More on topological vertex formalism for 5-brane webs with O5-plane

TL;DR

This paper introduces an O-vertex function for topological vertex formalism involving O5-planes, enabling computation of Nekrasov partition functions for 5d theories on 5-brane webs with O5-planes, and verifies its effectiveness through known gauge theories.

Contribution

It proposes a new O-vertex formalism extension for 5-brane webs with O5-planes, allowing explicit calculation of Nekrasov partition functions for various 5d gauge theories.

Findings

Successfully computed partition functions for SO(N) and G_2 gauge theories.

Validated results against known literature.

Extended the formalism to include Chern-Simons level in SU(3) theories.

Abstract

We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of arXiv:1709.01928. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO($N$) gauge theories and the pure $G_2$ gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level $9$. At the end we rewrite the O-vertex in a form of a vertex operator.