Chiral algebra

In mathematics, a chiral algebra is an algebraic structure introduced by {{harvtxt|Beilinson|Drinfeld|2004}} 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{{cite book |last1=Ben-Zvi |first1=David |last2=Frenkel |first2=Edward |title=Vertex algebras and algebraic curves |date=2004 |publisher=American Mathematical Society |location=Providence, Rhode Island |isbn=9781470413156 |page=339 |edition=Second}} 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

$\begin{array}{lcl}
& \Omega \boxtimes \mathcal{A}(\infty\Delta) & \rightarrow & \mathcal{A} \boxtimes \mathcal{A}(\infty \Delta) & \\
& \downarrow && \downarrow \\
& \Delta_!\mathcal A & \rightarrow & \Delta_! \mathcal A & \\
\end{array}$

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:

$\mathcal M \boxtimes \mathcal N (\infty \Delta) = \varinjlim \mathcal{M} \boxtimes \mathcal{N} (n \Delta),$

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

$\Delta_!\mathcal{M} = \frac{\Omega \boxtimes \mathcal{M}(\infty \Delta)}{\Omega \boxtimes \mathcal{M}}.$

Relation to other algebras

= 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.

References

{{Citation | last1=Beilinson|first1=Alexander|authorlink1=Alexander Beilinson|last2=Drinfeld|first2=Vladimir|authorlink2=Vladimir Drinfeld | title=Chiral algebras | url=https://books.google.com/books?id=yHZh3p-kFqQC | publisher=American Mathematical Society | location=Providence, R.I. | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-3528-9 | mr=2058353 | year=2004 | volume=51}}