Rationality of the vertex algebra $V_L^+$ when $L$ is a nondegenerate even lattice of arbitrary rank

TL;DR

This paper proves the rationality of the vertex algebra $V_L^+$ for certain classes of nondegenerate even lattices, extending previous results to include negative definite and indefinite lattices of finite rank.

Contribution

It establishes the rationality of $V_L^+$ for negative definite and indefinite even lattices, broadening the understanding beyond positive definite cases.

Findings

$V_L^+$ is rational for negative definite lattices.

Zhu algebras of $V_L^+$ are semisimple in these cases.

Extends previous positive definite lattice results.

Abstract

In this paper we prove that the vertex algebra $V_L^+$ is rational if $L$ is a negative definite even lattice of finite rank, or if $L$ is a non-degenerate even lattice of a finite rank that is neither positive definite nor negative definite. In particular, for such even lattices $L$, we show that the Zhu algebras of the vertex algebras $V_L^+$ are semisimple. This extends the result of Abe which establishes the rationality of $V_L^+$ when $L$ is a positive definite even lattice of finite rank.