Chiral algebra

In mathematics, a chiral algebra is an algebraic structure introduced by Beilinson and 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 $𝒜$, 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

$$\begin{array}{ccccc}& \mathrm{\Omega }⊠𝒜(\mathrm{\infty }\mathrm{\Delta })& \to & 𝒜⊠𝒜(\mathrm{\infty }\mathrm{\Delta })& \\ & \downarrow & & \downarrow \\ & {\mathrm{\Delta }}_{!}𝒜& \to & {\mathrm{\Delta }}_{!}𝒜& \end{array}$$ 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 $X={𝔸}^1$ 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

• Template:Cite journal

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