Deformation rigidity of some simple affine VOAs

TL;DR

This paper proves that simple affine vertex operator algebras with positive integral levels are deformation rigid, and extends this to certain non-integral levels, showing rigidity is not dependent on $C_2$-cofiniteness or rationality.

Contribution

It establishes deformation rigidity for simple affine VOAs at positive integral levels and certain non-integral levels, confirming the conjecture in these cases.

Findings

Simple affine VOAs at positive integral levels are deformation rigid.

Deformation rigidity also holds for $rak{sl}_2$ at level $-4/3$.

Rigidity does not require $C_2$-cofiniteness or rationality.

Abstract

In this paper, we prove that simple affine vertex operator algebras with positive integral levels admit only trivial first-order deformations. Therefore, the deformation rigidity conjecture of strongly rational vertex operator algebras holds for these cases. We also show that the same holds simple affine vertex operator algebra of $\mathfrak{sl}_2$ at the non-integral admissible level $-4/3$. Therefore, neither $C_2$-cofiniteness nor rationality is a necessary condition for deformation rigidity of VOAs. We conjecture that the same should hold for every simple affine VOA that does not coincide with the corresponding universal affine VOA.