Higgs bundles (associated to GL(n,C)) are complex vector bundles on algebraic curves along with the extra data of an endomorphism, twisted by the canonical line bundle. They arose originally by a correspondence with solutions to the Hitchin equations on connections, but are also related to representations of the fundamental group of the curve to GL(n,C), by the non-abelian Hodge correspondence. The spectral correspondence, relating Higgs bundles to line bundles on a spec-tral curve, provides geometric information about the moduli space of Higgs bundles. There are also notions of G-Higgs bundles where G is a real form of a complex reductive group, which in particular provides a non-abelian Hodge correspondence for representations to G. For quasi-split real forms, a spectral correspondence is provided by the abelianization of García-Prada and Peón-Nieto. Spectral correspondences are known in the non-quasi-split examples SO*(m), SU*(m), SO(p,q) and Sp(p,q), where the spectral data is, in a certain sense, 'non-abelian'. We calculate spectral data for the non-quasi split group G = U(p,q), expanding on work of Schaposnik for U(m,m) and Peón-Nieto for U(m+1,m), and find 'non-abelian' behaviour, in particular similar to that of SO(p,q)." .

Día

Mércores, 1 de febreiro de 2024

Lugar

Aula 10
Facultade de Matemáticas