Classical freeness of orthosymplectic affine vertex superalgebras

Abstract:

The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness" by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra $L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{m|2r})$ for all positive integers $m,n,r$ satisfying $-\frac {m}{2} + r +n+1 > 0$. In particular, it holds for the rational vertex superalgebras $L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{1|2r})$ for all positive integers $r,n$.