Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence

TL;DR

This paper develops an asymptotic expansion for matrix integrals that includes oscillatory terms, relates these to holomorphic anomalies in string theory, and demonstrates background independence of the expansion.

Contribution

It introduces a novel asymptotic expansion formula for matrix integrals incorporating oscillatory terms and connects these to holomorphic anomaly equations, showing background independence.

Findings

Oscillatory series resummed into a theta function

Coefficients match those in holomorphic anomaly equations

Expansion is background independent

Abstract

We propose an asymptotic expansion formula for matrix integrals, including oscillatory terms (derivatives of theta-functions) to all orders. This formula is heuristically derived from the analogy between matrix integrals, and formal matrix models (combinatorics of discrete surfaces), after summing over filling fractions. The whole oscillatory series can also be resummed into a single theta function. We also remark that the coefficients of the theta derivatives, are the same as those which appear in holomorphic anomaly equations in string theory, i.e. they are related to degeneracies of Riemann surfaces. Moreover, the expansion presented here, happens to be independent of the choice of a background filling fraction.