Chiral algebra

In mathematics, a chiral algebra is an algebraic structure introduced by (Beilinson Drinfeld) 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 on a smooth algebraic curve [math]\displaystyle{ X }[/math] is a right D-module [math]\displaystyle{ \mathcal{A} }[/math], equipped with a D-module homomorphism [math]\displaystyle{ \mu : \mathcal{A} \boxtimes \mathcal{A}(\infty \Delta) \rightarrow \Delta_! \mathcal{A} }[/math] on [math]\displaystyle{ X^2 }[/math] and with an embedding [math]\displaystyle{ \Omega \hookrightarrow \mathcal{A} }[/math], satisfying the following conditions

• [math]\displaystyle{ \mu = -\sigma_{12} \circ \mu \circ \sigma_{12} }[/math] (Skew-symmetry)
• [math]\displaystyle{ \mu_{1\{23\}} = \mu_{\{12\}3} + \mu_{2\{13\}} }[/math] (Jacobi identity)
• The unit map is compatible with the homomorphism [math]\displaystyle{ \mu_\Omega: \Omega \boxtimes \Omega (\infty \Delta) \rightarrow \Delta_!\Omega }[/math]; that is, the following diagram commutes

[math]\displaystyle{ \begin{array}{lcl} & \Omega \boxtimes \mathcal{A}(\infty\Delta) & \rightarrow & \mathcal{A} \boxtimes \mathcal{A}(\infty \Delta) & \\ & \downarrow && \downarrow \\ & \Delta_!\mathcal A & \rightarrow & \Delta_! \mathcal A & \\ \end{array} }[/math] Where, for sheaves [math]\displaystyle{ \mathcal{M}, \mathcal{N} }[/math] on [math]\displaystyle{ X }[/math], the sheaf [math]\displaystyle{ \mathcal{M}\boxtimes\mathcal{N}(\infty \Delta) }[/math] is the sheaf on [math]\displaystyle{ X^2 }[/math] whose sections are sections of the external tensor product [math]\displaystyle{ \mathcal{M}\boxtimes\mathcal{N} }[/math] with arbitrary poles on the diagonal: [math]\displaystyle{ \mathcal M \boxtimes \mathcal N (\infty \Delta) = \varinjlim \mathcal{M} \boxtimes \mathcal{N} (n \Delta), }[/math] [math]\displaystyle{ \Omega }[/math] is the canonical bundle, and the 'diagonal extension by delta-functions' [math]\displaystyle{ \Delta_! }[/math] is [math]\displaystyle{ \Delta_!\mathcal{M} = \frac{\Omega \boxtimes \mathcal{M}(\infty \Delta)}{\Omega \boxtimes \mathcal{M}}. }[/math]

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 [math]\displaystyle{ X = \mathbb{A}^1 }[/math] equivariant with respect to the group [math]\displaystyle{ T }[/math] of translations.

Factorization algebra

Chiral algebras can also be reformulated as factorization algebras.

See also

• Chiral homology
• Chiral Lie algebra

References

• Beilinson, Alexander; Drinfeld, Vladimir (2004), Chiral algebras, American Mathematical Society Colloquium Publications, 51, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3528-9, https://books.google.com/books?id=yHZh3p-kFqQC 

Further reading

• Francis, John; Gaitsgory, Dennis (2012). "Chiral Koszul duality". Sel. Math.. New Series 18 (1): 27–87. doi:10.1007/s00029-011-0065-z. http://nrs.harvard.edu/urn-3:HUL.InstRepos:10043337.