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.

We present a new method to solve the holomorphic anomaly equations governing the free energies of type B topological strings. The method is based on direct integration with respect to the non-holomorphic dependence of the amplitudes, and relies on the interplay between non-holomorphicity and modularity properties of the topological string amplitudes. We develop a formalism valid for any Calabi-Yau manifold and we study in detail two examples, providing closed expressions for the amplitudes at low genus, as well as a discussion of the boundary conditions that fix the holomorphic ambiguity. The first example is the non-compact Calabi-Yau underlying Seiberg-Witten theory and its gravitational corrections. The second example is the Enriques Calabi-Yau, which we solve in full generality up to genus six. We discuss various aspects of this model: we obtain a new method to generate holomorphic automorphic forms on the Enriques moduli space, we write down a new product formula for the fiber amplitudes at all genus, and we analyze in detail the field theory limit. This allows us to uncover the modularity properties of SU(2), N = 2 super Yang-Mills theory with four massless hypermultiplets.

Abstract:We discuss various aspects of multi-instanton configurations in generic multi-cut matrix models. Explicit formulae are presented in the two-cut case and, in particular, we obtain general formulae for multi-instanton amplitudes in the one-cut matrix model case as a degeneration of the two-cut case. These formulae show that the instanton gas is ultra-dilute, due to the repulsion among the matrix model eigenvalues. We exemplify and test our general results in the cubic matrix model, where multi-instanton amplitudes can be also computed with orthogonal polynomials. As an application, we derive general expressions for multi-instanton contributions in two-dimensional quantum gravity, verifying them by computing the instanton corrections to the string equation. The resulting amplitudes can be interpreted as regularized partition functions for multiple ZZ-branes, which take into full account their back-reaction on the target geometry. Finally, we also derive structural properties of the trans-series solution to the Painlevé I equation.

We consider d=4, N = 2 compactifications of heterotic strings with an arbitrary number of Wilson lines. In particular, we focus on known chains of candidate heterotic/type II duals. We give closed expressions for the topological amplitudes F (g) in terms of automorphic forms of SO(2 + k, 2, Z), and find agreement with the geometric data of the dual K3 fibrations wherever those are known.a marlene.weiss@cern.ch

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