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 [Lectures on $\mathcal{W}$-algebras, Australian Representation Theory Workshop 2016, University of Melbourne], and was given the name "classical freeness" by Jethro van Ekeren and Reimundo Heluani in their work on chiral homology [Comm. Math. Phys. 386 (2021), no. 1, 495-550]. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), 1689-1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra $L_n({\mathfrak{osp}}_{m|2r})$ for all positive integers $m,n,r$ satisfying $-\frac{m}{2}+r+n+1>0$. In particular, it holds for the rational vertex superalgebras $L_n({\mathfrak{osp}}_{1|2r})$ for all positive integers $r,n$.

References and addresses updated, final version to appear in Proc. Am. Math. Soc. arXiv admin note: text overlap with arXiv:2109.09050

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