Floer-theoretic filtration on Painlev\'e Hitchin systems

TL;DR

This paper classifies certain symmetries on moduli spaces of Higgs bundles related to Painlevé equations and shows that Floer-theoretic filtrations match multiplicity filtrations, deepening understanding of their geometric structure.

Contribution

It provides a classification of equivariant a7^*-actions on Painleve9 Higgs moduli spaces and proves the equivalence of Floer-theoretic and multiplicity filtrations.

Findings

Floer-theoretic filtration coincides with multiplicity filtration.

Classification of equivariant a7^*-actions on Painleve9 moduli spaces.

Comparison of different filtrations on Higgs moduli cohomology.

Abstract

We classify equivariant $\mathbb{C}^*$-actions on moduli spaces of Higgs bundles corresponding to the Painlev\'e equations. Using this, we compute the Floer-theoretic filtrations on the cohomology of these spaces, introduced by Ritter and the second author in arXiv:2304.13026. We compare it with the ``$P=W$'' and the filtration obtained by multiplicities of the irreducible components of the nilpotent cone, ultimately deducing that the Floer-theoretic filtration coincides with the multiplicity filtration, for all 2-dimensional Higgs moduli.