The Deformed Hermitian–Yang–Mills Equation in Geometry and Physics

Abstract

To Nigel Hitchin, with admiration, on the occasion of his 70th birthday.Abstract. We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kähler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, discuss some recent progress towards its solution. In dimension 3 we prove a new Chern number inequality and discuss the relationship with algebraic stability conditions.

Concave elliptic equations and generalized Khovanskii-Teissier inequalities

“…Remark 2.24. The Lagrangian phase equation, also called the deformed Hermitian-Yang-Mills (dHYM) equation, has recently seen a great deal of interest, owing to its importance in mirror symmetry [23,22,7,4,17]. In particular, the author, with Jacob and Yau [4], and the author and Yau [8] have formulated conjectures relating algebraic geometry and the solvability of the dHYM which would, in principle, determine the set Sol ω (Θ, Γ (n−1) π…”

A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation

“…The both are complex Hessian equations on Kähler manifolds and related to each other via the "small radius limit" as observed in [CXY17,CY18], i.e. we replace ω with sω (s ∈ (0, ∞)) and consider the limit s → ∞.…”

Hypercritical deformed Hermitian-Yang-Mills equation

“…It is believed that the dHYM equation plays a fundamental role in mirror symmetry and its solvability is expected to be related to deep notions of stability in algebraic geometry. We refer the reader to [11,13] and the references therein for an introduction to the physical and mathematical aspects of the dHYM equation.…”

“…We refer readers to the arxiv version [14] of [15] for more discussion on the relation between the GIT approach and the algebraic obstruction on the existence. In this work, we will continue the study of the existence problem to the dHYM equation from the viewpoint of H. We assume H to be non-empty so that θ 0 is well-defined modulo 2π, see [30,13]. Without loss of generality, we will assume 0 ∈ H. And we will work on the case when [α] has hypercritical phase, i.e., θ 0 ∈ (0, π 2 ).…”