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On uniform large genus asymptotics of Witten's intersection numbers

Jindong Guo, Di Yang, Don Zagier

Abstract

Following ideas from [14], we give a uniform large genus asymptotics for primitive psi-class intersection numbers on the moduli space of stable algebraic curves, and extend this result including insertions of zeros in a certain uniform way. Application to a particular formal solution of the Painlevé I equation is given. We also use a method from [14] to give a new proof of the polynomiality conjecture on large genus asymptotic expansions of psi-class intersection numbers.

On uniform large genus asymptotics of Witten's intersection numbers

Abstract

Following ideas from [14], we give a uniform large genus asymptotics for primitive psi-class intersection numbers on the moduli space of stable algebraic curves, and extend this result including insertions of zeros in a certain uniform way. Application to a particular formal solution of the Painlevé I equation is given. We also use a method from [14] to give a new proof of the polynomiality conjecture on large genus asymptotic expansions of psi-class intersection numbers.
Paper Structure (6 sections, 14 theorems, 122 equations, 2 tables)

This paper contains 6 sections, 14 theorems, 122 equations, 2 tables.

Table of Contents

  1. Introduction and statements of the main results
  2. An explicit formula for Witten's intersection numbers
  3. Uniform large genus asymptotics of Witten's intersection numbers
  4. Lower Bound
  5. Upper Bound
  6. Polynomiality in large genus

Key Result

Theorem 1

For $n\ge1$ and $\mathbf d\in (\mathbb{Z}_{\ge1})^n$ satisfying $g(\mathbf d)\in\mathbb{Z}_{\ge1}$, we have uniformly as $g(\mathbf d)\to\infty$.

Theorems & Definitions (31)

  • Conjecture 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Remark 4
  • Theorem 3
  • Proposition 1: GY
  • ...and 21 more