Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion

Abstract

The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications, for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal power series in $\hslash$. We apply the Laplace transformed formula to the Airy curve and the Lambert curve.

20 pages

References in corpus (4)

• Remodeling the B-model
• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• On the $x$-$y$ Symmetry of Correlators in Topological Recursion via Loop Insertion Operator
• A simple formula for the $x$-$y$ symplectic transformation in topological recursion

Cited by in corpus (7)

• KP integrability through the $x-y$ swap relation
• Degenerate and irregular topological recursion
• Log topological recursion through the prism of $x-y$ swap
• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• Symplectic (Non-)Invariance of the Free Energy in Topological Recursion
• GW/DT invariants and 5D BPS indices for strips from topological recursion
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants