Direct Integration and Non-Perturbative Effects 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 $\Gamma (2)$. 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.

51 pages, 8 figures

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Cited by in corpus (18)

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• Resurgence theory, ghost-instantons, and analytic continuation of path integrals
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
• Holomorphic Anomaly and Quantum Mechanics
• Finite N from Resurgent Large N
• Exact multi-instantons in topological string theory
• Argyres-Douglas theories, Painlevé II and quantum mechanics
• Stokes Phenomena and Non-perturbative Completion in the Multi-cut Two-matrix Models
• High order perturbation theory for difference equations and Borel summability of quantum mirror curves
• Analytic Study for the String Theory Landscapes via Matrix Models
• Large N instantons from topological strings
• Dijkgraaf-Vafa conjecture and beta-deformed matrix models
• Superpotentials, quantum parameter space and phase transitions in N=1 supersymmetric gauge theories