Rationality, quasirationality and finite W-algebras

TL;DR

This paper investigates the implications of the C_2 condition in conformal field theories, demonstrating finiteness properties, quasirationality of representations, and the structure of associated W-algebras.

Contribution

It proves that conformal field theories satisfying the C_2 condition have finitely many n-point functions and that their representations are quasirational, establishing a link to finite W-algebras.

Findings

Finiteness of n-point functions in C_2 conformal field theories

Quasirationality of all representations under the C_2 condition

Convergence of characters and finiteness of fusion rules

Abstract

Some of the consequences that follow from the C_2 condition of Zhu are analysed. In particular it is shown that every conformal field theory satisfying the C_2 condition has only finitely many n-point functions, and this result is used to prove a version of a conjecture of Nahm, namely that every representation of such a conformal field theory is quasirational. We also show that every such vertex operator algebra is a finite W-algebra, and we give a direct proof of the convergence of its characters as well as the finiteness of the fusion rules.