Moduli spaces of parabolic Higgs bundles on compact Riemann surfaces are a vast and geometrically rich class of non-compact hyperkähler manifolds. While originally described as infinite dimensional hyperkähler quotients parametrizing equivalence classes of solutions to Hitchin's equations, they also admit a parallel and independent construction in terms of the powerful algebro-geometric techniques of Geometric Invariant Theory. An interesting phenomenon that characterizes thir geometry is their dependence on a convex polytope of stability parameters called parabolic weights. The intrinsic convex geometry of this polytope leads to the idea of "wall-crossing".
Although wall-crossing for moduli spaces of parabolic Higgs bundles on Riemann surfaces is fairly understood from the algebro-geometric perspective, analytic invariants such as their hyperkähler structures lead to interesting non-trivial problems linked to the topology of their associated tautological ring, and conjecturally related to the semi-classical analysis of conformal field theories. Understanding these relations requires the introduction of an alternative complex-analytic approach, which is fully tractable in the case of low genera. Namely, the construction of complex-analytic geometric models for these moduli spaces in the genus 0 case is a recent proposal of the author crafted with this precise purpose in mind: the explicit nature of the geometric models provides a comprehensive understanding of wall-crossing in a relatively elementary (though intricate) combinatorial formulation.
In particular, we expect these results to shed light on the rich asymptotic geometry "at infinity" for these hyperkähler manifolds, which in the simplest case correspond to non-trivial examples of gravitational instantons of ALG type, which would broaden their understanding and applicability.
The project involves the following (ongoing and planned) research subjects:
- Geometric models for moduli of parabolic Higgs bundles in genus 0, their variation of parabolic weights, and wall-crossing.
WZNW actions and hyperkähler structures for moduli of parabolic Higgs bundles.
- Projective structures and asymptotic hyperkähler geometry.
We construct explicit geometric models for moduli spaces of stable parabolic Higgs bundles on the Riemann sphere, in the case of rank two, four marked points, any degree, and arbitrary weights. The construction mechanism relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on carefully crafted spaces. These techniques are not exclusive to the case we examine. Therefore, this work elucidates a general approach to construct arbitrary moduli spaces of stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. Moreover, we present a comprehensive analysis of the geometric models' behavior under variation of weights and wall-crossing. This analysis is concentrated on their nilpotent cones, and is applicable to the study of the hyperkähler geometry of Hitchin metrics as gravitational instantons of ALG type.
69Wall-crossing and hyperkähler geometry of moduli spaces