New Instantons for Matrix Models

TL;DR

This work provides a complete nonperturbative framework for random matrix models by embedding resurgent-transseries into matrix-integral language. It resolves the long-standing puzzle of resonant multi-instantons by showing that anti-eigenvalues, living on the non-physical sheet of the spectral curve, generate the backward Stokes data and a full transseries structure when combined with standard eigenvalue tunneling. The authors develop determinant-correlator techniques, uniformization of spectral curves, and a detailed contour analysis to compute all resonant sectors, validating them in cubic and quartic models and in their Painlevé I double-scaling limits. The results yield explicit Stokes data and match known transseries predictions, including logarithmic sectors, establishing a direct matrix-model route to resurgent structures and their double-scaling universality. The framework also connects to supermatrix models, suggesting broader implications for nonperturbative string theory and topological recursion.

Abstract

The complete, nonperturbative content of random matrix models is described by resurgent-transseries -- general solutions to their corresponding string-equations. These transseries include exponentially-suppressed multi-instanton amplitudes obtained by eigenvalue tunneling, but they also contain exponentially-enhanced and mixed instanton-like sectors with no known matrix model interpretation. This work shows how these sectors can be also described by eigenvalue tunneling in matrix models -- but on the non-physical sheet of the spectral curve describing their large-N limit. This picture further explains the full resurgence of random matrices via analysis of all possible eigenvalue integration-contours. How to calculate these "anti" eigenvalue-tunneling amplitudes is explained in detail and in various examples, such as the cubic and quartic matrix models, and their double-scaling limit to Painleve I. This further provides direct matrix-model derivations of their resurgent Stokes data, which were recently obtained by different techniques.

Figures (22)

• Figure 1: Pictorial description of Borel resummation of a divergent series into a function. Start with the left-downward arrow performing a Borel transform, which yields a power-series convergent on a disk. Moving with the rightward arrow implements analytic continuation beyond the radius of convergence of the power-series. This results in a function on the complex plane, albeit with singularities. The final step with the right-upward arrow is resummation via Laplace transform (inverse Borel transform); albeit this is ill-defined when crossing any of the aforementioned singularities upon the complex Borel plane (sitting along rays known as Stokes lines).
• Figure 2:
• Figure 3: The transseries sectors in \ref{['eq:transseriesexpansionGeneric']} represented as a two-dimensional semi-positive lattice. Resurgence relates different sectors to each other, essentially via the Stokes automorphisms \ref{['eq:forwardStokesAutomorphism']}-\ref{['eq:backwardStokesAutomorphism']}. The arrows in the figure symbolize Borel residues $\mathsf{S}_{\boldsymbol{n}\to\boldsymbol{m}}$ and exemplify how to reach, for instance, the $(2,2)$-sector via single-step moves on this lattice. We have colored each step differently (as first, second, third, and fourth steps). The dashed brown lines symbolize resonant directions, whereas the dashed arrows have associated vanishing Borel residues. Resurgence relations are fairly simple as long as we only move along the edges of the lattice; but as soon as one moves inwards, resonant contributions start appearing. For example, the third step reaches two distinct endpoints, implying that on the fourth step there will be Borel residues starting at both these $(2,1)$ and $(1,0)$ sectors. Upon iterated use of Stokes transitions, one hence finds a multitude of different sectors that are spawned by resonance. Throughout this paper we shall refer to similar diagrams in order to better understand the resurgence relations at play.
• Figure 4: Stokes vectors for a two-parameter resonant transseries may be conveniently arranged on a two-dimensional lattice that extends infinitely along two negative directions but is bounded upon the positive ones bssv22. These vectors naturally group themselves into two sets (orange and blue), separated by an empty diagonal, exactly related to the forward and backward Stokes automorphisms of figure \ref{['fig:PictureTwoAutomorphisms']}. We shall later compute some of these entries from matrix integrals.
• Figure 5: Illustration of the spectral curve for the cubic matrix model. There are two sheets separated by the cut (green) which further touch at the pinched cycle $x^{\star}$: the physical sheet (blue) containing the integration contour for the action (blue), and the non-physical sheet (orange).