First cohomologies of affine, Virasoro and lattice vertex operator algebras

TL;DR

This paper computes the first cohomologies of specific vertex operator algebras, confirming a conjecture that they are given by zero-mode derivations for a broad class of modules, including some negative energy cases.

Contribution

It proves that the first cohomology groups of affine, Virasoro, and lattice VOAs are given by zero-mode derivations, extending previous conjectures and including negative energy modules.

Findings

First cohomology equals zero-mode derivations for affine, Virasoro, and lattice VOAs.

Confirmation of Huang and the author's 2018 conjecture.

Extension of results to negative energy Virasoro modules with certain grading.

Abstract

In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For negative energy representations of Virasoro VOA, the same conclusion holds when $W$ is $L(0)$-graded with lowest weight greater or equal to $-3$. Relationship between the first cohomology of the VOA and that of the associated Zhu's algebra is also discussed.