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A positive formula for volumes of moduli spaces of flat unitary connections on compact surfaces

Quentin François, David García-Zelada, Thierry Lévy, Pierre Tarrago

Abstract

We provide a manifestly positive expression for the volume of the moduli spaces of flat $\mathrm{U}(n)$-valued connections on punctured compact oriented surfaces. This volume is obtained by summing volumes of explicit polytopes describing coloured honeycombs on a polygon, in the spirit of the work of Knutson and Tao describing the spectrum of the sum of two hermitian matrices. As a corollary, we also provide a positive formula for marginals of the $\mathrm{U}(n)$-valued Yang-Mills measure on a compact oriented surface in terms of the probability distribution of an explicit path process.

A positive formula for volumes of moduli spaces of flat unitary connections on compact surfaces

Abstract

We provide a manifestly positive expression for the volume of the moduli spaces of flat -valued connections on punctured compact oriented surfaces. This volume is obtained by summing volumes of explicit polytopes describing coloured honeycombs on a polygon, in the spirit of the work of Knutson and Tao describing the spectrum of the sum of two hermitian matrices. As a corollary, we also provide a positive formula for marginals of the -valued Yang-Mills measure on a compact oriented surface in terms of the probability distribution of an explicit path process.
Paper Structure (23 sections, 23 theorems, 142 equations, 9 figures, 1 table)

This paper contains 23 sections, 23 theorems, 142 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and statement of the main result
  3. Conjugacy classes of the unitary group
  4. Triangular honeycomb
  5. (g, p)-honeycombs
  6. Volume formula for the moduli space of flat connections
  7. Yang-Mills marginal for disjoint curves
  8. Graphs, divergence of flows and volume measures
  9. Canonical measure
  10. Boundary and sieving of graphs
  11. The three-holed sphere
  12. Parametrization of triangular honeycombs
  13. Dual hive
  14. From dual hive to triangular honeycomb
  15. From triangular honeycomb to dual hive
  16. ...and 8 more sections

Key Result

Theorem 2.6

For $\alpha_1, \dots, \alpha_p\in\mathcal{H}_{reg}$ such that $\sum_{i=1}^p\vert \alpha_i\vert\in \mathbb{N}$, where $c_{0,3}=\frac{2^{(n+1)[2]}(2\pi)^{(n-1)(n-2)}}{n!}$.

Figures (9)

  • Figure 1: The possible angles between segments. The colors of the edges are given in the couple below the configuration.
  • Figure 2: Boundaries of a honeycomb in $\texttt{HONEY}_{n, d}(\alpha, \beta, \gamma)$.
  • Figure 3: A $(2, 3)$ honeycomb for a surface with genus 2 and 3 boundaries. On each upper triangle, the endpoints of the segments of the honeycomb coincide on the left and on the right edge.
  • Figure 4: A disjoint loops configuration and its corresponding edge-labelled tree
  • Figure 5: Two adjacent vertices in $\mathop{\mathrm{int}}\nolimits(V)$ for which $\ell = \ell'=2$.
  • ...and 4 more figures

Theorems & Definitions (60)

  • Definition 2.1: Non-degenerate honeycomb
  • Definition 2.2
  • Definition 2.3: Triangular honeycomb
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6: Volume formula for $(g, p)$ partition function
  • Remark 2.7: Volume and conditioned processes
  • Corollary 2.8: Yang-Mills partition function
  • Remark 2.9: Yang-Mills marginals and conditioned processes
  • Proposition 3.1
  • ...and 50 more