From 6D superconformal field theories to dynamic gauged linear sigma models

Compactifications of six-dimensional (6D) superconformal field theories (SCFTs) on four- manifolds generate a large class of novel two-dimensional (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 Kähler surfaces, we present evidence that this provides a method to realize 2D SCFTs with $\mathcal{N}=(0,2)$ 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 antichiral two-form of the 6D theory. These extra sectors include both chiral and antichiral currents, as well as spacetime filling noncritical 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.

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/antichiral 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 *. 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 nonlinear sigma model. The match in anomalies for the two 2D theories so obtained suggests that * 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 antichiral currents which are depicted by a thickened shell around each quiver node.