A universal formula for the $x-y$ swap in topological recursion

Abstract

We prove a recent conjecture of Borot et al. that a particular universal closed algebraic formula recovers the correlation differentials of topological recursion after the swap of $x$ and $y$ in the input data. We also show that this universal formula can be drastically simplified (as it was already done by Hock). As an application of this general $x-y$ swap result, we prove an explicit closed formula for the topological recursion differentials for the case of any spectral curve with unramified $y$ and arbitrary rational $x$.

51 pages; various corrections and clarifications

Cited by in corpus (6)

• KP integrability through the $x-y$ swap relation
• Symplectic (Non-)Invariance of the Free Energy in Topological Recursion
• $x-y$ swap for $(2,2p+1)$ minimal string
• GW/DT invariants and 5D BPS indices for strips from topological recursion
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants
• Any topological recursion on a rational spectral curve is KP integrable