Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

Abstract: We develop techniques to compute multi-instanton corrections to the 1/N expansion in matrix models described by orthogonal polynomials. These techniques are based on finding trans-series solutions, i.e. formal solutions with exponentially small corrections, to the recursion relations characterizing the free energy. We illustrate this method in the Hermitian, quartic matrix model, and we provide a detailed description of the instanton corrections in the Gross-Witten-Wadia (GWW) unitary matrix model. Moreover, we use Borel resummation techniques and results from the theory of resurgent functions to relate the formal multi-instanton series to the nonperturbative definition of the matrix model. We study this relation in the case of the GWW model and its double-scaling limit, providing in this way a nice illustration of various mechanisms connecting the resummation of perturbative series to nonperturbative results, like the cancellation of nonperturbative ambiguities. Finally, we argue that trans-series solutions are also relevant in the context of topological string theory. In particular, we point out that in topological string models with both a matrix model and a large N gauge theory description, the nonperturbative, holographic definition involves a sum over the multi-instanton sectors of the matrix model

Nonperturbative Effects and Nonperturbative Definitions in Matrix Models and Topological Strings

The paper authored by Marcos Mariño, titled "Nonperturbative effects and nonperturbative definitions in matrix models and topological strings," presents a comprehensive study of nonperturbative phenomena in matrix models, highlighting techniques to compute multi-instanton corrections to the $1/N$ expansion and their implications in topological string theory. This paper explores how formal solutions characterized by trans-series expansions can be systematically related to nonperturbative definitions using Borel resummation and the theory of resurgence.

Matrix Models and Nonperturbative Effects

Matrix models provide a versatile framework for addressing nonperturbative effects, serving as simplified models for complex quantum field theories and string theories. The paper builds on the established understanding that nonperturbative effects are crucial in capturing the full behavior of matrix models, particularly focusing on eigenvalue tunneling mechanisms. These are expertly analyzed through trans-series expansions that incorporate exponentially small corrections expressing multi-instanton amplitudes.

1. Techniques Utilized: The research develops computational techniques leveraging orthogonal polynomials associated with matrix models. By finding trans-series solutions to recursion relations, it systematically derives formal multi-instanton expansions. The Hermitian quartic matrix model and the Gross-Witten-Wadia (GWW) unitary matrix model serve as key examples.

2. Results in Matrix Models: Strong numerical results detail the instanton corrections explicitly and validate the approach through classical examples like the quartic matrix model's instanton action, matching earlier results obtained through spectral curve analysis.

Nonperturbative Definitions Using Trans-series

The paper speculates that the formal multi-instanton trans-series can be related to actual solutions by using Borel resummation. This technique is pivotal in overcoming factorial divergence often seen in perturbative expansions. The article illustrates these concepts through:

1. Resurgence Theory: The approach draws heavily on resurgence theory and exponential asymptotics to relate lateral Borel resummations of trans-series solutions to true convergent solutions, demonstrating that nonperturbative ambiguities cancel reliably, thus yielding accurate real solutions.

2. Double-Scaling Limits: Special focus is given to double-scaling limits, especially in Painlevé I and II systems, where the formal solutions provide detailed insights into the nonperturbative behavior of matrix models and minimal string theories.

Topological Strings and Implications

The study extends its framework to topological string theories, suggesting that topological string theory possesses a trans-series structure akin to matrix models.

1. Topological String Theory: The paper conjectures that nonperturbative topological string partition functions potentially involve sums over instanton sectors, expanding beyond traditional perturbative calculations.

2. Holographic Duals: Instances of topological strings with large $N$ duals, such as the topological string on $A_{p-1}$ fibrations over $\mathbb{P}^1$, provide partial confirmation. The presence of large $N$ Chern-Simons duals suggests that topological string models in these contexts must consider multi-instanton corrections to comprehensively account for nonperturbative effects.

Future Prospects and Conclusion

This research opens several avenues for future exploration, notably the potential for matrix models and topological string theories to uncover novel insights into nonperturbative phenomena, including:

• Further exploration of nonperturbative definitions in complex matrix models and related Toda-type equations.
• Extension into other minimal models, possibly yielding a semiclassical understanding of exact solutions.
• Clarifying nonperturbative effects in topological strings and their implications for background independence and large $N$ dualities.

In conclusion, the paper posits that understanding nonperturbative effects through specialized mathematical techniques significantly contributes to the ongoing development in both matrix model and topological string theories. This approach heralds a framework that integrates perturbative and nonperturbative descriptions into a unified model, advancing both theoretical insights and computational strategies.