Peter Gothen will speak on

A general Cayley correspondence for Higgs bundles and higher Teichmüller theory

Abstract: Let G be a non-compact Lie group of Hermitian type such that the associated Hermitian symmetric space is of tube type. Then the Cayley correspondence for G-Higgs bundles produces special connected components in the moduli space of maximal G-Higgs bundles, which can be seen as modulis spaces of twisted Higgs bundles for a different group. We generalise this correspondence to a larger class of real reductive Lie groups G. Our generalisation encompasses Hitchin components when G is a split real form and special components discovered recently for SO(p,q), and also constructs previously unknown components for the quaternionic real forms of E6, E7, E8 and F4. The construction is based on a new class of sl_2-triples in a complex simple Lie algebra, which we call magical. The classification of magical sl_2-triples is in bijection with the set of theta-positive structures in the sense of Guichard-Wienhard, thus our construction conjecturally detects all examples of higher Teichmüller spaces.

The talk is based on arXiv:2101.09377, which is joint work with Steve Bradlow, Brian Collier, Oscar Garcia-Prada, and André Oliveira