Rationality, quasirationality and finite W-algebras
Matthias R. Gaberdiel, Andrew Neitzke
TL;DR
Gaberdiel and Neitzke prove that Zhu's $C_2$ condition ($A_{[2]}$ finite-dimensional) enforces strong finiteness properties in chiral CFTs: only finitely many $n$-point functions, quasirational highest-weight representations, finite fusion rules, and convergence of characters. They introduce generalized Zhu quotients $A_{u}$ and $A_{[k]}$, and establish a key vacuum-spanning set that yields a finite $W$-algebra structure and a bound on the central charge when Zhu's algebra is semisimple. The Nahm conjecture is established in this setting: every irreducible representation is quasirational and each $A_{[p]}^{R}$ is finite-dimensional, with tight control on coupling via injections to $A_{[p,1]}^*$. They also provide a refined interpretation of $A_{[p,1]}$ and prove a central-charge bound, highlighting the deep link between vacuum data and the full representation theory. The results advance the understanding of rational CFTs and furnish tools for analyzing singular limits and modular properties in vertex operator algebras.
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.