Brane webs, $5d$ gauge theories and $6d$ $\mathcal{N}$$=(1,0)$ SCFT's

TL;DR

This work develops a systematic framework to identify 6d N=(1,0) SCFT UV completions of 5d gauge theories realized by brane webs, using Higgs-branch flows and brane manipulations to map 5d theories to 6d lifts. It then analyzes torus compactifications of these 6d theories, matching resulting 4d class S theories and verifying central charges via anomaly polynomials and puncture data. The paper provides numerous concrete examples, including (N+2)F+SU_0(N)^k+(N+2)F and its generalizations, T_N with extra flavors, and antisymmetric-hypertended SU quivers, demonstrating consistent 5d–6d–4d connections and illustrating symmetry enhancements from instantons and higher fluxes. The findings support a broad program of deriving 4d isolated SCFTs from higher-dimensional theories and offer a practical method to classify 6d lifts of a wide class of 5d fixed points. The approach also highlights the interplay between brane constructions, anomaly polynomials, and class S technology as a unifying tool across dimensions.

Abstract

We study $5d$ gauge theories that go in the UV to $6d$ $\mathcal{N}$$=(1,0)$ SCFT. We focus on these theories that can be engineered in string theory by brane webs. Given a theory in this class, we propose a method to determine the $6d$ SCFT it goes to. We also discuss the implication of this to the compactification of the resulting $6d$ SCFT on a torus to $4d$. We test and demonstrate this method with a variety of examples.

Figures (51)

• Figure 1: A graphical summary of the main idea of this paper. The major relation we explore is between a $6d$$(1,0)$ SCFT and a $5d$ gauge theory generated by compactifying the former on a circle of radius $R_6$. This is represented in the figure by the wide blue arrow. We can employ this relationship to study the compactification of the $6d$$(1,0)$ SCFT to $4d$ on a torus. We first mass deform the $5d$ gauge theory, corresponding to taking the $R_6\rightarrow 0$ limit while keeping the $6d$ global symmetry intact. This leads to a $5d$ SCFT. We then compactify this SCFT on a circle of radius $R_5$, and take $R_5\rightarrow 0$. This leads to a $4d$ class S SCFT, which can in turn be thought of as a result of compactifying a $6d$$(2,0)$ SCFT on a Riemann sphere with three punctures. We can use this description as a consistency check by calculating the properties of this $4d$ SCFT when thought of as a compactification of a $6d$$(2,0)$ SCFT, known as class S technology, and comparing against what is expected from the compactification of the $6d$$(1,0)$ SCFT.
• Figure 2: The $6d$ quiver theories we consider.
• Figure 3: The $6d$ quiver theory we consider. The arrow in the second quiver stands for gauging a part of the global symmetry of the shown $6d$ SCFT, in this case an $SU(8)$ subgroup of $E_8$.
• Figure 4: The brane description of the $6d$ theory in figure \ref{['Img1']}. The horizontal lines represent D$6$-branes, and the number above the lines stand for the number of $6$-branes. The black circles represent NS$5$-branes, and their number is given below. Finally, the vertical line stands for the $O8^-$ plane. The configuration also include $2N+4l$ D$8$-branes, parallel to the $O8^-$ plane, on which the asymptotic D$6$-branes end. For clarity we have suppressed them in the figure.
• Figure 5: The web we end up with after performing T-duality on the brane configuration of figure \ref{['Img2']} and resolving the $O7^-$ planes.