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

TL;DR

This paper constructs boundary algebras in 3D ${ m N}=4$ superconformal theories that realize vertex algebras associated with Deligne-Cvitanović series, linking 3D field theories with complex algebraic structures.

Contribution

It introduces a 3D ${ m N}=4$ field theory framework that produces vertex algebras related to the Deligne-Cvitanović series as boundary algebras, connecting physical theories with algebraic structures.

Findings

Boundary local operators realize minimal W-algebras $W_{-h^2/6}(\u2206, ext{min})$

Holomorphic-topological twist yields affine algebras of intermediate Lie algebras

Provides a 3D origin for complex vertex algebra structures

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-Cvitanovi\'c (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.