Integrally graded conformal module

Let $V$ be a conformal vertex algebra of CFT type. An integrally graded conformal $V$-module is a module $M$ for which the conformal field has modes $Y_M(\omega ,z)={\sum }_{n\in \mathbb{Z}}{L}_M(n)z^{-n-2}$, $L_M(0)$ is diagonalizable, and $M={⨁}_{d\in \mathbb{Z},d\ge d_0}{M}_d,L_M(0){\mathrm{\mid }}_{M_d}=d\mathrm{i}\mathrm{d},L_M(n)M_d\subseteq M_{d-n}$ for some integer lower bound $d_0$. The operators $L_M(n)$ satisfy the Virasoro relations with the same central charge as $V$. The sum is algebraic: each vector has finitely many nonzero homogeneous components. No finite-dimensionality of $M_d$ is required. Integer weights make $a^{L_M(0)}$ single-valued for $a\in {\mathbb{C}}^{\ast }$; the lower bound makes each positive-mode exponential used in the formal-coordinate action locally finite. Modules with fractional $L(0)$ weights require additional tangent or root data and are outside this definition.

Conformal vertex algebra of CFT type

Virasoro algebra in the $L_m$ convention

Module for an ordinary vertex algebra

Virasoro modes give an Aut O action with field covariance