$C_2$-cofiniteness of the vertex algebra $V_L^+$ when $L$ is a non-degenerate even lattice

TL;DR

This paper extends the understanding of $C_2$-cofiniteness in vertex algebras, proving it for $V_L^+$ when $L$ is a non-degenerate even lattice, including negative definite cases.

Contribution

It demonstrates $C_2$-cofiniteness of $V_L^+$ for non-degenerate even lattices beyond positive definite cases, broadening previous results.

Findings

$V_L^+$ is $C_2$-cofinite for negative definite lattices.

$V_L^+$ is $C_2$-cofinite for non-degenerate lattices not positive or negative definite.

Irreducible weak modules of $V_L^+$ are also $C_2$-cofinite in these cases.

Abstract

It was shown by Abe, Buhl and Dong that the vertex algebra $V_L^+$ and its irreducible weak modules satisfy the $C_2$-cofiniteness condition when $L$ is a positive definite even lattice. In this paper, we extend their results by showing that the vertex algebra $V_L^+$ and its irreducible weak modules are $C_2$-cofinite when $L$ is a negative definite even lattice and when $L$ is a non-degenerate even lattice that is neither negative definite nor positive definite.