Chiral algebra

In mathematics, a chiral algebra is an algebraic structure introduced by Beilinson and Drinfeld (2004)^[1] 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^[2] 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

^ Beilinson, Alexander (2004). Chiral algebras. Colloquium Publications. Vladimir G. Drinfeld (Online-Ausg ed.). Providence, R.I: American Mathematical Society. ISBN 978-0-8218-3528-9.
^ Ben-Zvi, David; Frenkel, Edward (2004). Vertex algebras and algebraic curves (Second ed.). Providence, Rhode Island: American Mathematical Society. p. 339. ISBN 9781470413156.