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Convergence of integrals on the moduli spaces of curves and cographical matroids

Alexander Polishchuk, Nicholas Proudfoot

TL;DR

The paper analyzes the convergence of local integrals on moduli spaces of curves near degenerate boundary points, expressing the convergence threshold through the degeneration graph $\Gamma$. It connects the leading asymptotics of the period matrix $\tau$ to the Kirchhoff polynomial $\psi_{\Gamma}$ and, via matroid-theoretic reductions, reduces the problem to optimal graphs with $c(\Gamma)=\frac{e(\Gamma)}{b(\Gamma)}$. The main result, Theorem A, shows that $J_{\Gamma}(s)$ converges for $\operatorname{Re}(s)>c(\Gamma)$ and diverges at $s=c(\Gamma)$, with extensions to $I_{\Gamma}(\varphi,s)$ when all components are rational; crucially, for genus $g\ge 6$ there exist $\Gamma$ with $I_{\Gamma}(5)$ diverging, indicating the need for additional regularization in superstring vacuum amplitudes. This work thus provides a precise combinatorial criterion for convergence of moduli-space integrals and links graph polynomials, period-matrix degenerations, and string-theoretic regularization.

Abstract

We determine the convergence regions of certain local integrals on the moduli spaces of curves in neighborhoods of fixed stable curves in terms of the combinatorics of the corresponding graphs.

Convergence of integrals on the moduli spaces of curves and cographical matroids

TL;DR

The paper analyzes the convergence of local integrals on moduli spaces of curves near degenerate boundary points, expressing the convergence threshold through the degeneration graph . It connects the leading asymptotics of the period matrix to the Kirchhoff polynomial and, via matroid-theoretic reductions, reduces the problem to optimal graphs with . The main result, Theorem A, shows that converges for and diverges at , with extensions to when all components are rational; crucially, for genus there exist with diverging, indicating the need for additional regularization in superstring vacuum amplitudes. This work thus provides a precise combinatorial criterion for convergence of moduli-space integrals and links graph polynomials, period-matrix degenerations, and string-theoretic regularization.

Abstract

We determine the convergence regions of certain local integrals on the moduli spaces of curves in neighborhoods of fixed stable curves in terms of the combinatorics of the corresponding graphs.
Paper Structure (8 sections, 9 theorems, 44 equations)

This paper contains 8 sections, 9 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Kirchoff polynomial and the asymptotics of the period matrix
  3. Kirchoff polynomial
  4. Asymptotics of the period matrix
  5. Convergence region
  6. Elementary observations
  7. Combinatorial statement
  8. Proof of Theorem A

Key Result

Proposition 2.1

(i) One has $\det(B)=\sum_{S\in {\mathcal{B}}} \prod_{e\in S} x_e$. (ii) Let us view $B$ as a nondegenerate form over the field $QR$, the fraction field over $R$, and let $B^{-1}$ denote the corresponding bilinear form on the dual space $H^\vee$ with values in $QR$. Then setting $x_e=\ln(y_e)$, wher

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 11 more