From the Donaldson-Uhlenbeck-Yau theorem to stability in mirror symmetry

Abstract

We give an introduction to the mathematical formulation of Yang-Mills theory. In particular we derive the Hermitian-Yang-Mills equation and show that Hermitian-Yang-Mills connections can be described as the zeroes of the corresponding moment map. We then introduce deformed Hermitian- Yang-Mills equations by considering arbitrary moment maps. Furthermore we introduce slope stability of holomorphic vector bundles and show that holomorphic vector bundles have a unique Harder-Narasimhan ltration. We then give a proof of the Donaldson-Uhlenbeck-Yau theorem in the case of algebraic surfaces, which states that there is a one-to-one correspondence between stable bundles and bundles that admit an irreducible Hermitian- Yang-Mills connection. Finally we look at stability in the context of homological mirror symmetry. We discuss Bridgeland stability conditions on the bounded derived category of coherent sheafs Db Coh(M) over a K ahler manifold M and discuss how it connects to the deformed Hermitian-Yang-Mills equation.