W-algebras of the Deligne-Cvitanović Exceptional series and the minimal 3d ${\mathcal N}=4$ SCFT

Abstract

We propose a three-dimensional field theory construction that realizes the vertex algebras associated with the intermediate Lie algebras and the related $C_2$-cofinite minimal $W$-algebras of the Deligne-Cvitanović (DC) series as boundary algebras. The construction is based on the minimal three-dimensional ${\mathcal N}=4$ superconformal field theory coupled to a topological field theory. For a Neumann-type boundary condition compatible with the topological $A$-twist, the algebra of boundary local operators realizes the minimal $W$-algebra $W_{-h^\vee/6}(\mathfrak{g},f_{\text{min}})$. While this boundary condition is not deformable to the $B$-twist, we argue that a holomorphic-topological ($HT^B$) twist instead realizes the level-one affine algebras of the intermediate Lie algebras, providing a uniform three-dimensional origin for these vertex algebra structures.

Figures (2)

• Figure 5.1: A decomposition of $\mathfrak{e}_8$. The lattice points are labeled by their $\mathfrak{so}_{12}$ representations. We have ${\bf U} = \mathfrak{g}_{0,0} = {\bf 66} + {\bf 1} + {\bf 1}$.
• Figure 7.1: Quiver diagrams for the ${\mathcal{N}}=2$ gauge theory description for $\mathbb{T}_\mathfrak{g}$. Circular nodes denote $U(1)$ gauge groups, interacting through the CS couplings with level matrix given by $K=C(\mathfrak{g})$, represented by dotted lines. Solid lines together with square nodes denote chiral multiplets of unit charge. In case of $\mathbb{T}_{\mathfrak{a}_1}$, a double solid line indicates a chiral multiplet of charge 2. These complete the quivers to the $\mathfrak{g}$-affine Dynkin diagrams.