Holomorphic Lie algebroids, stacks of twisted modules and applications to the Hitchin system and Poisson geometry.

Abstract

The Hitchin system is a completely integrable system which plays a fundamental role in several areas of mathematics (algebraic geometry and representation theory) and of theoretical physics (gauge theory).The main goal of this research project is to recast the geometry of this integrable system in the framework of the theory of holomorphic Lie algebroids. In particular, using the theory of the $\lambda$-connections we want to analyse the twistor space of the hyperK\"ahler structure naturally associated to the Hitchin system and, at the same time, we want to investigate the special K\"ahler structure defined on the base of such an integrable system. Further directions of this investigation will be the stacky generalization of the theory of the $\Lambda$-modules and study of the Calogero-Moser-KP corespondence (via the so called adelic Grassmannian) in terms of the theory of the $\lambda$-connections.