Higgs bundles: geometry, algebra and physics

This project focuses on the geometry of moduli spaces of Higgs bundles, central objects in modern mathematics that bridge geometry, topology, representation theory, and physics. Introduced by Hitchin, Higgs bundles on a Riemann surface consist of a holomorphic bundle together with a Higgs field. As usually happens in algebraic geometry, these objects are parameterized by an algebraic variety, the so-called moduli space of Higgs bundles. The notion of HIggs bundle depends on a choice of a Lie group G, with the corresponding moduli space being denoted by M(G). These moduli spaces M(G) play crucial roles in gauge theory, integrable systems, the Langlands program, etc., and are the main objects of this project.

Indeed, the general goal of this project is to advance the understanding of the geometry of the moduli spaces M(G) by developing new methods and perspectives, fostering deeper insights into their geometry and interrelations with representation theory, mirror symmetry, and Lie algebroid connections.

The project explores mirror symmetry by constructing new branes in moduli spaces using Hecke modifications, investigates the nilpotent cone through these modifications, and examines Slodowy maps from Lie theory to detect novel geometric loci in M(G). Additionally, it studies moduli spaces of integrable Lie algebroid connections, generalizing Higgs bundles and aiming to construct Hitchin-like maps in such spaces. It also studies moduli spaces of Hodge bundles and quiver bundles, deeply connected to the moduli spaces M(G). All these topics will contribute to advancing the geometry of Higgs bundles and related fields.

A key component of this project is its training aspect. We will hire a master student to be introduced to Higgs bundle theory and will organize a conference/workshop to promote collaboration and engage the team with the international Higgs bundle community.