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

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.

26 pages, 11 figures, an assumptions is added to the second version

References in corpus (2)

• Generalised ordinary vs fully simple duality for $n$-point functions and a proof of the Borot--Garcia-Failde conjecture
• Complete solution of the LSZ Model via Topological Recursion

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