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

Topological Recursion generates a family of symmetric differential forms (correlators) from some initial data \((\Sigma ,x,y,B)\) $(\mathrm{\Sigma },x,y,B)$ . We give a functional relation between the correlators of genus \(g=0\) $g=0$ generated by the initial data \((\Sigma ,x,y,B)\) $(\mathrm{\Sigma },x,y,B)$ and by the initial data \((\Sigma ,y,x,B)\) $(\mathrm{\Sigma },y,x,B)$ , 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 Borot et al. (2021) in case of \(g=0\) $g=0$ . Consequently, we are providing a simplified functional relation between generating series of higher order free cumulants and moments in higher order free probability.