Resurgent Transseries and the Holomorphic Anomaly

The gauge theoretic large N expansion yields an asymptotic series which requires a nonperturbative completion to be well defined. Recently, within the context of random matrix models, it was shown how to build resurgent transseries solutions encoding the full nonperturbative information beyond the ’t Hooft genus expansion. On the other hand, via large N duality, random matrix models may be holographically described by B-model closed topological strings in local Calabi–Yau geometries. This raises the question of constructing the corresponding holographically dual resurgent transseries, tantamount to nonperturbative topological string theory. This paper addresses this point by showing how to construct resurgent transseries solutions to the holomorphic anomaly equations. These solutions are built upon (generalized) multi-instanton sectors, where the instanton actions are holomorphic. The asymptotic expansions around the multi-instanton sectors have both holomorphic and anti-holomorphic dependence, may allow for resonance, and their structure is completely fixed by the holomorphic anomaly equations in terms of specific polynomials multiplied by exponential factors and up to the holomorphic ambiguities—which generalizes the known perturbative structure to the full transseries. In particular, the anti-holomorphic dependence has a somewhat universal character. Furthermore, in the non-perturbative sectors, holomorphic ambiguities may be fixed at conifold points. This construction shows the nonperturbative integrability of the holomorphic anomaly equations and sets the ground to start addressing large-order analysis and resurgent nonperturbative completions within closed topological string theory.