Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings (0711.1954v2)

Abstract: This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large-order behavior of the 1/N expansion. We study instanton configurations in generic one-cut matrix models, obtaining explicit results for the one-instanton amplitude at both one and two loops. The holographic description of topological strings in terms of matrix models implies that our nonperturbative results also apply to topological strings on toric Calabi-Yau manifolds. This yields very precise predictions for the large-order behavior of the perturbative genus expansion, both in conventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve, and Hurwitz theory. In all these cases we provide extensive numerical checks which heavily support our nonperturbative analytical results. Moreover, since all these models have a critical point describing two-dimensional gravity, we also obtain in this way the large-order asymptotics of the relevant solution to the Painleve I equation, including corrections in inverse genus. From a mathematical point of view, our results predict the large-genus asymptotics of simple Hurwitz numbers and of local Gromov-Witten invariants.

Nonperturbative Effects and the Large--Order Behavior of Matrix Models and Topological Strings

The paper explores the intricate web of nonperturbative effects in matrix models and their interplay with topological string theories, employing advanced analytical tools to unpack these relationships. By exploring instanton configurations within a class of one-cut matrix models, the authors derive explicit results for one-instanton amplitudes at both one and two-loop levels. This work not only illuminates the holographic connections between matrix models and topological strings on toric Calabi-Yau manifolds but also makes significant strides in predicting the large-order behavior of perturbative expansions in both arenas.

In the domain of large N gauge theories, nonperturbative corrections typically emerge as instanton effects within their $1/N$ expansions. These effects correlate with large-order phenomena in matrix models, where duals of string theories are explored. The paper rigorously evaluates instanton effects through precise analytic techniques, validated via extensive numerical checks across diverse scenarios, from quartic matrix models to more complex systems such as topological strings on the local curve and Hurwitz theory.

A significant theme of this research lies in the confirmation of nonperturbative predictions by leveraging the holomorphic structure of the associated spectral curves. This approach extends to topological string theories interpreted through the matrix model framework, contingent upon the spectral curve characteristics. Such insights grant the ability to address broader implications, particularly in evaluating noncritical string backgrounds and providing enhanced predictions for large-order asymptotics, a novel trajectory that refines mathematical understanding of Hurwitz numbers and Gromov-Witten invariants.

Specific emphasis is placed on the numerical testing of these predictions against classical results for matrix models, notably the quartic model and its double-scaled limit embodying two-dimensional gravity. This testing is pivotal in aligning theoretical conjectures with computed data, leveraging techniques like Richardson extrapolation to verify the asymptotic behavior with a high degree of accuracy.

The paper further explores speculative future directions, including potential expansions into multi-cut matrix models to redefine the understanding of nonperturbative completions in matrix model theory. These explorations propose a comprehensive analytical paradigm that, while technical, sheds light on broader physical phenomena prevalent in both matrix theories and their topological string duals. Such work not only bolsters comprehension of nonperturbative dynamics but also establishes a groundwork for continued research into holographically driven interpretations of string theory.

Overall, this research embodies a significant step towards reconciling perturbative subtleties with nonperturbative phenomena, setting the stage for advanced interpretive frameworks that bridge computationally complex spheres within theoretical physics.