The Refined Topological Vertex

TL;DR

This paper introduces a refined topological vertex with an additional parameter, enabling advanced computations in gauge theories and potential links to knot invariants, extending the original topological vertex framework.

Contribution

It defines a new refined topological vertex that generalizes previous models and connects to instanton expansions and knot invariants.

Findings

Computed a two-parameter instanton expansion of gauge theories.

Reproduced Nekrasov's results using the refined vertex.

Suggested a relation to Khovanov knot invariants.

Abstract

We define a refined topological vertex which depends in addition on a parameter, which physically corresponds to extending the self-dual graviphoton field strength to a more general configuration. Using this refined topological vertex we compute, using geometric engineering, a two-parameter (equivariant) instanton expansion of gauge theories which reproduce the results of Nekrasov. The refined vertex is also expected to be related to Khovanov knot invariants.