A generalized topological recursion for arbitrary ramification

TL;DR

This paper extends the Eynard-Orantin topological recursion to handle cases where the function x has arbitrary ramification points, broadening its applicability in geometric and mathematical physics contexts.

Contribution

The authors propose a generalized topological recursion applicable to arbitrary ramification, supported by degeneration analysis and compatibility checks with key invariance properties.

Findings

Generalized recursion works for arbitrary ramification points.

Compatibility with (x,y) -> (y,x) invariance is demonstrated in examples.

Counterexamples show subtlety in invariance properties.

Abstract

The Eynard-Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple ramification points. In this paper we propose a generalized topological recursion that is valid for x with arbitrary ramification. We justify our proposal by studying degenerations of Riemann surfaces. We check in various examples that our generalized recursion is compatible with invariance of the free energies under the transformation (x,y) -> (y,x), where either x or y (or both) have higher order ramification, and that it satisfies some of the most important properties of the original recursion. Along the way, we show that invariance under (x,y) -> (y,x) is in fact more subtle than expected; we show that there exists a number of counter examples, already in the case of the original Eynard-Orantin recursion, that deserve further study.