# On the $x$-$y$ symmetry in Topological Recursion via loop insertion operator

Roughly speaking, Topological Recursion is a universal procedure which generates recursively from some initial data $(x,y)$ an infinite family of differential forms $\omega_{g,n}$. Depending on the initial data $(x,y)$, the generated differential forms are related to different areas in mathematics, e.g. counting surfaces, intersection numbers of the moduli space of complex curves, volume of the moduli space of hyperbolic Riemann surfaces, Hurwitz theory, Gromov-Witten theory etc.

I will focus in my talk on the question: what happens if we interchange the initial data $(x,y)\to (y,x)$ and how are the differential forms generated by $(x,y)$ related to the differential forms generated by $(y,x)$? We will derive via loop insertion operator (under some assumptions) for $g=0$ a functional relation between these two sets of differential forms. The functional relation has a combinatorial description in terms of decorated trees, which will be introduced. Interestingly, this result gives in free probability theory the functional relation between higher order moments and higher order free cumulants.