Chiral algebra

Template:Format footnotes In mathematics, a chiral algebra is an algebraic structure introduced by Template:Harvtxt 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 $$\mu :𝒜⊠𝒜(\mathrm{\infty }\Delta )\to {\Delta }_{!}𝒜$$ on $X^2$ and with an embedding $\Omega \hookrightarrow 𝒜$, 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 ${\mu }_{\Omega }:\Omega ⊠\Omega (\mathrm{\infty }\Delta )\to {\Delta }_{!}\Omega$; that is, the following diagram commutes

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

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

References

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Further reading

• Template:Cite journal

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