Abstract:

The Hitchin moduli space associated to a compact Riemann surface and a complex reductive group is naturally a hyperkähler manifold. In one of its complex structures, this space is identified with the moduli space of Higgs bundles, while in another one it is identified with the moduli space of holomorphic bundles with holomorphic connection. Moreover, it is endowed with a fibration by abelian varieties, known as the Hitchin fibration. The geometry of the Hitchin moduli space is known to be very rich and has been a very active topic of research in the last 4 decades.

In particular, the Hitchin fibration is one of the contexts where Langlands duality seems to appear more naturally.

In recent years, there have appeared several moduli spaces in the literature that share some properties with Hitchin moduli space, like having an associated "Hitchin-type" fibration. Some of these spaces arise naturally in the context of non-abelian Hodge theory, while others have a different origin. A general framework for the study of these spaces and their associated fibrations was proposed recently in unpublished work by B. Morrissey and Ngô B.C.

One of these analogues of the Hitchin moduli space is the space of monopoles with Dirac-type singularities in the product of a Riemann surface and a circle, which was shown by B. Charbonneau and J. Hurtubise to be in bijective correspondence with the space of multiplicative Higgs bundles on the Riemann surface. These multiplicative Higgs bundles are pairs very similar to usual Higgs bundles, but where the Higgs field is meromorphic and takes values on the group, rather than on its Lie algebra.

In the first part of the talk, we will review some basic properties of Hitchin moduli space, introduce the general framework of Morrissey--Ngô, and give some examples of these analogues and generalizations of the Hitchin fibration. The second part of the talk is centered around multiplicative Higgs bundles. We will give the basic definitions and explain the Charbonneau-Hurtubise correspondence and the point of view in terms of the Vinberg monoid developed by A. Bouthier, J. Chi and X.G. Wang.

We conclude the talk by giving a list of open problems related to multiplicative Higgs bundles.