Chiral algebra

In mathematics, a chiral algebra is an algebraic structure introduced by as a rigorous version of the rather vague concept of a chiral algebra in physics. In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-modules. They give a 'coordinate independent' notion of vertex algebras, which are based on formal power series. Chiral algebras on curves are essentially conformal vertex algebras.

Definition
A chiral algebra on a smooth algebraic curve $$X$$ is a right D-module $$\mathcal{A}$$, equipped with a D-module homomorphism $$\mu : \mathcal{A} \boxtimes \mathcal{A}(\infty \Delta) \rightarrow \Delta_! \mathcal{A}$$ on $$X^2$$ and with an embedding $$\Omega \hookrightarrow \mathcal{A}$$, satisfying the following conditions
 * $$\mu = -\sigma_{12} \circ \mu \circ \sigma_{12}$$ (Skew-symmetry)
 * $$\mu_{1\{23\}} = \mu_{\{12\}3} + \mu_{2\{13\}}$$ (Jacobi identity)
 * The unit map is compatible with the homomorphism $$\mu_\Omega: \Omega \boxtimes \Omega (\infty \Delta) \rightarrow \Delta_!\Omega$$; that is, the following diagram commutes

Where, for sheaves $$\mathcal{M}, \mathcal{N}$$ on $$X$$, the sheaf $$\mathcal{M}\boxtimes\mathcal{N}(\infty \Delta)$$ is the sheaf on $$X^2$$ whose sections are sections of the external tensor product $$\mathcal{M}\boxtimes\mathcal{N}$$ with arbitrary poles on the diagonal:

$$\Omega$$ is the canonical bundle, and the 'diagonal extension by delta-functions' $$\Delta_!$$ is

Vertex algebra
The category of vertex algebras as defined by Borcherds or Kac is equivalent to the category of chiral algebras on $$X = \mathbb{A}^1$$ equivariant with respect to the group $$T$$ of translations.

Factorization algebra
Chiral algebras can also be reformulated as factorization algebras.