A universal formula for the $x-y$ swap in topological recursion

TL;DR

This paper proves a universal algebraic formula that describes how topological recursion correlation differentials transform under swapping the input functions x and y, simplifying the process and providing explicit formulas for certain spectral curves.

Contribution

It establishes a universal formula for the x-y swap in topological recursion and simplifies its application, including explicit formulas for specific spectral curves.

Findings

Universal formula recovers correlation differentials after x-y swap

Simplification of the universal formula as done by Hock

Explicit closed formula for spectral curves with unramified y and rational x

Abstract

We prove a recent conjecture of Borot et al. that a particular universal closed algebraic formula recovers the correlation differentials of topological recursion after the swap of $x$ and $y$ in the input data. We also show that this universal formula can be drastically simplified (as it was already done by Hock). As an application of this general $x-y$ swap result, we prove an explicit closed formula for the topological recursion differentials for the case of any spectral curve with unramified $y$ and arbitrary rational $x$.