First cohomologies of affine, Virasoro, and lattice vertex operator algebras

In this paper, we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated with a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; and (iii) the lattice VOA associated with a positive definite even lattice. We prove that in all these cases, the first cohomology H-1 (V, W) consists of zero-mode derivations for every N-graded V-module W (where the grading is not necessarily given by the L(0) operator). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. The relationship between the first cohomology of the VOA and that of the associated Zhu's algebra is also discussed.