[Paper Review] Resurgent large genus asymptotics of intersection numbers

Benoît Eynard, Elba Garcia‐Failde+3 more|arXiv (Cornell University)|Sep 6, 2023

Algebraic Geometry and Number TheoryMathematics3 citations

TL;DR

This paper introduces a resurgent analysis framework to compute large genus asymptotics of intersection numbers across ψ-class, Θ-class, and r-spin theories. By leveraging determinantal formulae and quantum curves, it derives full trans-series expansions, proving the Guo–Yang conjecture and uncovering oscillatory and exponentially subleading corrections in r-spin systems, extending Aggarwal's results with all subleading terms.

ABSTRACT

In this paper, we present a novel approach for computing the large genus asymptotics of intersection numbers. Our strategy is based on a resurgent analysis of the $n$-point functions of such intersection numbers, which are computed via determinantal formulae, and relies on the presence of a quantum curve. With this approach, we are able to extend the recent results of Aggarwal for Witten-Kontsevich intersection numbers with the computation of all subleading corrections, proving a conjecture of Guo-Yang, and to obtain new results on $r$-spin and Theta-class intersection numbers.

Motivation & Objective

• To develop a systematic method for computing large genus asymptotics of intersection numbers beyond leading-order approximations.
• To extend Aggarwal's recent proof of the Witten–Kontsevich asymptotic formula by including all subleading corrections.
• To resolve the Guo–Yang conjecture on the structure of subleading terms in ψ-class intersection numbers.
• To derive new asymptotic expansions for Θ-class and r-spin intersection numbers using resurgent techniques.
• To characterize the trans-series structure of n-point functions via Borel plane singularity analysis and quantum curve data.

Proposed method

• Apply the Borel transform to the generating series of intersection numbers to analyze their singularity structure in the Borel plane.
• Use resurgent function theory to extract trans-series expansions from the singularities of the Borel transform.
• Leverage determinantal formulae for n-point functions derived from the underlying quantum curve structure.
• Identify instanton actions and their contributions via the Riemann–Hilbert problem associated with the quantum curve.
• Construct sequences to visualize the dominance of specific singularities in the Borel plane, such as |A(xi)| for correlators.
• Use the quantum curve to derive the full asymptotic expansion, including oscillatory and exponentially suppressed corrections.

Experimental results

Research questions

• RQ1 What is the complete trans-series structure of ψ-class intersection numbers in the large genus limit?
• RQ2 How do subleading corrections in the asymptotics of Witten–Kontsevich numbers behave, and do they confirm the Guo–Yang conjecture?
• RQ3 What is the nature of the large genus asymptotics for Θ-class intersection numbers, and how do they differ from ψ-class?
• RQ4 How do r-spin intersection numbers behave asymptotically for r ≥ 3, especially regarding oscillatory and exponentially subleading terms?
• RQ5 How do the relative contributions of different Borel plane singularities compete in determining the dominant asymptotic behavior of n-point correlators?

Key findings

• The paper proves the Guo–Yang conjecture by computing all subleading corrections to the large genus asymptotics of Witten–Kontsevich intersection numbers.
• For r-spin systems with r ≥ 3, the asymptotics exhibit oscillatory behavior due to complex-conjugate instanton actions, with leading-order terms given by equation (1.12).
• For r ≥ 4, exponentially subleading corrections appear due to isolated singularities in the Borel plane, not from pairs of instantons, and are captured via sequences like J^r-spin_{d,a,K} in equation (C.4).
• The dominance of specific instanton actions in n-point correlators depends on the values of the variables x_i, with the minimal |A(x_i)| dominating, as shown in figure 7.
• The asymptotic behavior of 1-point 4-spin numbers is visualized in figure 6a and 6b, confirming the leading oscillatory term and the first subleading correction, respectively.
• The trans-series structure of the n-point functions is fully characterized by the singularity structure of the Borel transform, with contributions from multiple instanton actions and their interference.