Log topological recursion through the prism of $x-y$ swap

TL;DR

This paper introduces a logarithmic extension of topological recursion that handles logarithmic singularities, proves the universal $x-y$ swap relation, and generalizes recent conjectures, offering a unified conceptual framework.

Contribution

It develops a new logarithmic topological recursion framework and proves the universal $x-y$ swap relation, generalizing and rectifying previous approaches.

Findings

Proves the universal $x-y$ swap relation for the new recursion.

Provides a generalization of Hock's conjecture.

Rectifies previous formulas for $n$-point functions.

Abstract

We introduce a new concept of logarithmic topological recursion that provides a patch to topological recursion in the presence of logarithmic singularities and prove that this new definition satisfies the universal $x-y$ swap relation. This result provides a vast generalization and a proof of a very recent conjecture of Hock. It also uniformly explains (and conceptually rectifies) an approach to the formulas for the $n$-point functions proposed by Hock.