Topological Vertex for Symmetric matter

A bstract We propose a novel topological vertex formalism for 5d $$ \mathcal{N}=1 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> 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 O7 + ∼ ℤ 2 + 4D7, we introduce two new types of vertices: the ℤ 2 -vertex, which implements the 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 JHEP 04 (2025) 182 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.