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Irregular connections and Kac-Moody root systems

Philip Boalch

TL;DR

This work builds a precise bridge between irregular connections and Nakajima quiver varieties by encoding irregular data into quiver representations and then using hyperkähler quotients to realize connection moduli as quiver varieties. It introduces a detailed dictionary via abstract data (I,J), their representations, and realizations, including cycling and twisting operations, to read a single connection in multiple quiver readings, yielding numerous isomorphisms of moduli spaces. The paper extends stability and existence criteria for connections from Crawley-Boevey to the irregular setting, via deformed preprojective algebras and root-system language, and highlights dualities such as Harnad’s reading-duality as natural reflection functors. This framework advances the understanding of isomonodromy, wild Hitchin spaces, and the geometric Langlands program by providing concrete, graph-based models with rich symmetry and non-emptiness criteria, applicable to poles of order up to three (and beyond in appendix).

Abstract

Some moduli spaces of irregular connections on the trivial bundle over the Riemann sphere will be identified with Nakajima quiver varieties. In particular this enables us to associate a Kac-Moody root system to such connections (yielding many isomorphisms between such moduli spaces, via the reflection functors for the corresponding Weyl group). The possibility of 'reading' a quiver in different ways also yields numerous isomorphisms between such moduli spaces, often between spaces of connections on different rank bundles and with different polar divisors. Finally some results of Crawley-Boevey on the existence of stable connections will be extended to this more general context.

Irregular connections and Kac-Moody root systems

TL;DR

This work builds a precise bridge between irregular connections and Nakajima quiver varieties by encoding irregular data into quiver representations and then using hyperkähler quotients to realize connection moduli as quiver varieties. It introduces a detailed dictionary via abstract data (I,J), their representations, and realizations, including cycling and twisting operations, to read a single connection in multiple quiver readings, yielding numerous isomorphisms of moduli spaces. The paper extends stability and existence criteria for connections from Crawley-Boevey to the irregular setting, via deformed preprojective algebras and root-system language, and highlights dualities such as Harnad’s reading-duality as natural reflection functors. This framework advances the understanding of isomonodromy, wild Hitchin spaces, and the geometric Langlands program by providing concrete, graph-based models with rich symmetry and non-emptiness criteria, applicable to poles of order up to three (and beyond in appendix).

Abstract

Some moduli spaces of irregular connections on the trivial bundle over the Riemann sphere will be identified with Nakajima quiver varieties. In particular this enables us to associate a Kac-Moody root system to such connections (yielding many isomorphisms between such moduli spaces, via the reflection functors for the corresponding Weyl group). The possibility of 'reading' a quiver in different ways also yields numerous isomorphisms between such moduli spaces, often between spaces of connections on different rank bundles and with different polar divisors. Finally some results of Crawley-Boevey on the existence of stable connections will be extended to this more general context.

Paper Structure

This paper contains 24 sections, 11 theorems, 73 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Connections on $G$-bundles
  3. Relation to quivers
  4. Connections and quivers
  5. Abstract data $(I,J)$.
  6. Representations of $(I,J)$.
  7. Realizations of $(I,J, V, \alpha, {\bf B})$.
  8. Passage to complete data
  9. Cycling
  10. Passage to incomplete data
  11. Different realizations
  12. Quivers
  13. Stability
  14. Non-emptiness conditions
  15. Quiver varieties
  16. ...and 9 more sections

Key Result

Theorem 1

For any such quiver $\mathcal{Q}$ there is a moduli space $\mathcal{M}^*$ of meromorphic connections with fixed formal types on a trivial holomorphic vector bundle over the Riemann sphere isomorphic to a Nakajima quiver variety for the quiver $\mathcal{Q}$, for some ${\bf d}, \lambda$. Moreover the

Figures (6)

  • Figure 1: Graphs from partitions of $N\le 6$
  • Figure 2: Four readings of $\Gamma(221)=2\cdot\Gamma(21)=1\cdot\Gamma(22)$
  • Figure 3: Deconstructing a triangle with legs.
  • Figure 4: Some four dimensional cases.
  • Figure 5: A leg
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 1
  • Lemma 2
  • Proposition 3
  • Lemma 4
  • Lemma 5
  • Theorem 2: CB-mmap
  • Theorem 3
  • Lemma 6
  • Lemma 7
  • ...and 1 more