Green-Schwarz Automorphisms and 6D SCFTs

TL;DR

The paper identifies Aut$(\Lambda)$, the automorphism group of the lattice of string charges on the 6D SCFT tensor branch, as the key discrete data governing global/gauge symmetries and moduli-space structure. Using F-theory realizations, it decomposes Aut$(\Lambda)$ into endpoint/capacitance factors and analyzes how Higgs/Higgs-branch tuning and RG flows affect these symmetries. It then explores the implications for compactifications, including Seiberg-like dualities in 4D, twisted sectors in 2D, and consistent discrete quotients/orientifolds that yield new lower-dimensional SCFTs, with explicit treatments of A-, D-, and E-type class $\mathcal{S}_{\Gamma}$ theories. The work provides a framework to understand how discrete lattice automorphisms shape the landscape of lower-dimensional theories descended from 6D SCFTs and highlights rich avenues for future study of dualities and orientifold-like constructions in these systems.

Abstract

All known interacting 6D superconformal field theories (SCFTs) have a tensor branch which includes anti-chiral two-forms and a corresponding lattice of string charges. Automorphisms of this lattice preserve the Dirac pairing and specify discrete global and gauge symmetries of the 6D theory. In this paper we compute this automorphism group for 6D SCFTs. This discrete data determines the geometric structure of the moduli space of vacua. Upon compactification, these automorphisms generate Seiberg-like dualities, as well as additional theories in discrete quotients by the 6D global symmetries. When a perturbative realization is available, these discrete quotients correspond to including additional orientifold planes in the string construction.

Figures (5)

• Figure 1: The two quiver phases of A-type class $\mathcal{S}_\Gamma$ theories compactified on a $T^2$. In addition, the effect of quotienting by the global symmetries associated to the outer automorphisms is shown. On the left the reduction on the tensor branch yields $(N-1)$$SU(k+1)$ gauge nodes and two $SU(k+1)$ flavor symmetry factors. On the right the affine quiver with $(k+1)$$SU(N)$ gauge nodes. The segments represent standard $\mathcal{N} =2$ hypermultiplets in the bifundamental representation.
• Figure 2: Suspended brane configurations for discrete quotients of the A-type class $\mathcal{S}_\Gamma$ theories on the tensor branch. We depict linear symmetric suspended branes configurations of D4 (in black) filling $x^{0},\ldots, x^{3}$ and extended along $x_{6}$, NS5 branes (in blue) filling $x^{0},\ldots, x^{3}$, wrapping $x^{4},x^{5}$ (i.e. the two torus directions), and probing $x^{6}$, O6$^{-}$$+$ 2D6 (in red) filling $x^{0},\ldots, x^{3}$ and extended along $x_{7},x_{8}, x_{9}$. The other configurations are given by moving the O6$^{-}$$+$$2$D6 on top of an NS5 brane, or by moving one NS5 brane inside the O6$^{-}$$+$$2$D6 system depending on whether $k$ and $N$ are even or odd. In this figure, only the physical branes have been illustrated. However in the presence of orientifold $O6^{-}$ there is an equivalent mirror image of the brane system, which in the affine cases makes the quiver circular, and the $x_{6}$ direction compact.
• Figure 3: Resulting quiver theories for A-type class $\mathcal{S}_\Gamma$ theories associated with the brane systems in \ref{['fig:Bs1']}. On the left the reduction on the tensor branch to 4D, on the right the reduction at the fixed point. The possible cases are listed depending on $k$ and $N$ even or odd. The double box labeling some matter on the right or left gauge nodes stands for a full antisymmetric hypermultiplet.
• Figure 4: D-type quiver theories. On top the unquotiented theories in the tensor branch phase (left) and affine phase (right). On the bottom the two cases depending on $N$ being even or odd for the two corresponding phases. The segments represents standard $\mathcal{N}=2$ hypermultiplets, when the $1/2$ appears on a link it stands for half hypermultiplet. The label “ CM” means that it is not just an hypermultiplet, but generalized matter coming from compactification of the conformal matter given by the two connected gauge nodes.
• Figure 5: Quiver theories obtained from a discrete quotient of $E_6$-type class $\mathcal{S}_\Gamma$ theories reduced on a $T^2$ with a discrete quotient. On top the unquotiented theories in the tensor branch phase (left) and affine phase (right). On the bottom the two cases depending on $N$ being even or odd for the two corresponding phases. The segments represents standard $\mathcal{N}=2$ hypermultiplets, when the $1/2$ appears on a link it stands for half hypermultiplet. The label “ CM” denotes generalized matter coming from compactification of the 6D conformal matter.