Basic Representations of Genus Zero Nonabelian Hodge Spaces

In some previous work, we defined an invariant of genus zero nonabelian Hodge spaces taking the form of a diagram. Here, enriching the diagram by fission data to obtain a refined invariant, the enriched tree, including a partition of the core diagram into $k$ subsets, we show that this invariant contains sufficient information to reconstruct $k+1$ different classes of admissible deformations of wild Riemann surfaces, that are all representations of one single nonabelian Hodge space, so that the isomonodromy systems defined by these representations are expected to be isomorphic. This partially generalises to the case of arbitrary singularity data the picture of the simply-laced case featuring a diagram with a complete $k$-partite core. We illustrate this framework by discussing different Lax representations for Painlevé equations.

💡 Research Summary

The paper develops a combinatorial invariant that fully captures genus‑zero wild non‑abelian Hodge spaces, extending previous diagrammatic approaches to arbitrary irregular types. Starting from the known picture in the simply‑laced case—where a complete k‑partite core graph (the “supernova quiver”) together with a set of linear legs encodes k + 1 isomorphic moduli spaces—the author observes that for general irregular singularities the core is no longer complete, and the diagram alone does not determine the Hodge space.

To remedy this, the author introduces the “enriched tree”: a pair consisting of a short fission tree (a metrised tree whose non‑root vertices have height ≤ 2) and a collection of conjugacy classes in GLₙ(C) attached to each leaf. Theorem 1.4 proves that any algebraic connection on a Zariski‑open subset of ℙ¹ determines a unique enriched tree, and that this tree is invariant under the full SL₂(C) symplectic action (Fourier–Laplace, Möbius transformations, rank‑one twists). Consequently, all connections related by these basic operations share the same enriched tree.

The enriched tree encodes a pair (g,F) where g is the genus (here always 0) and F is a fission datum—a set of fission trees describing admissible deformations of the underlying wild Riemann surface. Two wild Riemann surfaces are admissibly equivalent precisely when they have the same (g,F). Hence the enriched tree classifies equivalence classes of “weak representations” of a given non‑abelian Hodge space M(F,C), where C records the conjugacy classes attached to the leaves.

From a single enriched tree one can read off k + 1 “nearby representations”. One is the generic representation of maximal rank, featuring a single irregular singularity at infinity; the remaining k representations are obtained by selecting one of the k subsets of the core diagram and interpreting those vertices as regular singularities at finite points, while the rest remain exponential factors at infinity. This reproduces the multiduality observed by Boalch in the simply‑laced case, but now applies to arbitrary irregular data.

The paper then applies this framework to Painlevé equations. Each Painlevé equation corresponds to a wild connection whose enriched tree is the same for all known Lax pairs. Thus the various Lax representations are merely different nearby representations of a single non‑abelian Hodge space, explaining why their isomonodromic deformation equations (the Painlevé equations) are identical. This unifies Harnad duality, Boalch’s multiduality, and earlier results on Fourier–Laplace induced isomorphisms of wild character varieties.

In summary, the enriched tree provides a complete, SL₂(C)‑invariant combinatorial encoding of genus‑zero wild non‑abelian Hodge spaces. It allows one to reconstruct all admissible deformations (k + 1 classes) from a single diagrammatic object, thereby giving a systematic method to identify and classify isomorphic Hodge spaces and their associated isomonodromic systems. The work opens the way to extend these ideas beyond genus 0, to other reductive groups, and to a deeper understanding of the relationship between Stokes data and the combinatorial fission structures.