Rational Vertex Operator Algebras and the Effective Central Charge

TL;DR

This paper investigates the structure of rational vertex operator algebras, establishing bounds on their Lie algebra rank and characterizing lattice VOAs through effective central charge and related invariants.

Contribution

It proves that the Lie algebra of weight one states in rational VOAs is reductive and bounded by the effective central charge, and characterizes lattice VOAs via these invariants.

Findings

Lie algebra of weight one states is reductive in rational VOAs

Lie rank is bounded above by the effective central charge

Lattice VOAs are characterized by equalities involving effective central charge, Lie rank, and central charge

Abstract

We establish that the Lie algebra of weight one states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank is bounded above by the effective central charge. We show that lattice vertex operator algebras may be characterized by the equalities of the effective central charge, the Lie rank and the central charge, and in particular holomorphic lattice theories may be characterized among all holomorphic vertex operator algebras by the equality of the Lie rank and the central charge.