O-vertex, O7$^+$-plane, and Topological Vertex

TL;DR

This paper develops a topological vertex formalism incorporating O7$^+$-planes to compute instanton partition functions for 5d SO(N) gauge theories, offering a new perspective on brane web diagrams with orientifold planes.

Contribution

It introduces an identity that rewrites the unrefined partition function in terms of Nekrasov factors, enabling the incorporation of O7$^+$-planes into the topological vertex formalism.

Findings

Derived a new expression for the partition function using Nekrasov factors.

Proposed a topological vertex formalism with O7$^+$-planes.

Interpreted the O7-plane as a frozen O7-plane in the web diagram.

Abstract

We revisit the instanton partition function for 5d $\mathcal{N}=1$ SO($N$) gauge theories compactified on S$^1$, computed from the topological vertex formalism with the O-vertex based on a 5-brane web diagram with an O5-plane. We introduce an identity that enables us to rewrite the unrefined partition function into a new expression in terms of the Nekrasov factors summed over Young diagrams, which can be interpreted as the freezing of an O7-plane. Based on this, we propose topological vertex formalism with an O7$^+$-plane.