KP integrability through the $x-y$ swap relation

Abstract

We discuss a universal relation that we call the $x-y$ swap relation, which plays a prominent role in the theory of topological recursion, Hurwitz theory, and free probability theory. We describe in a very precise and detailed way the interaction of the $x-y$ swap relation and KP integrability. As an application, we prove a recent conjecture that relates some particular instances of topological recursion to the Mironov-Morozov-Semenoff matrix integrals.

28 pages; minor corrections

References in corpus (4)

• Think globally, compute locally
• A simple formula for the $x$-$y$ symplectic transformation in topological recursion
• Laplace transform of the $x-y$ symplectic transformation formula in Topological Recursion
• A universal formula for the $x-y$ swap in topological recursion

Cited by in corpus (7)

• $x-y$ swap for $(2,2p+1)$ minimal string
• GW/DT invariants and 5D BPS indices for strips from topological recursion
• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• 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
• Blobbed topological recursion and KP integrability
• Any topological recursion on a rational spectral curve is KP integrable