This is the text of a Master Thesis of the Master of Advanced Mathematics at Universidad Complutense de Madrid.
A Higgs bundle over a compact Riemann surface X is a pair (E,f), where E is a holomorphic vector bundle E over X and f is an endomorphism twisted by the canonical bundle K. These objects were introduced by Nigel Hitchin more than 30 years ago. The moduli space M(n,d) of stable Higgs bundles of fixed rank n and degree d has a very rich geometric structure, in particular it is an algebraically integrable system. Evaluating f in a base of invariant polynomials by GL(n,C) in the space of complex nxn matrices one gets the Hitchin map, whose generic fibre is the Jacobian of the spectral curve that lives within the total space of K.
In this dissertation we explore a generalization of the theory of Higgs bundles over compact Riemann surfaces, in which the canonical line bundle is replaced by a vector bundle of arbitrary rank. This kind of situation appears naturally in the study of supersymmetric gauge theories. From the mathematical point of view, we study several topics: analogous of Hitchin’s equations, stability, Hitchin–Kobayashi correspondence and a generalization of the spectral curve.
Keywords: Moduli space, vector bundle, Higgs bundle, Hitchin’s equations, Hitchin–Kobayashi correspondence, Hitchin fibration, spectral curve