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arXiv · 2201.05357

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

Alexander Hock
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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.

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Alexander Hock. 2022-01-14. On the $x$-$y$ Symmetry of Correlators in Topological Recursion via Loop Insertion Operator. https://doi.org/10.1007/s00220-024-05043-1

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