry implications.

Contribution

It introduces a variational framework for the dHYM equation, proves existence of smooth and weak geodesics, and connects algebraic invariants to stability conditions in mirror symmetry.

Findings

Existence of smooth approximate geodesics in hypercritical phase

Weak geodesics with $C^{1,eta}$ regularity established

Algebraic invariants provide obstructions to dHYM solutions

Abstract

We study the deformed Hermitian-Yang-Mills (dHYM) equation, which is mirror to the special Lagrangian equation, from the variational point of view via an infinite dimensional GIT problem mirror to Thomas' GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold $\mathcal{H}$ mirror to Solomon's space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with $C^{1,\alpha}$ regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. We apply these results to the infinite dimensional GIT problem for deformed Hermitian-Yang-Mills. We associate algebraic invariants to certain birational models of $X\times \Delta$, where $\Delta \subset \mathbb{C}$ is a disk. Using the existence of regular weak geodesics we prove that these invariants give rise to obstructions to the existence of solutions to the dHYM equation. Furthermore, we show that these invariants fit into a stability framework closely related to Bridgeland stability. Finally, we use a Fourier-Mukai transform on toric K\"ahler manifolds to describe degenerations of Lagrangian sections of SYZ torus fibrations of Landau-Ginzburg models $(Y,W)$. We speculate on the resulting algebraic invariants, and discuss the implications for relating Bridgeland stability to the existence of special Lagrangian sections of $(Y,W)$.