Classical freeness of orthosymplectic affine vertex superalgebras

TL;DR

This paper proves the classical freeness property for a broad class of orthosymplectic affine vertex superalgebras, extending previous results to all positive integer parameters satisfying a specific inequality.

Contribution

It establishes the classical freeness of simple affine vertex superalgebras $L_n(\mathfrak{osp}_{m|2r})$ for all positive integers meeting a certain inequality, generalizing prior work.

Findings

Proves classical freeness for $L_n(\mathfrak{osp}_{m|2r})$ with specified parameters.

Includes all rational vertex superalgebras $L_n(\mathfrak{osp}_{1|2r})$.

Extends previous results to a wider class of 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 $r,n$.