CFT and topological recursion

TL;DR

This paper demonstrates the equivalence between the conformal field theory approach and the topological recursion method in analyzing the quasiclassical expansion of complex curves, particularly in the context of matrix integrals.

Contribution

It establishes a formal connection between CFT and topological recursion, showing they produce equivalent expansions for symplectic invariants on complex curves.

Findings

CFT and topological recursion are equivalent methods for complex curve expansions.

One graph expansion can be obtained as a partial resummation of the other.

The approaches provide different but related frameworks for matrix integral expansions.

Abstract

We study the quasiclassical expansion associated with a complex curve. In a more specific context this is the 1/N expansion in U(N)-invariant matrix integrals. We compare two approaches, the CFT approach and the topological recursion, and show their equivalence. The CFT approach reformulates the problem in terms of a conformal field theory on a Riemann surface, while the topological recursion is based on a recurrence equation for the observables representing symplectic invariants on the complex curve. The two approaches lead to two different graph expansions, one of which can be obtained as a partial resummation of the other.