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Floer-theoretic filtration on Painlevé Hitchin systems

Szilárd Szabó, Filip Živanović

TL;DR

The paper analyzes contracting holomorphic ${\mathbb C}^*$-actions on 2-dimensional Higgs moduli, focusing on Painlevé spaces and related parabolic moduli. Using Ritter–Živanović Floer-theoretic machinery, it computes the resulting cohomological filtrations $\mathscr{F}$, encodes weight data via Puiseux polynomials $P^{PX}$, and compares with the $P=W$ (perverse) and multiplicity filtrations arising from the central Hitchin fiber. Through explicit case-by-case constructions for Painlevé I–VI and related affine-root moduli, it shows that in dimension two the Floer filtration coincides with the multiplicity filtration, while both refine $P=W$ in the parabolic setting; at Painlevé VI all three filtrations coincide. A complete classification is provided: any 2D Higgs moduli with a contracting ${\mathbb C}^*$-action is birational to a Painlevé moduli or a parabolic affine-root moduli, and the Hitchin fibration extends to an elliptic fibration on a blown-up rational surface with at most two singular fibers. These results illuminate deep links between Floer theory, Higgs moduli topology, and mirror-symmetry considerations for these integrable systems.

Abstract

We classify equivariant $\mathbb{C}^*$-actions on moduli spaces of Higgs bundles corresponding to the Painlevé 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.

Floer-theoretic filtration on Painlevé Hitchin systems

TL;DR

The paper analyzes contracting holomorphic -actions on 2-dimensional Higgs moduli, focusing on Painlevé spaces and related parabolic moduli. Using Ritter–Živanović Floer-theoretic machinery, it computes the resulting cohomological filtrations , encodes weight data via Puiseux polynomials , and compares with the (perverse) and multiplicity filtrations arising from the central Hitchin fiber. Through explicit case-by-case constructions for Painlevé I–VI and related affine-root moduli, it shows that in dimension two the Floer filtration coincides with the multiplicity filtration, while both refine in the parabolic setting; at Painlevé VI all three filtrations coincide. A complete classification is provided: any 2D Higgs moduli with a contracting -action is birational to a Painlevé moduli or a parabolic affine-root moduli, and the Hitchin fibration extends to an elliptic fibration on a blown-up rational surface with at most two singular fibers. These results illuminate deep links between Floer theory, Higgs moduli topology, and mirror-symmetry considerations for these integrable systems.

Abstract

We classify equivariant -actions on moduli spaces of Higgs bundles corresponding to the Painlevé 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 ``'' 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.
Paper Structure (12 sections, 29 theorems, 135 equations, 5 figures, 1 table)

This paper contains 12 sections, 29 theorems, 135 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Filtration from the Floer theory of a $\mathbb C^*$-action
  3. Construction of $\mathbb C^*$-actions and filtration computation
  4. Painlevé I case
  5. Painlevé II case
  6. Painlevé II (Flaschka--Newell) case
  7. Painlevé IV (degenerate) case
  8. Painlevé VI case
  9. Remaining Painlevé cases
  10. Further Higgs moduli of $\dim_\mathbb C=2$
  11. Classification of 2-dimensional $\mathbb C^*$-equivariant Higgs moduli spaces
  12. Three filtrations compared

Key Result

Theorem 1.1

The equivariant $\mathbb C^*$-actions on Hitchin moduli spaces in the Painlevé cases have invariants given in Table table.

Figures (5)

  • Figure 1: Spectral sequence for $X=I$
  • Figure 2: Spectral sequence for $X=II$
  • Figure 3: Spectral sequence for $X=IV$.
  • Figure 4: Spectral sequence for $X=VI$
  • Figure 5: Imaginary root labellings for (a) $\widetilde{D}_n$ (total number of vertices is equal to $n+1$) (b) $\widetilde{E}_8$ (c) $\widetilde{E}_7$ (d) $\widetilde{E}_6$.

Theorems & Definitions (48)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 38 more