Chiral algebra

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 an '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[1] on a smooth algebraic curve is a right D-module , equipped with a D-module homomorphism

on and with an embedding , satisfying the following conditions

  • (Skew-symmetry)
  • (Jacobi identity)
  • 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.

Factorization algebra

Chiral algebras can also be reformulated as factorization algebras.

See also

References

  • Beilinson, Alexander; Drinfeld, Vladimir (2004), Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3528-9, MR 2058353
  1. Ben-Zvi, David; Frenkel, Edward (2004). Vertex algebras and algebraic curves (Second ed.). Providence, Rhode Island: American Mathematical Society. p. 339. ISBN 9781470413156.

Further reading


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