From 6D SCFTs to Dynamic GLSMs

TL;DR

This work probes the compactification of 6D $\mathcal{N}=(1,0)$ SCFTs on four-manifolds, focusing on rank-one simple non-Higgsable clusters, and uncovers a dynamic 2d picture: a dynamic GLSM (DGLSM) on the tensor branch whose couplings are promoted to dynamical fields and which couples to extra non-Lagrangian sectors arising from the 6D anti-chiral two-form. By twisting on Kähler surfaces and reducing the 6D anomaly polynomial, the authors derive a 2d $\mathcal{N}=(0,2)$ theory whose anomaly data matches the tensor-branch reduction, suggesting that the UV SCFT and the tensor-branch description flow to the same 2d fixed point. The DGLSM framework naturally includes gauge couplings as dynamical scalars, a non-compact target space, and a rich current algebra inherited from the 6D anti-chiral two-form, together with spacetime filling strings that ensure anomaly cancellation. The analysis provides concrete checks in the simple NHC class, delineates the parameter space and geometric constraints on the internal four-manifold, and sets a stage for further exploration of 2d SCFTs arising from higher-dimensional theories and their holographic duals.

Abstract

Compactifications of 6D superconformal field theories (SCFTs) on four-manifolds generate a large class of novel 2d quantum field theories. We consider in detail the case of the rank one simple non-Higgsable cluster 6D SCFTs. On the tensor branch of these theories, the gauge group is simple and there are no matter fields. For compactifications on suitably chosen Kahler surfaces, we present evidence that this provides a method to realize 2d SCFTs with N = (0,2) supersymmetry. In particular, we find that reduction on the tensor branch of the 6D SCFT yields a description of the same 2d fixed point that is described in the UV by a gauged linear sigma model (GLSM) in which the parameters are promoted to dynamical fields, that is, a "dynamic GLSM" (DGLSM). Consistency of the model requires the DGLSM to be coupled to additional non-Lagrangian sectors obtained from reduction of the anti-chiral two-form of the 6D theory. These extra sectors include both chiral and anti-chiral currents, as well as spacetime filling non-critical strings of the 6D theory. For each candidate 2d SCFT, we also extract the left- and right-moving central charges in terms of data of the 6D SCFT and the compactification manifold.

Figures (3)

• Figure 1: Compactification of the tensor branch deformation of a 6D SCFT on a four-manifold yields a DGLSM, i.e. a gauged linear sigma model in which the couplings are dynamical. These theories are also coupled to chiral / anti-chiral currents and spacetime filling strings.
• Figure 2: Depiction of the different theories starting from compactification of a 6D theory. We can either begin with a 6D SCFT and compactify directly to two dimensions, obtaining a strongly coupled candidate 2d SCFT indicated by $\ast$. Alternatively, we can instead consider compactification of the 6D theory far out on its tensor branch, arriving at a weakly coupled dynamic GLSM (the coupling is indicated by the radius of the circle) fibered over the tensor non-linear sigma model. The match in anomalies for the two 2d theories so obtained suggests that $\ast$ is at finite distance in the appropriate sigma model metric.
• Figure 3: Depiction of the DGLSM defined by compactifying the 6D SCFT defined a $-4$ curve with $SO(8)$ gauge group on a Kähler surface. In two dimensions, we obtain a GLSM with gauge group $SO(8) \times Sp(N)$, and matter transforming in representations of the two gauge groups. Additionally, the GLSM sector is coupled to a collection of chiral and anti-chiral currents which are depicted by a thickened shell around each quiver node.