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Counting the fission trees and nonabelian Hodge graphs (untwisted case)

Philip Boalch

TL;DR

This work develops a comprehensive combinatorial framework to count untwisted fission trees arising from irregular connections on curves, which classify deformation spaces of wild character varieties and their nonabelian Hodge moduli. By encoding fission trees with height-based sets and surjective parent maps, and applying the Euler transform, the authors derive recursive counts: slope-2 trees correspond to partitions, slope-3 to double partitions, and higher slopes via iterative transforms. The paper also situates fission trees within broader structures—fission graphs, supernova graphs, and new multiplicative quiver varieties—highlighting their role as building blocks for 2d gauge theory moduli and wild nonabelian Hodge spaces. The counting results yield explicit tables and sequences, providing a practical toolkit for classifying deformation spaces and their associated geometric objects.

Abstract

Any algebraic connection on a vector bundle on a smooth complex algebraic curve determines an irregular class and in turn a fission tree at each puncture. The fission trees are the discrete data classifying the admissible deformation classes. Here we explain how to count the fission trees with given slope and number of leaves, in the untwisted case. This also leads to a clearer picture of the ``periodic table'' of the atoms that play the role of building blocks in 2d gauge theory.

Counting the fission trees and nonabelian Hodge graphs (untwisted case)

TL;DR

This work develops a comprehensive combinatorial framework to count untwisted fission trees arising from irregular connections on curves, which classify deformation spaces of wild character varieties and their nonabelian Hodge moduli. By encoding fission trees with height-based sets and surjective parent maps, and applying the Euler transform, the authors derive recursive counts: slope-2 trees correspond to partitions, slope-3 to double partitions, and higher slopes via iterative transforms. The paper also situates fission trees within broader structures—fission graphs, supernova graphs, and new multiplicative quiver varieties—highlighting their role as building blocks for 2d gauge theory moduli and wild nonabelian Hodge spaces. The counting results yield explicit tables and sequences, providing a practical toolkit for classifying deformation spaces and their associated geometric objects.

Abstract

Any algebraic connection on a vector bundle on a smooth complex algebraic curve determines an irregular class and in turn a fission tree at each puncture. The fission trees are the discrete data classifying the admissible deformation classes. Here we explain how to count the fission trees with given slope and number of leaves, in the untwisted case. This also leads to a clearer picture of the ``periodic table'' of the atoms that play the role of building blocks in 2d gauge theory.

Paper Structure

This paper contains 8 sections, 2 theorems, 20 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Topological classification of irregular types
  3. Nonabelian Hodge graphs
  4. New multiplicative quiver varieties
  5. Counting
  6. Encoding via surjective maps
  7. Counting
  8. Changing the question

Key Result

Lemma 3

$\phi(k,n) = \Phi(k,n)-\Phi(k-1,n).$

Figures (7)

  • Figure 1: Example fission tree $\mathbb{T}$ (with $4$ leaves and slope $3$).
  • Figure 2: $1+3+9$ fission trees with slope $3$ and $2,3$ or $4$ leaves.
  • Figure 3: The diagrams of the Painlevé moduli spaces 1,2,3,4,5,6 (see diags).
  • Figure 4: Fission graphs with $4$ nodes and maximal edge multiplicity $2$.
  • Figure 5: Fission construction of the fission graph of the fission tree of Fig. 1 (as drawn in rims15slides p.79).
  • ...and 2 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Lemma 3
  • Lemma 4