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 a '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 $X$ is a right D-module ${\mathcal {A}}$, equipped with a D-module homomorphism

$$\mu :{\mathcal {A}}\boxtimes {\mathcal {A}}(\infty \Delta )\rightarrow \Delta _{!}{\mathcal {A}}$$

on $X^{2}$ and with an embedding $\Omega \hookrightarrow {\mathcal {A}}$, satisfying the following conditions

• $\mu =-\sigma _{12}\circ \mu \circ \sigma _{12}$ (Skew-symmetry)
• $\mu _{1\{23\}}=\mu _{\{12\}3}+\mu _{2\{13\}}$ (Jacobi identity)
• The unit map is compatible with the homomorphism $\mu _{\Omega }:\Omega \boxtimes \Omega (\infty \Delta )\rightarrow \Delta _{!}\Omega$; that is, the following diagram commutes

$${\begin{array}{lcl}&\Omega \boxtimes {\mathcal {A}}(\infty \Delta )&\rightarrow &{\mathcal {A}}\boxtimes {\mathcal {A}}(\infty \Delta )&\\&\downarrow &&\downarrow \\&\Delta _{!}{\mathcal {A}}&\rightarrow &\Delta _{!}{\mathcal {A}}&\\\end{array}}$$

Where, for sheaves ${\mathcal {M}},{\mathcal {N}}$ on $X$, the sheaf ${\mathcal {M}}\boxtimes {\mathcal {N}}(\infty \Delta )$ is the sheaf on $X^{2}$ whose sections are sections of the external tensor product ${\mathcal {M}}\boxtimes {\mathcal {N}}$ with arbitrary poles on the diagonal:

$${\mathcal {M}}\boxtimes {\mathcal {N}}(\infty \Delta )=\varinjlim {\mathcal {M}}\boxtimes {\mathcal {N}}(n\Delta ),$$

$\Omega$ is the canonical bundle, and the 'diagonal extension by delta-functions' $\Delta _{!}$ is

$$\Delta _{!}{\mathcal {M}}={\frac {\Omega \boxtimes {\mathcal {M}}(\infty \Delta )}{\Omega \boxtimes {\mathcal {M}}}}.$$

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 $X=\mathbb {A} ^{1}$ 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

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

Further reading

• Francis, John; Gaitsgory, Dennis (2012). "Chiral Koszul duality". Sel. Math. New Series. 18 (1): 27–87. arXiv:1103.5803. doi:10.1007/s00029-011-0065-z. S2CID 8316715.