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October 10, 2026
AI Meets Nonabelian Hodge
As many of you know, I have spent far too much time thinking about families of Hitchin systems over the moduli space of (punctured) Riemann surfaces, . In my defense, the Coulomb branch geometry of a 4d theory is encoded in the data of a complex integrable system. For class-S, that’s a Hitchin system and the “conformal manifold” of a family of SCFTs is the aforementioned moduli space. So it’s quite natural to consider the family of Hitchin systems as we vary the exactly-marginal parameters of the SCFT.
A striking feature of Hitchin’s integrable system is that it’s actually hyperKähler; it has a whole worth of complex structures. We always work in the complex structure where it has the interpretation as the moduli space of Higgs bundles. This is the physically-interesting case where the base of the Hitchin system (the space of “action variables” of the integrable system) is the Coulomb branch of the 4d theory.
One of the main points of our work is that the bundle of Hitchin bases has a natural extension to the Deligne-Mumford compactification , and that extension has an interesting interpretation in the physics.
In another complex structure, Hitchin’s system has the interpretation as a character variety: the space of homomorphisms up to overall conjugation.
The map from semi-stable parabolic Higgs bundles to the irreducible character variety for is called the Nonabelian Hodge Correspondence (NAH). See Simpson’s paper for more details of the correspondence. We’ll be interested in the strongly-parabolic case, where the residues of are nilpotent.
We had some interesting conjectures about NAH in Appendix B of our first paper. So you can imagine that I was quite excited to see a recent paper which claims to construct families of character varieties over and examines how they behave as the curve degenerates.