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Conformal vertex algebra of CFT type

Definition

A conformal vertex algebra of CFT type is a complex vertex algebra $V$ in the sense of Vertex algebra in the state–field convention, with a specified vector $ω \in V$ and scalar $c \in C$ , such that, writing

Y (ω, z) = \sum_{n \in Z} L (n) z^{- n - 2},

the operators $L (n)$ and scalar central action $c$ satisfy the Virasoro relations of Virasoro algebra in the L_m convention, and $L (- 1) = T$ . In addition, $L (0)$ is diagonalizable, $V_{n} : = {v \in V : L (0) v = n v}$ gives $V = ⨁_{n \geq 0} V_{n}$ , every $V_{n}$ is finite dimensional, and $V_{0} = C 1$ . Under this page's convention, this is a vertex operator algebra of CFT type. The grading and the CFT-type condition are part of the definition; they do not follow from the vertex-algebra axioms alone.

The conformal weight of $v \in V_{n}$ is $n$ . A conformal vector, a grading, and the nonnegative grading condition are additional data beyond an ordinary vertex algebra.

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
The one-dimensional vertex operator algebra is rational assuming Choice Example
The rank-one Heisenberg vacuum space is a conformal VOA of central charge one Theorem

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Sources

Christophe Nozaradan, Introduction to Vertex Algebras, §§4.2–4.3 (standard reference, not scraped)
Yi-Zhi Huang, James Lepowsky, and Lin Zhang, Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I, §2 (standard reference, not scraped)
Chongying Dong and Cuipo Jiang, Rationality of Vertex Operator Algebras, §2 (standard reference, not scraped)