Communications in Number Theory and Physics

Volume 2 (2008)

Number 2

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

Pages: 349 – 419



Marcos Mariño (Department of Physics, CERN, Genève, Switzerland)

Ricardo Schiappa (Department of Physics, CERN, Genève, Switzerland)

Marlene Weiss (ITP, ETH Zürich, Switzerland)


This work addresses nonperturbative effects in both matrix models andtopological strings, and their relation with the large-order behaviorof the $1/N$ expansion. We study instanton configurations in genericone-cut matrix models, obtaining explicit results for theone-instanton amplitude at both one and two loops. The holographicdescription of topological strings in terms of matrix models impliesthat our nonperturbative results also apply to topological strings ontoric Calabi–Yau manifolds. This yields very precise predictions forthe large-order behavior of the perturbative genus expansion, both inconventional matrix models and in topological string theory. We testthese predictions in detail in various examples, including thequartic matrix model, topological strings on the local curve and theHurwitz theory. In all these cases, we provide extensive numericalchecks which heavily support our nonperturbative analytical results.Moreover, since all these models have a critical point describingtwo-dimensional gravity, we also obtain in this way the large-orderasymptotics of the relevant solution to the Painlevé I equation,including corrections in inverse genus. From a mathematical point ofview, our results predict the large-genus asymptotics of simpleHurwitz numbers and of local Gromov–Witten invariants.

Published 1 January 2008