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Module for an ordinary vertex algebra

Definition

Let $V$ be a vertex algebra over $C$ in the ordinary, non-super convention of Vertex algebra in the state–field convention. A left $V$ -module is a complex vector space $M$ and a linear map

Y_{M} (-, z) : V ⟶ {formal fields on M}, a ⟼ Y_{M} (a, z) = \sum_{n \in Z} a_{(n)}^{M} z^{- n - 1},

with the following properties:

Lower truncation. For every $a \in V$ and $m \in M$ , $a_{(n)}^{M} m = 0$ for all sufficiently large integers $n$ , so $Y_{M} (a, z) m \in M ((z))$ in the support convention of Formal distributions and fields on a vector space.
Vacuum. $Y_{M} (1, z) = {id}_{M}$ .
Module Jacobi identity. For every $a, b \in V$ ,

z_{0}^{- 1} δ ⁣ (\frac{z_{1} - z_{2}}{z_{0}}) Y_{M} (a, z_{1}) Y_{M} (b, z_{2}) - z_{0}^{- 1} δ ⁣ (\frac{z_{2} - z_{1}}{- z_{0}}) Y_{M} (b, z_{2}) Y_{M} (a, z_{1}) = z_{2}^{- 1} δ ⁣ (\frac{z_{1} - z_{0}}{z_{2}}) Y_{M} (Y (a, z_{0}) b, z_{2}) .

Here $δ (x) = \sum_{r \in Z} x^{r}$ ; the first expansion is in nonnegative powers of $z_{2}$ , the second in nonnegative powers of $z_{1}$ , and the right-hand expansion is in nonnegative powers of $z_{0}$ . They use the formal binomial conventions of Formal distributions and fields on a vector space. The $Y$ inside the right-hand side is the state-field map of $V$ ; it is not the action map $Y_{M}$ .

A $V$ -module homomorphism $f : M \to N$ is a linear map such that $f Y_{M} (a, z) = Y_{N} (a, z) f$ for every $a \in V$ . This definition does not impose a conformal grading, finite-dimensional weight spaces, semisimplicity of $L_{0}$ , a right-module convention, or a twist by an automorphism. For a conformal vertex algebra, write $Y_{M} (ω, z) = \sum_{n \in Z} L_{M} (n) z^{- n - 2}$ when a conformal vector $ω$ is specified.

A $V$ -submodule of $M$ is a linear subspace stable under every mode $a_{(n)}^{M}$ . A module is simple (or irreducible) if it is nonzero and has no proper nonzero $V$ -submodule. A module is completely reducible (or semisimple) if it is an internal direct sum of simple $V$ -submodules; the empty direct sum is the zero module.

Depends on

Used by

Finite simple labels alone do not imply a semisimple module category Counterexample
Admissible module for a vertex operator algebra Definition
Grading-restricted generalized module Definition
Rationality in the admissible-module convention Definition
The one-dimensional vertex operator algebra is rational assuming Choice Example
Heisenberg normal-ordered fields give modules with an arbitrary zero-mode action Lemma

Dependency tree · two levels

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Sources

Yi-Zhi Huang, James Lepowsky, and Lin Zhang, Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I, Definition 2.9 (standard reference, not scraped)
Tomoyuki Arakawa and Anne Moreau, Arc Spaces and Vertex Algebras (standard reference, not scraped)