Determine the vertex operator algebras produced by the topological twists

Determine whether the H- and C-topological twists of the Abelian rank-0 theories T_{\ell,n} produce, respectively, the vertex operator algebras W-min_{n-1/2}(sp_{2\ell}) and L_n(osp_{1|2\ell}), including the precise interpretation of these algebras under the proposed symplectic–Abelian duality.

Background

The family T_{\ell,n} is proposed as the Abelian dual of the symplectic Chern–Simons theory U Sp(2n) at level n+1/2+ℓ with one fundamental chiral. The theories admit two topological twists associated with the two branches of the N=4 structure expected in the infrared.

The claimed vertex-algebra assignments extend the known ℓ=1 relation to non-unitary Virasoro minimal models. They remain conjectural in the cited work and are used in the paper as supporting structure rather than as a result derived from the Lagrangian analysis.