Intertwining operator and integrable hierarchies from topological strings

TL;DR

This paper explores the algebraic structures underlying topological string models, revealing how quantum symmetries and automorphisms relate to integrable hierarchies, and extends these concepts to refined models using advanced algebraic frameworks.

Contribution

It provides an algebraic perspective on topological string models, connecting quantum $W_{1+ obreak extstyle+ obreak ext{infty}}$ symmetry with integrable hierarchies and extending to refined models via quantum toroidal algebra.

Findings

Identifies the role of automorphisms in the algebraic structure of topological strings.

Extends the derivation to refined models using quantum toroidal algebra.

Lays groundwork for defining deformed hierarchies in refined topological string theory.

Abstract

In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum $W_{1+\infty}$ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner - or vertex operator - of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $\mathfrak{gl}(1)$ (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.