Topological Vertex for Symmetric matter

We propose a novel topological vertex formalism for 5d $\mathcal{N}=1$ SU($N$) gauge theory with a hypermultiplet in the symmetric tensor representation, whose Type IIB brane construction involves an NS5-brane attached to an O7$^+$-plane. Inspired by the identification $\mathrm{O7}^+\sim \mathbb{Z}_2 + 4 \mathrm{D7}$, we introduce two new types of vertices: the $\mathbb{Z}_2$-vertex, which implements the $\mathbb{Z}_2$ orbifold action, and the FD-vertex, which encodes the monodromy cut induced by the O7$^+$-plane. This formalism generalizes the framework presented in arXiv:2412.19655 and establishes a systematic method for computing partition functions for 5-brane configurations that incorporate an O7$^+$-plane. The resulting partition functions are expressed as sums over Young diagrams, providing a powerful computational tool for studying such gauge theories.