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

TL;DR

This paper develops a method to count fission trees associated with algebraic connections on complex curves, providing insights into the classification of deformation classes and the structure of gauge theory building blocks.

Contribution

It introduces a counting technique for fission trees with specified parameters, enhancing understanding of their role in algebraic and gauge theory contexts.

Findings

Counted fission trees with given slope and leaves

Clarified the structure of the 'periodic table' of gauge theory atoms

Linked fission trees to deformation class classification

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.