Floer-theoretic filtration on Painlev\'e Hitchin systems

We classify equivariant $\mathbb{C}^*$-actions on moduli spaces of Higgs bundles corresponding to the Painlev\'e equations. We deduce the Floer-theoretic filtrations on the cohomology of these spaces, introduced by Ritter and the second author in arXiv:2304.13026. Filtration is then compared with the $``P=W"$ and the filtration obtained by multiplicities of the irreducible components of the nilpotent cone, for all the 2-dimensional Higgs moduli.
Comment: 21 pages, 5 figures