(Ir)regular singularities and Quantum Field Theory
Character varieties and holomorphic bundles
Description
For some time, character varieties have been studied for their relevance in representation theory and the QFT interpretation of knot polynomials. More recently, the non-abelian Hodge theory identifies them with moduli spaces of G-Higgs bundles, which are certain holomorphic bundles with additional Lie group structure, over Kähler manifolds. In this theme we focus on the description of the geometry, topology, mixed Hodge structures and enumerative invariants of character varieties, and study their relation to Nakajima quiver varieties.
Goals
One of the objectives is to obtain topological, algebraic, and number theoretic information on character varieties of free groups, abelian groups, and surface groups, as these are the ones more relevant for the geometry of moduli spaces of G-Higgs bundles and the Hitchin system. Another more long term goal is to address the same kind of questions for quiver varieties and wild character varieties.
Results so far (January 2021)
Florentino and Silva explicitly computed the mixed Hodge structure of character varieties of free abelian groups, which also provides their Poincaré and Serre polynomials.
Florentino, Nozad and Zamora have shown a kind of mirror symmetry between character varieties of free groups, for the Langlands dual pair SL_n and PGL_n, which solves a conjecture of Lawton and Muñoz. A formula for generating series of E-polynomials was also found, which applies to a general character variety with a natural stratification.
