Deformation rigidity of some simple affine VOAs

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 ${\mathfrak{s}\mathfrak{l}}_2$ at the non-integral admissible level $-4\mathrm{/}3$. 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.