Vertex Operators, $\mathbb{C}^3$ Curve and Topological Vertex

TL;DR

This paper proves that Kodaira-Spencer theory for the topological vertex is a free fermion theory, using vertex operator methods and correlation functions to connect to known identities in knot theory.

Contribution

It introduces a vertex operator approach to the topological vertex and formulates a new correlation function identity conjecture related to Hopf links.

Findings

Proves Kodaira-Spencer theory for the topological vertex is a free fermion theory.

Constructs a generic three-leg correlation function using Boson-Fermion correspondence.

Proposes a conjecture linking the correlation function to Zhou's identity for Hopf links.

Abstract

In this article, we prove the conjecture that Kodaira-Spencer theory for the topological vertex is a free fermion theory. By dividing the $\mathbb{C}^3$ curve into core and asymptotic regions and using Boson-Fermion correspondence, we construct a generic three-leg correlation function which reformulates the topological vertex in a vertex operator approach. We propose a conjecture of the correlation function identity which in a degenerate case becomes Zhou\rq{}s identity for a Hopf link.