Graded Unitarity in the SCFT/VOA Correspondence

TL;DR

This paper introduces graded unitarity, a new concept capturing four-dimensional unitarity properties in vertex algebras, and explores classification constraints for certain classes of these algebras.

Contribution

It defines graded unitarity for vertex algebras and initiates a classification program, linking 4D unitarity to specific algebraic structures and central charges.

Findings

Only (2,p) Virasoro central charges are compatible with graded unitarity.

Boundary admissible levels for sl2 and sl3 Kac--Moody algebras are compatible with graded unitarity.

Graded unitarity captures 4D unitarity properties in vertex algebra structures.

Abstract

Vertex algebras that arise from four-dimensional, $\mathcal{N}=2$ 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 \emph{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 $\mathfrak{R}$-filtration for these vertex algebras, we show that only the $(2,p)$ central charges for Virasoro VOAs and boundary admissible levels for $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ 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.