Hopf link invariants and integrable hierarchies

TL;DR

This paper demonstrates that the generating function of HOMFLY-PT invariants for the Hopf link, colored with various representations, is a tau-function of the KP hierarchy, revealing integrable structures in topological string theory.

Contribution

It establishes the integrable hierarchy properties of Hopf link invariants and connects them to the KP and UC hierarchies, including the case of composite representations.

Findings

The Hopf link invariants generate a KP tau-function.

Composite representation invariants relate to the UC hierarchy.

A simple matrix model associated with the UC hierarchy is discussed.

Abstract

The goal of this note is to study integrable properties of a generating function of the HOMFLY-PT invariants of the Hopf link colored with different representations. We demonstrate that such a generating function is a $\tau$-function of the KP hierarchy. Furthermore, this Hopf generating function in the case of composite representations, which is a generating function of the 4-point functions in topological string (corresponding to the resolved conifold with branes on the four external legs), is a $\tau$-function of the universal character(UC) hierarchy put on the topological locus. We also briefly discuss a simple matrix model associated with the UC hierarchy.