Finite-energy-coset category O for a conformal vertex algebra

Let $V$ be a conformal vertex algebra of CFT type and $M$ a weak $V$-module. For $h\in \mathbb{C}$, its generalized $L_0$-eigenspace is $M_{[h]}:=\{v\in M:(L_M(0)-hI{)}^{q}v=0\text{ for some integer }q\ge 1\}$. The module $M$ lies in VOA category $\mathcal{O}$ when $L_M(0)$ is locally finite (each vector lies in a finite-dimensional $L_M(0)$-invariant subspace) and there are finitely many representatives $h_1,\dots ,h_s$ of distinct cosets in $\mathbb{C}\mathrm{/}\mathbb{Z}$ such that

$M={⨁}_{j=1}^s{⨁}_{n\in {\mathbb{Z}}_{\ge 0}}{M}_{[h_j+n]}\mathrm{.}$

The empty list is allowed only for $M=0$. The induced grading is $M(n):={⨁}_{j=1}^s{M}_{[h_j+n]}$. The module Jacobi identity gives $a_{(r)}^{M}M(n)\subseteq M(n+\mathrm{w}\mathrm{t}(a)-r-1)$ for homogeneous $a$, so this is an admissible grading. This is the weak-module category used in the affine regularity argument; it is distinct from Kac–Moody category $\mathcal{O}$, which imposes finite-dimensional Cartan weight spaces and finite-cone support.

Conformal vertex algebra of CFT type

Module for an ordinary vertex algebra

Admissible module for a vertex operator algebra