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 $$C_2$$ C 2 -cofinite minimal W -algebras of the Deligne–Cvitanovi? (DC) series as boundary algebras. The construction is based on the minimal three-dimensional $$\mathcal {N}=4$$ 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 $$W_{-h^\vee /6}(\mathfrak {g},f_{\text {min}})$$ W - h ∨ / 6 ( g , f min ) . While this boundary condition is not deformable to the B -twist, we argue that a holomorphic-topological ( $$HT^B$$ H T 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.