Symplectic duality via log topological recursion

TL;DR

This paper explores symplectic duality within log topological recursion, revealing its properties and providing a unified proof of topological recursion for various weighted double Hurwitz numbers, extending previous results.

Contribution

It demonstrates that symplectic duality can be expressed through $x-y$ dualities in log topological recursion, establishing its properties and applying them to prove topological recursion uniformly.

Findings

Symplectic duality can be decomposed into $x-y$ dualities.

Properties like invertibility and group structure are established.

A new proof of topological recursion for weighted double Hurwitz numbers is provided.

Abstract

We review the notion of symplectic duality earlier introduced in the context of topological recursion. We show that the transformation of symplectic duality can be expressed as a composition of $x-y$ dualities in a broader context of log topological recursion. As a corollary, we establish nice properties of symplectic duality: various convenient explicit formulas, invertibility, group property, compatibility with topological recursion and KP integrability. As an application of these properties, we get a new and uniform proof of topological recursion for large families of weighted double Hurwitz numbers; this encompasses and significantly extends all previously known results on this matter.