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The deformed Hermitian-Yang-Mills equation in geometry and physics. (English) Zbl 1421.35300

Andersen, Jørgen Ellegaard (ed.) et al., Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 1. Oxford: Oxford University Press. 69-90 (2018).
Summary: 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, and some recent progress towards its solution. In dimension 3 we prove a new Chern number inequality and discuss the relationship with algebraic stability conditions.
For the entire collection see [Zbl 1408.14005].

MSC:

35Q40 PDEs in connection with quantum mechanics
53C55 Global differential geometry of Hermitian and Kählerian manifolds
81T13 Yang-Mills and other gauge theories in quantum field theory
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14J33 Mirror symmetry (algebro-geometric aspects)
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