Quantum Drinfeld--Sokolov Reduction and the McRae--Yang Tensor Functors

Let $p,q\ge 2$ be relatively prime and set $k=-2+p\mathrm{/}q$ and $c_{p,q}=1-6(p-q{)}^2\mathrm{/}(pq)$. McRae and Yang constructed a right exact braided tensor functor $$F_{p,q}:K{L}^k({\mathfrak{s}\mathfrak{l}}_2)⟶{\mathcal{O}}_{c_{p,q}}$$ sending the standard Weyl module $V_2$ to the Virasoro Kac module ${\mathcal{K}}_{2,1}$, and conjectured that it is exact and naturally isomorphic to quantum Drinfeld–Sokolov reduction. We prove this conjecture. We first determine the images of every indecomposable projective under $F_{p,q}$ without assuming exactness of $F_{p,q}$. The proof combines the projective tensor recursions in $K{L}^k({\mathfrak{s}\mathfrak{l}}_2)$, exact fusion by ${\mathcal{K}}_{2,1}$, and generalized conformal-weight blocks. Projective faithfulness and the non-semisimple affine twists then imply that these images are logarithmic. Using Nakano’s logarithmic Virasoro extension theorem and his computation of the relevant ${\mathrm{Ext}}^1$-groups, we identify the projective images and compute their Hom spaces, proving full faithfulness on projectives. Quantum Drinfeld–Sokolov reduction has the same projective images. Compatibility with the non-standard affine twist fixes the scalars needed to construct a natural isomorphism on projectives, and uniqueness of right exact extension then gives $$F_{p,q}\cong H_{DS,+}^0{\mid }_{K{L}^k({\mathfrak{s}\mathfrak{l}}_2)}\mathrm{.}$$ In particular, $F_{p,q}$ is exact. The analogous assertion holds after interchanging $p$ and $q$.