Exactness and Drinfeld-Sokolov Realization of the McRae-Yang Tensor Functors

Abstract

Let $p,q\geq2$ be coprime, set $k=-2+p/q$, and let $c_{p,q}=1-6(p-q)^2/(pq)$. McRae and Yang constructed right exact braided tensor functors from the non-semisimple Kazhdan-Lusztig categories of $V^{-2+p/q}(\mathfrak{sl}_2)$ and $V^{-2+q/p}(\mathfrak{sl}_2)$ to the Virasoro category $\mathcal{O}_{c_{p,q}}$, and conjectured that these functors are exact and agree with the corresponding quantum Drinfeld-Sokolov reductions. We prove this conjecture. For the $(p,q)$ branch, we determine the images of all indecomposable projective objects without assuming exactness, using projective tensor-product recursions and exact generalized conformal-residue projections. Projective faithfulness is then combined with a one-row Virasoro extension analysis to identify the non-wall images and to prove full faithfulness on projectives. The Virasoro input is matched objectwise with Nakano's logarithmic extension theorem away from the vacuum edge; the exceptional vacuum edge is handled directly by the staggered-module theory of Kyt\"ol\"a-Ridout, including the higher prime singular-vector branch. Independently, principal Drinfeld-Sokolov reduction is shown to be exact and faithful on the whole finite-length affine category and to have the same Weyl, simple, and projective images. Compatibility with the non-standard affine twist identifies the canonical nilpotent endomorphisms on projectives and removes the remaining scalar ambiguity in the comparison of adjacent projective morphisms. Projective density for right exact functors then yields a natural isomorphism \[ F_{p,q}\cong H^0_{DS,+}\big|_{{KL}^k(\mathfrak{sl}_2)}. \] The same argument after interchanging $p$ and $q$ identifies the second McRae-Yang functor with the transposed Drinfeld-Sokolov reduction.