General Prescription for Global U(1)'s in 6D SCFTs

TL;DR

The paper develops a general, bottom-up prescription to identify global U(1) flavor symmetries in 6D SCFTs by analyzing the tensor-branch quiver and enforcing Adler–Bell–Jackiw anomaly cancellation. It complements this with a top-down perspective rooted in fission/fusion progenitor theories and their F-theory realizations, including Mordell–Weil and Higgs/tensor branch deformations. The main results show how anomaly constraints select anomaly-free linear combinations of candidate U(1) symmetries, with many U(1)’s arising from non-abelian flavor symmetry deformations or extra Mordell–Weil sections, while some do not descend from gravity-decoupled gauge sectors. The work provides concrete examples across A-type, D/E-type, and frozen SCFTs, demonstrating agreement between field-theoretic ABJ analyses and geometric/morphological interpretations, and clarifying how U(1) symmetries evolve under RG flows. Overall, the authors offer a systematic framework to read off and interpret global U(1) symmetries in 6D SCFTs and connect them to F-theory geometry and progenitor dynamics.

Abstract

We present a general prescription for determining the global U(1) symmetries of six-dimensional superconformal field theories (6D SCFTs). We use the quiver-like gauge theory description of the tensor branch to identify candidate U(1) symmetries which can act on generalized matter. The condition that these candidate U(1)'s are free of Adler-Bell-Jackiw (ABJ) anomalies provides bottom-up constraints for U(1)'s. This agrees with the answer obtained from symmetry breaking patterns induced by Higgs branch flows. We provide numerous examples illustrating the details of this proposal. In the F-theory realization of these theories, some of these symmetries originate from deformations of non-abelian flavor symmetries localized on a component of the discriminant, while others come from an additional generator of the Mordell-Weil group. We also provide evidence that some of these global U(1)'s do not arise from gauge symmetries, as would happen in taking a decoupling limit of a model coupled to six-dimensional supergravity.

Figures (5)

• Figure 1: Depiction of fission and fusion for 6D SCFTs. Progenitor theories arise from M5-brane probes of an ADE singularity $\mathbb{C}^2 / \Gamma_{ADE}$, and can correspond to cases with an $E_8$ nine-brane (heterotic $E_8$ small instantons) as well as cases without such a nine-brane. In both sets of progenitor theories, there is a $U(1)$ global symmetry factor for $\Gamma = \mathbb{Z}_N, N \geq 3$, whereas there is no abelian symmetry factor for D- and E-type singularities. For $\Gamma = \mathbb{Z}_2$, the $U(1)$ symmetry enhances to $SU(2)$. Deformations of these progenitor theories lead to "fission" products. Fission products can also be "fused" by gauging a common non-abelian global symmetry factors and adding an additional tensor multiplet.
• Figure 2: Depiction of a delocalized $U(1)$ symmetry in the F-theory model of line (\ref{['INITOUTIT']}).
• Figure 3: Depiction of the local quiver gauge theory associated with the tensor branch of the 6D SCFT described by line (\ref{['quiverbeforemepitifulmortals']}). We have also indicated the appearance of the candidate $U(1)$ global symmetries which act on bifundamental hypermultiplets.
• Figure 4: The $k=1$ case of the $(E_8,SU(N))$ collision.
• Figure 5: The $k=0$ case of the $(E_8,SU(N))$ collision, also known as $(E_8,SU(N))$ conformal matter.