In mathematics, a chiral algebra is an algebraic structure introduced by
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.
The unit map is compatible with the homomorphism ; that is, the following diagram commutes
Where, for
sheaves on , the sheaf is the sheaf on whose sections are sections of the external tensor product with arbitrary poles on the diagonal:
is the
canonical bundle, and the 'diagonal extension by delta-functions' is
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 equivariant with respect to the
group of
translations.