A simple formula for the $x$-$y$ symplectic transformation in topological recursion

TL;DR

This paper presents a straightforward formula linking correlators in topological recursion under $x$-$y$ symplectic transformation, simplifying the understanding of their relationship and connecting to free probability concepts.

Contribution

It provides a simple explicit formula for the $x$-$y$ symplectic transformation between correlators in topological recursion, extending previous functional relations.

Findings

Derived a simple formula for $W_{g,n}$ and $W^\/vee_{g,n}$ relation

Confirmed the formula applies to meromorphic $x,y$ with simple ramification points

Connected the results to moment-cumulant relations in free probability.

Abstract

Let $W_{g,n}$ be the correlators computed by Topological Recursion for some given spectral curve $(x,y)$ and $W^\vee_{g,n}$ for $(y,x)$, where the role of $x,y$ is inverted. These two sets of correlators $W_{g,n}$ and $W^\vee_{g,n}$ are related by the $x$-$y$ symplectic transformation. Bychkov, Dunin-Barkowski, Kazarian and Shadrin computed a functional relation between two slightly different sets of correlators. Together with Alexandrov, they proved that their functional relation is indeed the $x$-$y$ symplectic transformation in Topological Recursion. This article provides a fairly simple formula directly between $W_{g,n}$ and $W^\vee_{g,n}$ which holds by their theorem for meromorphic $x$ and $y$ with simple and distinct ramification points. Due to the recent connection between free probability and fully simple vs ordinary maps, we conclude a simplified moment-cumulant relation for moments and higher order free cumulants.