Topological Vertex for Symmetric matter

TL;DR

The paper develops an unrefined topological vertex formalism for 5d $\mathcal{N}=1$ $SU(N)$ gauge theories with a hypermultiplet in the symmetric representation, realized by an NS5-brane attached to an $O7^+$-plane. It introduces two new vertices, the $\mathbb{Z}_2$-vertex and the FD-vertex, to encode the $\mathbb{Z}_2$ orbifold action and the O7$^+$ monodromy cut, respectively, enabling partition functions to be computed as sums over Young diagrams. A reformulation of Nekrasov partition functions using a new Nekrasov factor $\tilde{n}_{\lambda\mu}(a;\hbar)$ and four frozen masses under the freezing prescription leads to a consistent framework that matches known 4d limits and Higgsing to $SO(2N)$. Consistency checks, including Higgsing to $SO(2N)$ and cross-comparisons with prior formalisms, validate the approach and demonstrate its capacity to compute $Z_{top}$ for $SU(N)_k$ with a symmetric hypermultiplet via a partition-function expression involving Schur functions and $R$-factors. The work generalizes earlier O7$^+$ constructions and provides a systematic tool for exploring 5d gauge theories with orientifolds and symmetric matter, with potential extensions to other orientifolds and representations.

Abstract

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.

Figures (9)

• Figure 1: The change of the 5-brane charge from $(p,1)$ to $(p+4,1)$ after going across the monodromy cut created by an O7$^+$-plane, drawn in the fundamental region taken on the left side of the O7$^+$-plane.
• Figure 2: The powers $n$ for the framing factor assigned to the edges connected to the $\mathbb{Z}_2$ orbifold fixed point (O7$^+$-plane): Left: $n=0$. Right: $n=1$.
• Figure 3: The FD-vertex and Young diagram assignment.
• Figure 4: The 5-brane web with O7$^+$-plane on which 5d SU($N$) gauge theory with a hypermultiplet in symmetric tensor representation is realized.
• Figure 5: Assignment of Young diagrams and decomposition of the diagrams into strips for the 5d SU($N$)$_k$ + 1 Sym.