Direct Integration and Non-Perturbative Effects in Matrix Models

TL;DR

This work develops and applies direct integration of the holomorphic anomaly equations to multi-cut matrix models, with a detailed treatment of the two-cut cubic model. It builds a modular, covariant framework that yields explicit $F_g$ up to high genus, including a Seiberg--Witten reduction on a special moduli submanifold and a complete genus-1 description via $\Gamma(2)$ modular forms. The paper demonstrates a precise link between large-order behavior and instanton effects, validating a leading instanton picture in the two-cut setting while arguing that new non-perturbative sectors are required for full large-genus asymptotics. It also provides a high-genus benchmark (up to genus 52) on the $S_1=-S_2$ slice and clarifies how gap conditions fully fix holomorphic ambiguities, highlighting the interplay between spectral-curve modularity, SW geometry, and non-perturbative physics in matrix models.

Abstract

We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $Γ(2)$. On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.

Figures (6)

• Figure 1: Choice of branch cuts and cycles on the elliptic geometry (\ref{['ycurve']}).
• Figure 2: The sequence $Q_g$ (•) and two Richardson transforms ($\blacksquare$, $\blacklozenge$) at $\tau_0=\frac{{\rm i}}{2}$ (left) and $\tau_0=\frac{2{\rm i}}{3}+\frac{1}{9}$ (right) which corresponds to $S\approx 0.139$ and $S\approx 0.117+0.016{\rm i}$, respectively. The leading asymptotics as predicted by the instanton action $|A|^{-2}$ is shown as a straight line. The error for genus 52 is about $10^{-8}$ % and $10^{-10}$ %, resp.
• Figure 3: The sequence $Q'_g$ (•) and two Richardson transforms ($\blacksquare$, $\blacklozenge$) at $\tau_0=\frac{{\rm i}}{2}$ (left) and $\tau_0=\frac{2{\rm i}}{3}+\frac{1}{9}$ (right) which corresponds to $S\approx 0.139$ and $S\approx 0.117+0.016{\rm i}$, respectively. The leading asymptotics as predicted by the parameter $b=-1$ is shown as a straight line. The error for genus 52 is about $10^{-8}$ % in both cases.
• Figure 4: $Q_3$ (•), $R_Q(1,2)$ ($\blacksquare$) and $|A|^{-2}$ ($\blacklozenge$) are plotted for several values of $S_1$ around the slice point $S_1=-S_2=S=0.004$. $Q_3$ and $R_Q(1,2)$ have a relative error of about $30$ % and $10$ %, respectively, as compared to the instanton action $|A|^{-2}$ throughout the data set.
• Figure 5: The asymptotics of the coefficients of the $\ell$-th instanton solution $u_{\ell}(z)$ of Painlevé I is determined by the two nearest neighbor instantons, which are obtained by eigenvalue tunneling, and by the generalized instanton amplitude $u_{\ell|1}$, which is represented here by the label $(\ell,1)$.