A simple formula for the $x$-$y$ symplectic transformation in topological recursion

Abstract

Let $W_{g,n}$ be the correlators computed by Topological Recursion for some given spectral curve $(x,y)$ and $W_{g,n}^{\vee }$ for $(y,x)$, where the role of $x,y$ is inverted. These two sets of correlators $W_{g,n}$ and $W_{g,n}^{\vee }$ are related by the $x$-$y$ symplectic transformation. Bychkov, Dunin-Barkowski, Kazarian and Shadrin computed a functional relation between two slightly different sets of correlators. Together with Alexandrov, they proved that their functional relation is indeed the $x$-$y$ symplectic transformation in Topological Recursion. This article provides a fairly simple formula directly between $W_{g,n}$ and $W_{g,n}^{\vee }$ which holds by their theorem for meromorphic $x$ and $y$ with simple and distinct ramification points. Due to the recent connection between free probability and fully simple vs ordinary maps, we conclude a simplified moment-cumulant relation for moments and higher order free cumulants.

37 pages, 6 figures; difference in version 3: coincides with the published version, corrected typos, added explanations

References in corpus (2)

• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• On discrete surfaces: Enumerative geometry, matrix models and universality classes via topological recursion

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• Resonance transformations for the $(2,2p+1)$ minimal string via $x-y$ swap: a proof of Artemev's conjecture