[Paper Review] Large genus behavior of topological recursion

Bertrand Eynard|arXiv (Cornell University)|May 27, 2019

Mathematical Dynamics and Fractals8 references4 citations

TL;DR

This paper establishes factorial upper bounds on the large-genus growth of topological recursion invariants $ F_g $ for generic regular spectral curves, showing $ |F_g| = O((5g)! r^{-g}) $ under smoothness assumptions. The result implies the generating series $ extstylerac{1}{eta} ext{Borel transform} $ converges in a disk, setting the stage for resurgence analysis in quantum invariants and enumerative geometry.

ABSTRACT

We show that for a rather generic set of regular spectral curves, the Topological-Recursion invariants F_g grow at most like $O((βg)! r^{-g}) $ with some $r>0$ and $β\leq 5$.

Motivation & Objective

• To determine the asymptotic large-genus behavior of topological recursion invariants $ F_g $ associated with generic spectral curves.
• To establish rigorous upper bounds on the growth rate of $ F_g $ as $ g \to \infty $, particularly in terms of factorial decay.
• To provide conditions under which the generating series $ \sum_g \hbar^{2g-2} F_g $ is asymptotic with factorially bounded coefficients.
• To lay the foundation for future analysis of resurgence and Borel resummation of the partition function in topological recursion.
• To quantify the divergence of $ F_g $ as ramification points coalesce, using the radius $ R $ of disjoint spectral discs.

Proposed method

• Imposes generic smoothness conditions on the spectral curve $ \mathcal{S} = (\Sigma, x, y, B) $, including simple ramification and analyticity of $ y $ near branch points.
• Introduces local coordinates $ \rho(p) = \sqrt{\prod_{a \in \mathfrak{R}} (x(p) - a)} $ to define a domain $ \Sigma_R $ of disjoint discs around ramification points.
• Defines key constants $ C $ and $ B $ from the topological recursion kernel $ K $ and bidifferential $ B $, ensuring finiteness under assumptions.
• Derives recursive bounds on the multidifferentials $ \omega_{g,n} $ via a recursive sequence $ C_{g,n} $, using combinatorial decomposition over stable pairs $ (g_i, n_i) $.
• Applies a refined estimate involving $ (d+k)^{d+k}/(k^k d^d) $ to control growth in the recursion, leading to factorial-type bounds.
• Uses contour integration and the $ \omega_{g,1} $-formula to bound $ F_g $, incorporating the $ \Phi $-form and $ \rho $-scaling to derive $ O((5g)! r^{-g}) $ decay.

Experimental results

Research questions

• RQ1 How fast can the topological recursion invariants $ F_g $ grow as $ g \to \infty $ for generic spectral curves?
• RQ2 Under what conditions is the generating series $ \sum_g \hbar^{2g-2} F_g $ asymptotic with factorially bounded coefficients?
• RQ3 Can the Borel transform of the $ F_g $-series be shown to converge in a disk, enabling potential resummation?
• RQ4 What is the precise dependence of $ F_g $ on the distance between ramification points, as measured by $ R $?
• RQ5 Is the growth of $ F_g $ compatible with resurgence, and what are the implications for the full partition function?

Key findings

• For generic regular spectral curves with simple ramification and analytic $ y $, the invariants satisfy $ |F_g| \leq \tilde{C} C^{2g-2} B^{g-1} \frac{1}{R^{6g-6}} \frac{C_{g,1}}{2g-2} $, with $ \tilde{C} $ depending on $ \Phi $-behavior.
• A refined bound yields $ |F_g| \leq \tilde{C} \frac{9}{80e} C^{2g-2} B^{g-1} \frac{1}{R^{6g-6}} r^{-g} \frac{(5g-2)!}{2g-2} $, showing at most $ O((5g)!) $ growth.
• The series $ \sum_g \hbar^{2g-2} F_g $ is asymptotic with factorially bounded coefficients, implying its Borel transform converges in a disk.
• The bound is sharp in the sense that $ \beta \leq 5 $ is optimal under the assumptions, and $ \beta = 2 $ is typical in known examples.
• The divergence of $ F_g $ as $ R \to 0 $ (ramification points coalescing) is captured by the $ R^{-6g+6} $ term, confirming known physical intuition.
• The paper sets the stage for future work on resurgence, suggesting that $ \hat{F}(s) = \sum_g \frac{s^{\beta g}}{(\beta g)!} F_g $ may be analytically continued, leading to instanton corrections of the form $ e^{-\hbar^{-1}} $.