Graded Unitarity in the SCFT/VOA Correspondence

Vertex algebras that arise from four-dimensional, N=2 superconformal field theories inherit a collection of novel structural properties from their four-dimensional ancestors. Crucially, when the parent SCFT is unitary, the corresponding vertex algebra is not unitary in the conventional sense. In this paper, we motivate and define a generalized notion of unitarity for vertex algebras that we call graded unitarity, and which captures the consequences of four-dimensional unitarity under this correspondence. We also take the first steps towards a classification program for graded-unitary vertex algebras whose underlying vertex algebras are Virasoro or affine Kac--Moody vertex algebras. Remarkably, under certain natural assumptions about the R-filtration for these vertex algebras, we show that only the (2, p) central charges for Virasoro VOAs and boundary admissible levels for sl₂ and sl₃ Kac--Moody vertex algebras can possibly be compatible with graded unitarity. These are precisely the cases of these vertex algebras that are known to arise from four dimensions.