[Paper Review] New Instantons for Matrix Models

Marcos Mariño, Ricardo Schiappa+1 more|arXiv (Cornell University)|Oct 24, 2022

Molecular spectroscopy and chirality17 citations

TL;DR

This paper resolves the long-standing puzzle of nonperturbative sectors in matrix models by showing that exponentially enhanced and mixed instanton-like amplitudes—previously without matrix model interpretation—arise from eigenvalue tunneling on the non-physical sheet of the spectral curve. Using contour integration over eigenvalues in the complex plane, the authors derive explicit formulas for these 'anti-instanton' amplitudes in cubic and quartic matrix models, providing direct matrix-model derivations of resurgent Stokes data previously obtained via ODE analysis.

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.

Motivation & Objective

• To resolve the lack of matrix model interpretation for exponentially enhanced and mixed instanton sectors in transseries solutions of random matrix models.
• To show that these nonperturbative sectors arise from eigenvalue tunneling on the non-physical sheet of the spectral curve in the large-N limit.
• To provide a systematic method for computing these 'anti-instanton' amplitudes using complex eigenvalue integration contours.
• To derive the full resurgent Stokes data for matrix models directly from matrix model path integrals, confirming earlier results from ODE analysis.
• To establish a complete resurgence picture for matrix models by analyzing all possible eigenvalue integration contours.

Proposed method

• Analyzing eigenvalue integration contours in the complex plane to identify non-physical sheet contributions to the matrix model partition function.
• Extending the standard eigenvalue tunneling picture to include contours that access the non-physical sheet of the spectral curve.
• Deriving explicit expressions for multi-instanton amplitudes Z^{(n,m)} in the cubic and quartic matrix models using contour deformation and saddle-point analysis.
• Computing transseries solutions for the free energy and specific heat in the double-scaling limit, matching the Painlevé I equation.
• Extracting Stokes data and Borel residues from the transseries coefficients, validating them against known results from ODE resurgence.
• Using backward-forward symmetries and logarithmic sector relations to constrain and verify the structure of the transseries.

Experimental results

Research questions

• RQ1 Can the exponentially enhanced and mixed instanton sectors in matrix model transseries be given a direct matrix model interpretation?
• RQ2 Do these nonperturbative sectors arise from eigenvalue tunneling, and if so, on which Riemann sheet of the spectral curve?
• RQ3 Can the full set of Stokes data for matrix models be derived directly from the matrix model path integral, without relying on ODE analysis?
• RQ4 How do the transseries coefficients and logarithmic sectors in the double-scaling limit relate to the underlying eigenvalue integration contours?
• RQ5 What is the role of contour deformation and non-physical sheet contributions in generating the full resurgence structure of matrix models?

Key findings

• The authors derive explicit formulas for multi-instanton amplitudes Z^{(n,m)} in the cubic and quartic matrix models, including the previously mysterious m>0 sectors.
• The amplitudes Z^{(n,m)} are obtained via eigenvalue tunneling on the non-physical sheet of the spectral curve, providing a physical interpretation for all transseries sectors.
• The double-scaling limit of the quartic matrix model reproduces the resonant transseries of the Painlevé I equation, with full agreement in logarithmic and power-law structures.
• The Stokes vectors and coefficients, such as S_{(0,0)→(1,0)} = -i√(3/πλ), are derived directly from matrix model integrals, confirming earlier ODE-based results.
• The transseries coefficients F^{(n,m)[k]}(x) exhibit a precise backward-forward symmetry, with F^{(n,m)[0]}_g(x) = (-1)^{g+⌊n/2⌋} F^{(m,n)[0]}_g(x) for n>m, as derived from contour analysis.
• The method successfully reproduces the full set of transcendental Stokes data, including the complex coefficient S_{(1,1)→(1,0)} = -i√(λ/3π)(γ_E + log 96√3), from first-principles matrix model calculations.