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 $X$ is a right D-module ${𝒜}$, equipped with a D-module homomorphism $$\mu :{𝒜}⊠{𝒜}(\mathrm{\infty }\mathrm{\Delta })\to {\mathrm{\Delta }}_{!}{𝒜}$$ on $X^2$ and with an embedding $\mathrm{\Omega }\hookrightarrow {𝒜}$, 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 ${\mu }_{\mathrm{\Omega }}:\mathrm{\Omega }⊠\mathrm{\Omega }(\mathrm{\infty }\mathrm{\Delta })\to {\mathrm{\Delta }}_{!}\mathrm{\Omega }$; that is, the following diagram commutes

$$\begin{array}{lclll}& \mathrm{\Omega }⊠{𝒜}(\mathrm{\infty }\mathrm{\Delta })& \to & {𝒜}⊠{𝒜}(\mathrm{\infty }\mathrm{\Delta })& \\ & \downarrow & & \downarrow \\ & {\mathrm{\Delta }}_{!}{𝒜}& \to & {\mathrm{\Delta }}_{!}{𝒜}& \\ \end{array}$$ Where, for sheaves ${ℳ},{𝒩}$ on $X$, the sheaf ${ℳ}⊠{𝒩}(\mathrm{\infty }\mathrm{\Delta })$ is the sheaf on $X^2$ whose sections are sections of the external tensor product ${ℳ}⊠{𝒩}$ with arbitrary poles on the diagonal: $${ℳ}⊠{𝒩}(\mathrm{\infty }\mathrm{\Delta })=\underrightarrow{\lim }{ℳ}⊠{𝒩}(n\mathrm{\Delta }),$$ $\mathrm{\Omega }$ is the canonical bundle, and the 'diagonal extension by delta-functions' ${\mathrm{\Delta }}_{!}$ is $${\mathrm{\Delta }}_{!}{ℳ}=\frac{\mathrm{\Omega }⊠{ℳ}(\mathrm{\infty }\mathrm{\Delta })}{\mathrm{\Omega }⊠{ℳ}}.$$

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, 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.