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Wall-crossing and hyperkähler geometry of moduli spaces

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.

## Publications

Moduli spaces of stable parabolic bundles of parabolic degree $$0$$ over the Riemann sphere are stratified according to the Harder-Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $$\mathscr{N}_{0}$$ of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $$\mathscr{S}$$ is defined as the regularized critical value of the non-compact Wess-Zumino-Novikov-Witten action functional. The definition of $$\mathscr{S}$$ depends on a suitable notion of parabolic bundle 'uniformization map' following from the Mehta-Seshadri and Birkhoff-Grothendieck theorems. It is shown that $$-\mathscr{S}$$ is a primitive for a (1,0)-form $$\vartheta$$ on $$\mathscr{N}_{0}$$ associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $$-\mathscr{S}$$ is a Kähler potential for $$(\Omega-\Omega_{\mathrm{T}})|_{\mathscr{N}_{0}}$$,  where $$\Omega$$ is the Narasimhan-Atiyah-Bott Kähler form in $$\mathscr{N}$$ and $$\Omega_{\mathrm{T}}$$ is a certain linear combination of tautological $$(1,1)$$-forms associated with the marked points. These results provide an explicit relation between the cohomology class $$[\Omega]$$ and tautological classes, which holds globally over certain open chambers of parabolic weights where $$\mathscr{N}_{0} = \mathscr{N}$$.

 Journal Commun. Math. Phys. Publisher Springer Volume 387, no. 2 Pages 649–680 Link to preprint version Link to published version

We survey several mathematical developments in the holonomy approach to gauge theory. A cornerstone of such approach is the introduction of group structures on spaces of based loops on a smooth manifold, relying on certain homotopy equivalence relations --- such as the so-called thin homotopy --- and the resulting interpretation of gauge fields as group homomorphisms to a Lie group G satisfying a suitable smoothness condition, encoding the holonomy of a gauge orbit of smooth connections on a principal G-bundle. We also prove several structural results on thin homotopy, and in particular we clarify the difference between thin equivalence and retrace equivalence for piecewise-smooth based loops on a smooth manifold, which are often used interchangeably in the physics literature. We conclude by listing a set of questions on topological and functional analytic aspects of groups of based loops, which we consider to be fundamental to establish a rigorous differential geometric foundation of the holonomy formulation of gauge theory.

 Journal Contemp. Math. Publisher Amer. Math. Soc. Volume 775 Pages 233–253 Link to preprint version

We define a functional $${\cal J}(h)$$ for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that $${\cal J}(h)$$ is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite-Yang-Mills metrics. We derive a formula relating $${\cal J}(h)$$ and another functional $${\cal I}(h)$$, closely related to the Yang-Mills-Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of $${\cal J}(h)$$, which is expressed as a certain $$L^{2}$$-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of $${\cal J}(h)$$ if and only if the corresponding Hitchin-Simpson mean curvature is parallel with respect to the Hitchin-Simpson connection.

 Journal International Journal of Geometric Methods in Modern Physics Publisher World Scientific Volume 17(13) Line art. no. 2050200 Link to preprint version Link to published version

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.

Related project(s):
69Wall-crossing and hyperkähler geometry of moduli spaces

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## Team Members

Dr. Claudio Meneses Torres