
## 5. Hyperkähler geometry of Hitchin systems

1. ### Volume decay properties for $L^{2}$-metrics

A conjecture of Hitchin–Hausel states that the $L^{2}$-metric in Hitchin’s moduli space QALG. This has been verified for the toy model (i.e., for moduli of stable rank $r=2$ parabolic Higgs bundles over $\mathbb{CP}^{1}$ with $n=4$ marked points) by Fredrickson–Mazzeo–Swoboda–Weiss [arXiv:2001.03682].

#### Problem 5.1.

[Steven Rayan] Compute the volume decay of the $L^{2}$-metric in Hitchin’s moduli space, and verify that it is QALG.
• ### Validity of the Gaiotto–Moore–Neitzke programme in the absence of singularities

The Gaiotto–Moore–Neitzke programme to construct hyperkähler metrics as "instanton-corrected" semi-flat metrics is based in the construction of Darboux coordinates for the holomorphic symplectic form in terms of spectral networks, together with Twistor methods, and integral equations. The recent mathematical construction of Maximilian Holdt [arXiv:2106.16017] works in the weakly parabolic case, and it is not immediately clear how it could be specialized to the simpler case of Higgs bundles without parabolic structures.

#### Problem 5.2.

[Hartmut Weiss] Verify the validity of the Gaiotto–Moore–Neitzke programme for moduli of Higgs bundles without parabolic structures. Moreover, are the resulting metrics the same as the Hitchin $L^{2}$-metric?
• ### Compactification of Hitchin moduli spaces

A natural problem motivated by the construction of limiting configurations [arXiv:2001.03682] is to find extensions of the Hitchin fibration to a (singular) torus fibration over a compactification of the Hitchin base to a manifold with corners. Ideally, these asymptotics should correspond to an analog of the compactified monopole moduli space constructed by Fritzsch–Kottke–Singer [arXiv:1811.00601].

In the case of $\mathrm{SU}(2)$ gauge theory on the 4-punctured sphere, this is easily understood over the radial compactification of the Hitchin base $\mathbb{C}$, since the discriminant locus coincides with the nilpotent cone. For $\mathrm{SU}(2)$ gauge theory on the 5-punctured sphere a potential compactification of the Hitchin base is the radial compactification of the Hitchin base $\mathbb{C}^2$ blown up at the intersection of the discriminant locus with this radial boundary.

#### Problem 5.3.

[Arnav Tripathy] To determine the right algebro-geometric and complex-analytic interpretation of the so-called "fiducial solutions" as limiting Hitchin data in terms of a suitable compactification of Hitchin moduli spaces.
• ### Extension of semi-flat metrics along singular loci

#### Problem 5.4.

[Sergey A. Cherkis, Jacques Hurtubise, Hartmut Weiss] To determine an extension of the semi-flat metric along the singular loci of the Hitchin fibration, in terms of the integrable subsystems studied by Hitchin [MR4307206]. What is the local metric near the singular fibers (in analogy to the Oogury–Vafa metric in the case of $\mathbb{CP}^{1}$, rank 2, and 5 marked points)?
• ### Explicit local models near singular fibers

#### Problem 5.5.

[Sergey A. Cherkis] Find good local models for Hitchin metrics for all loci where the semi-flat metric is ill-defined. On the other hand, determine and classify the singularities of semi-flat metrics along all stratification components of the singular locus in Hitchin’s fibration.
• ### Classification of 4-dimensional Hitchin systems

Moduli spaces of parabolic Higgs bundles are a broad class of complete hyperkähler manifolds with a rich asymptotic geometry "at infinity" of their Hitchin fibrations. Of particular interest are the complex two-dimensional examples such as the so-called toy model [MR1650276] which has been recently proved to be an ALG-gravitational instanton [arXiv:2001.03682]. It has been conjectured that all ALG-gravitational instantons can be realized as Hitchin moduli spaces with concrete genus 0 models in ranks 2, 3, 4 and 6 [MR2500553], [MR3931781]. These examples are “primitive”, in the sense that each of them gives rise to an infinite family by rescaling the rank and the corresponding flag type at the marked points of $\mathbb{CP}^{1}$.

It is easy to verify that there exist many more (and possibly infinitely many) primitive sets of parameters in genus 0 (i.e., rank, number of marked points, and flag types) for which the expected complex dimension of the moduli space is equal to two, beyond those listed by Boalch. Therefore, these could potentially lead to complete 4-dimensional hyperkähler manifolds with an asymptotic geometry at infinity.

#### Problem 5.6.

[Claudio Meneses] Determine a combinatorial classification of primitive sets of parameters leading to a 2-dimensional Hitchin system in genus 0. Prove or disprove the existence of semi-stable elements for an arbitrary choice of primitive sets of parameters. For all non-empty moduli, classify their elliptic fibration type in accordance to Kodaira’s list, and moreover, classify their asymptotic hyperkähler geometry at infinity.
• ### Explicit solutions to Hitchin’s equations

#### Problem 5.7.

[Sergey A. Cherkis] Construction of explicit examples of solutions to Hitchin’s equations (i.e., either singular hermitian metrics or their associated logarithmic Chern connections) on parabolic Higgs bundles $(E_{*},\Phi)$ over $\mathbb{CP}^{1}$, beyond the toy model, for example in the simplest case of $n=3$ marked points and rank $r=3$. A natural starting point is to consider limiting configurations, which are explicit in nature.

Cite this as: AimPL: Geometry and physics of ALX metrics in gauge theory, available at http://aimpl.org/geomphysalx.