Large N instantons from topological strings

Abstract

The 1/N1/N expansion of matrix models is asymptotic, and it requires non-perturbative corrections due to large NN instantons. Explicit expressions for large NN instanton amplitudes are known in the case of Hermitian matrix models with one cut, but not in the multi-cut case. We show that the recent exact results on topological string instanton amplitudes provide the non-perturbative contributions of large NN instantons in generic multi-cut, Hermitian matrix models. We present a detailed test in the case of the cubic matrix model by considering the asymptotics of its 1/N1/N expansion, which we obtain at relatively high genus for a generic two-cut background. These results can be extended to certain non-conventional matrix models which admit a topological string theory description. As an application, we determine the large NN instanton corrections for the free energy of ABJM theory on the three-sphere, which correspond to D-brane instanton corrections in superstring theory. We also illustrate the applications of topological string instantons in a more mathematical setting by considering orbifold Gromov-Witten invariants. By focusing on the example of {\mathbb C}^3/{\mathbb Z}_3ℂ3/ℤ3, we show that they grow doubly-factorially with the genus and we obtain and test explicit asymptotic formulae for them.