Chiral algebra

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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.

Definition

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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

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Vertex algebra

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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

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Chiral algebras can also be reformulated as factorization algebras.

See also

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References

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  • 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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