Z-graded weak modules and regularity

TL;DR

This paper establishes that the complete reducibility of Z-graded weak modules for a vertex operator algebra V characterizes V as regular, linking module structure to algebraic properties like rationality and C_2-cofiniteness.

Contribution

It provides a natural characterization of regular vertex operator algebras through the complete reducibility of Z-graded weak modules.

Findings

V is regular if all Z-graded weak modules are completely reducible.

Regularity is equivalent to rationality and C_2-cofiniteness for V.

Characterization simplifies understanding of vertex operator algebra properties.

Abstract

It is proved that if any Z-graded weak module for vertex operator algebra V is completely reducible, then V is rational and C_2-cofinite. That is, V is regular. This gives a natural characterization of regular vertex operator algebras.