Chiral algebra explained

In mathematics, a chiral algebra is an algebraic structure introduced by 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

X is a right D-module l{A} , equipped with a D-module homomorphism$$\backslash mu\; :\; \backslash mathcal\; \backslash boxtimes\; \backslash mathcal(\backslash infty\; \backslash Delta)\; \backslash rightarrow\; \backslash Delta\_!\; \backslash mathcal$$on X ^{2} and with an embedding \Omega \hookrightarrow l{A} , satisfying the following conditions\mu = - \sigma _{12} \circ \mu \circ \sigma _{12} (Skew-symmetry)\mu _{1\{23\} } = \mu_ + \mu_ (Jacobi identity)

• The unit map is compatible with the homomorphism

\mu _{\Omega:} \Omega \boxtimes \Omega ( inf ty \Delta) → \Delta _{!\Omega} ; that is, the following diagram commutesWhere, for sheaves l{M}, l{N} on X , the sheaf l{M}\boxtimes l{N}(infty \Delta) is the sheaf on X ^{2} whose sections are sections of the external tensor product l{M}\boxtimes l{N} with arbitrary poles on the diagonal:$$\backslash mathcal\; M\; \backslash boxtimes\; \backslash mathcal\; N\; (\backslash infty\; \backslash Delta)\; =\; \backslash varinjlim\; \backslash mathcal\; \backslash boxtimes\; \backslash mathcal\; (n\; \backslash Delta),$$\Omega is the canonical bundle, and the 'diagonal extension by delta-functions' \Delta _{!} is$$\backslash Delta\_!\backslash mathcal\; =\; \backslash frac.$$

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 = A ^{1} equivariant with respect to the group T of translations.

Factorization algebra

Chiral algebras can also be reformulated as factorization algebras.

See also

• Chiral homology
• Chiral Lie algebra

Further reading

• Francis . John . Gaitsgory . Dennis . Chiral Koszul duality . Sel. Math. . New Series . 18 . 2012 . 1 . 27–87 . 10.1007/s00029-011-0065-z . 1103.5803 . 8316715 .

Notes and References

1. Book: Ben-Zvi . David . Frenkel . Edward . Vertex algebras and algebraic curves . 2004 . American Mathematical Society . Providence, Rhode Island . 9781470413156 . 339 . Second.