On the $x$-$y$ Symmetry of Correlators in Topological Recursion via Loop Insertion Operator

TL;DR

This paper establishes a functional relation between correlators generated by topological recursion when interchanging the roles of initial data functions, simplifying the connection between free cumulants and moments in higher order free probability.

Contribution

It introduces a new functional relation between correlators under the exchange of initial data functions using the loop insertion operator, extending previous results to higher order free probability.

Findings

Derived a functional relation for genus 0 correlators with interchanged initial data.

Showed equivalence to recent results in the literature for genus 0 case.

Provided a simplified relation between free cumulants and moments.

Abstract

Topological Recursion generates a family of symmetric differential forms (correlators) from some initial data $(\Sigma,x,y,B)$. We give a functional relation between the correlators of genus $g=0$ generated by the initial data $(\Sigma,x,y,B)$ and by the initial data $(\Sigma,y,x,B)$, where $x$ and $y$ are interchanged. The functional relation is derived with the loop insertion operator by computing a functional relation for some intermediate correlators. Additionally, we show that our result is equivalent to the recent result of \cite{Borot:2021thu} in case of $g=0$. Consequently, we are providing a simplified functional relation between generating series of higher order free cumulants and moments in higher order free probability.