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

Abstract

Let Wg,n be the correlators computed by Topological Recursion for some given spectral curve (x,y) and Wg,n∨ for (y,x), where the role of x,y is inverted. These two sets of correlators Wg,n and Wg,n∨ 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 Wg,n and Wg,n∨ 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.