Nonperturbative effects and the large-order behavior of matrix models and topological strings

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 the 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 Painlevé 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.

Resurgence of the cusp anomalous dimension

“…This can be done by using the Richardson extrapolation. Let us consider the n-th Richardson transform [48] of a sequence f ,…”

Topological Strings on Local Curves

“…In matrix models, instanton effects are associated to eigenvalue tunneling [29,10], and in fact it is possible to write down explicit formulae for the instanton amplitudes up to two loops by using only information from the spectral curve [25]. Let us describe the instanton amplitudes for a matrix model described by a curve of the form…”

“…In this note we review some properties of topological string theory on X p discovered in [8,24,25]. We summarize them briefly in this introduction by emphasizing both their connections to general aspects of topological string theory, as well as their idiosyncrasies.…”

“…The matrix model formalism makes possible to compute spacetime instanton corrections to the partition function [25]. Using the connection between instantons and large order behavior, one obtains conjectural, precise descriptions of the asymptotic behavior of the couplings F g at large g [24,25] which can be tested numerically. This aspect is reviewed in section 4.…”

“…This is established in section 2.2. Most of the properties above can be seen to be inherited by this Hurwitz model (like in particular the critical behavior and the instanton effects describing large order), and we refer for this to the original papers [8,25].…”