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.

arXiv comment26 pages, 11 figures, an assumptions is added to the second version

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• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• Symplectic duality for topological recursion
• Symplectic (Non-)Invariance of the Free Energy in Topological Recursion
• Blobbed topological recursion from extended loop equations
• GW/DT invariants and 5D BPS indices for strips from topological recursion
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants
• Resonance transformations for the $(2,2p+1)$ minimal string via $x-y$ swap: a proof of Artemev's conjecture