Restricted contragredient of a generalized VOA module

Let $U$ be a conformal vertex algebra of CFT type and let $M={⨁}_{h\in \mathbb{C}}{M}_{[h]}$ be a grading-restricted generalized left $U$-module. Its restricted contragredient vector space is $M^{\mathrm{\prime }}={⨁}_{h\in \mathbb{C}}{M}_{[h]}^{\ast },$ where each algebraic functional is supported on one finite-dimensional generalized weight space and every element of the direct sum has finite support in $h$. The conformal vector defines operators $L_M(n)$ on $M$. The proposed contragredient action is determined, for homogeneous $u\in U$ and $\lambda \in M^{\mathrm{\prime }}$, by $⟨Y_{M^{\mathrm{\prime }}}(u,z)\lambda ,m⟩=⟨\lambda ,Y_M\text{\hspace{0.17em}⁣}(e^{zL(1)}(-z^{-2}{)}^{L(0)}u,z^{-1})m⟩,m\in M\mathrm{.}$ For homogeneous $u$ of integer weight $d$, $(-z^{-2}{)}^{L(0)}u=(-1{)}^d{z}^{-2d}u$. The formal exponential $e^{zL(1)}u$ is finite on each homogeneous $u$ because positive Virasoro modes lower its nonnegative weight. For the component variant fix a positive integer $k$ and $k$ commuting left $V$-module actions on $M$, with conformal operators $L_j(n)$. For $\lambda =({\lambda }_1,\dots ,{\lambda }_k)\in {\mathbb{C}}^k$ define $M_{[\lambda ]}=\{m\in M:\text{for each }j,(L_j(0)-{\lambda }_{j}I{)}^{N_j}m=0\text{ for some integer }{N}_j\ge 1\}\mathrm{.}$ Assume $M={⨁}_{\lambda \in {\mathbb{C}}^k}{M}_{[\lambda ]}$, each joint space is finite dimensional, and there are real numbers $b_j$ such that $M_{[\lambda ]}\mathrm{\ne }0$ implies $\mathrm{Re}{\lambda }_j\ge b_j$. This is an algebraic direct sum, so every vector has finite joint-weight support. Set $M^{\mathrm{\prime }}={⨁}_{\lambda \in {\mathbb{C}}^k}{M}_{[\lambda ]}^{\ast }$, pairing each summand with its matching joint space and with zero on the others. Define each candidate component action by the same opposite-field pairing formula, using the $j$th action on $M$ and the state operators of $V$. No decomposition of $M$ as a tensor product is required. This definition specifies the candidate action; its module axioms and grading-restrictedness are asserted only by the separate theorem item.

Conformal vertex algebra of CFT type

Module for an ordinary vertex algebra

Grading-restricted generalized module