Think globally, compute locally

TL;DR

This paper introduces a global formulation of topological recursion on compact Riemann surfaces, proving its equivalence to the generalized recursion and showing how correlation functions for arbitrary ramification can be derived as limits from simple cases.

Contribution

It presents a new global formulation of topological recursion and establishes its equivalence to existing generalized recursion, extending properties to arbitrary ramification cases.

Findings

Global formulation is equivalent to generalized recursion.

Correlation functions for arbitrary ramification are limits of simple cases.

Properties proved for simple ramification also hold for arbitrary ramification.

Abstract

We introduce a new formulation of the so-called topological recursion, that is defined globally on a compact Riemann surface. We prove that it is equivalent to the generalized recursion for spectral curves with arbitrary ramification. Using this global formulation, we also prove that the correlation functions constructed from the recursion for curves with arbitrary ramification can be obtained as suitable limits of correlation functions for curves with only simple ramification. It then follows that they both satisfy the properties that were originally proved only for curves with simple ramification.