### Thursday 14 May 2015 from 12:00–13:00 in Carslaw 535A

## Higgs bundles, spectral data, and applications

Laura Schaposnik (UIUC)

Abstract

Higgs bundles (introduced by N. Hitchin in 1987) are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle. The moduli space of Higgs bundles carries a natural Hyperkahler structure, through which we can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes) with respect to each structure. Notably, these A and B-branes have gained significant attention in string theory.

We shall begin the talk by first introducing Higgs bundles for complex Lie groups and the associated Hitchin fibration, and recalling how to realize Langlands duality through spectral data. We shall then look at a natural construction of families of subspaces which give different types of branes. Finally, by means of spectral data, we shall relate these subspaces to the study of 3-manifolds and surface group representations. We shall conclude with some conjectures related to Langlands duality.