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.
A chiral algebra[1] on a smooth algebraic curve
is a right D-module
, equipped with a D-module homomorphism
, satisfying the following conditions
{\displaystyle {\begin{array}{lcl}&\Omega \boxtimes {\mathcal {A}}(\infty \Delta )&\rightarrow &{\mathcal {A}}\boxtimes {\mathcal {A}}(\infty \Delta )&\\&\downarrow &&\downarrow \\&\Delta _{!
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'
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
Chiral algebras can also be reformulated as factorization algebras.
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