Oscar García-Prada : Non-abelian Hodge correspondence and higher Teichmüller spaces

Posted by Ben Bakker , part of the Departmental Colloquium.

At

March 18, 2022, 3 p.m.

In

636 SEO

Abstract

Given a closed orientable surface S and a Lie group G, one can consider the set of equivalence classes of representationsof the fundamental group of S in G, which is often referred as the G-character variety of the fundamental group of S. Choosing a complex structure on S, one can identify the elements in the G-character variety with certain holomorphic objects on the corresponding Riemann surface. If G is the circle, this gives the identification of the character variety with the Jacobian of the Riemann surface --- correspondence that goes back to the 19th century. In this talk we will focus on the case in which G is a non-abelian non-compact semisimple Lie group. In this situation, the non-abelian Hodge correspondence identifies the G-character variety with the space of some holomophic objects on the Riemann surface known as G-Higgs bundles. Using this correspondence, we give a classification of the simple Lie groups G for which the G-character variety has components that generalize the Teichmüller space of S parameterizing complex structures on S, regarded as a topological component of the character variety for G=PSL(2,R).