5-brane webs for 5d $\mathcal{N}=1$ $G_2$ gauge theories

TL;DR

The paper constructs and analyzes 5d $ abla 1$ supersymmetric gauge theories with gauge group $G_2$ via 5-brane webs in type IIB string theory, using both $ ilde{O5}$- and $O5$-plane realizations. By Higgsing from $SO(7)$ with a spinor and from $SO(8)$ with spinor data, it produces two distinct webs realizing the pure $G_2$ theory, and shows how flavors can be incorporated through further Higgsing; the webs reproduce expected prepotential relations and monopole tensions. The Nekrasov partition functions are computed using a generalized topological vertex for webs with O5/$ ilde{O5}$ planes, with perturbative and instanton parts for the pure $G_2$ theory and for $G_2$ with one flavor, and results agree with known one- and two-instanton data and flavor decoupling limits. The work extends the topological vertex framework to nontrivial orientifold backgrounds and provides robust cross-checks, offering new tools to study 5d UV fixed points and Seiberg-Witten-type structures for exceptional gauge groups. Overall, it broadens the landscape of 5d SCFTs realized by brane webs and demonstrates precise quantitative agreement with field-theoretic expectations.

Abstract

We propose 5-brane webs for 5d $\mathcal{N}=1$ $G_2$ gauge theories. From a Higgsing of the $SO(7)$ gauge theory with a hypermultiplet in the spinor representation, we construct two types of 5-brane web configurations for the pure $G_2$ gauge theory using an O5-plane or an $\widetilde{\text{O5}}$-plane. Adding flavors to the 5-brane web for the pure $G_2$ gauge theory is also discussed. Based on the obtained 5-brane webs, we compute the partition functions for the 5d $G_2$ gauge theories using the recently suggested topological vertex formulation with an O5-plane, and we find agreement with known results.

Figures (32)

• Figure 1: A Higgsing procedure of the 5-brane web of the $SO(8)$ gauge theory with one flavor to the 5-brane web of the pure $SO(7)$ gauge theory. (a): Moving the flavor D7-brane to the middle of the diagram. The branch cut is denoted by the dashed line. (b): Lowering the flavor D7-brane as well as the bottom color D5-brane to the O5$^-$-plane. (c): Splitting the D7-branes into two half D7-branes. We have an $\widetilde{\text{O5}}^-$-plane between the half D7-branes and there are effectively three fractional D5-branes between the half D7-branes. We also have monodromy branch cuts for the half D7-branes represented by the red and orange dashed lines. (d): Removing the two fractional D5-branes. We have a half D5-brane denoted by the blue line stretched between the $(2, 1)$ 5-brane and the $(1, -1)$ 5-brane. The diagram gives the pure $SO(7)$ gauge theory.
• Figure 2: Another 5-brane web diagram for the 5d pure $SO(7)$ gauge theory by removing the half D7-branes in the opposite directions. The diagram is constructed with an $\widetilde{\text{O5}}$-plane. The $\widetilde{\text{O5}}$-plane is realized by the O5-plane with the half monodromy cut which is denoted by the orange dashed line.
• Figure 3: A 5-brane web diagram for the $[1] - SO(9) - USp(2) - \left[\frac{3}{2}\right]$ quiver theory.
• Figure 4: A 5-brane web diagram for a $SO(7)$ gauge theory with spinor matter obtained by removing the D5-brane from the diagram for the $[1] - SO(9) - USp(2) - \left[\frac{3}{2}\right]$ quiver theory.
• Figure 5: 5-brane web diagrams for the $SO(7)$ gauge theory with one spinor. (a): The 5-brane web diagram which is obtained by Higgsing the diagram of the $SO(8)$ gauge theory with one vector and one spinor. Compared to the diagram in Figure \ref{['fig:SO8w1flvrHiggsd']}, the $(2, 1)$ 5-brane is attached on the right, yielding the spinor matter. Removing the half D7-branes in the opposite direction gives the diagram in Figure \ref{['fig:SO7wspinor']}. (b): An equivalent diagram to the one in Figure \ref{['fig:SO7wspinor2a']} (and also to the one in Figure \ref{['fig:SO7wspinor']}). We move the half D7-branes in the left direction compared to the diagram in Figure \ref{['fig:SO7wspinor2a']}. Then the monodromy cut disappears and the diagram is constructed with an O5-plane.