Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces

Abstract

This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.

Higgs Bundles—Recent Applications

“…However, the independence holds in direct calculations of the Betti numbers in low rank, and was recently shown for GL(n, C) and SL(n, C)-Higgs bundles by Groechenig-Wyss-Ziegler in [GWZ17]. This suggests that some topological properties of Hitchin systems are independent of the hyperkähler geometry (see references in [Hau13,Ray18] for more details).…”

“…For instance, one may ask how do those metrics behave at the boundaries of the moduli space, or how do the energy densities of the corresponding harmonic maps at different points of the Hitchin fibration determine each other (the reader may be interested in the reviews [Li19] and [Fre19], and references therein). From Hitchin's work, the moduli space of Higgs bundles has a natural C * -action λ · (E, Φ) = (E, λΦ), whose fixed point sets allow one to study different aspects of the topology and the geometry of the space, as done in [Hit87b] (see also [Ray18,Col19]). Moreover, as shown by Simpson, the the fixed points by this action are complex variations of Hodge structure (VHS).…”

Trimming the permutahedron to extend the parking space

“…Recall that in [11,Theorem 6.1], the number of orbits was computed by way of explicit character values. See also [17,Section 5]…”

Moduli spaces of generalized hyperpolygons

“…This leads to a localization for the Higgs bundle cohomology, as originally done in rank 2 in [21] (cf. [41] for a survey in arbitary rank). On the Higgs bundle moduli space, the action leaves vector bundles invariant while rotating Higgs fields -and in particular, the moduli space of stable bundles is invariant.…”