# Three-partition Hodge integrals and the topological vertex

## TL;DR

This paper proves a conjecture linking cubic Hodge integrals and the topological vertex in string theory by using algebraic symmetries, revealing their generating functions as tau functions of integrable hierarchies.

## Contribution

It introduces generalized shift symmetries in fermionic realizations to connect Hodge integrals with the topological vertex, resolving a key conjecture.

## Key findings

- Generated functions are tau functions of Gelfand-Dickey hierarchies.
- Fermionic operator relations convert between Hodge integrals and topological vertex.
- The conjecture relating cubic Hodge integrals and the topological vertex is proven.

## Abstract

A conjecture on the relation between the cubic Hodge integrals and the topological vertex in topological string theory is resolved. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters therein is shown to be a tau function of the generalized KdV (aka Gelfand-Dickey) hierarchies.