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 {\displaystyle X} is a right D-module A {\displaystyle {\mathcal {A}}} , equipped with a D-module homomorphism
μ : A ⊠ A ( ∞ Δ ) → Δ ! A {\displaystyle \mu :{\mathcal {A}}\boxtimes {\mathcal {A}}(\infty \Delta )\rightarrow \Delta _{!}{\mathcal {A}}}
on X 2 {\displaystyle X^{2}} and with an embedding Ω ↪ A {\displaystyle \Omega \hookrightarrow {\mathcal {A}}} , satisfying the following conditions
μ = − σ 12 ∘ μ ∘ σ 12 {\displaystyle \mu =-\sigma _{12}\circ \mu \circ \sigma _{12}} (Skew-symmetry)
μ 1 { 23 } = μ { 12 } 3 + μ 2 { 13 } {\displaystyle \mu _{1\{23\}}=\mu _{\{12\}3}+\mu _{2\{13\}}} (Jacobi identity) The unit map is compatible with the homomorphism μ Ω : Ω ⊠ Ω ( ∞ Δ ) → Δ ! Ω {\displaystyle \mu _{\Omega }:\Omega \boxtimes \Omega (\infty \Delta )\rightarrow \Delta _{!}\Omega } ; that is, the following diagram commutes
Ω ⊠ A ( ∞ Δ ) → A ⊠ A ( ∞ Δ ) ↓ ↓ Δ ! A → Δ ! A {\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}}}
Where, for sheaves M , N {\displaystyle {\mathcal {M}},{\mathcal {N}}} on X {\displaystyle X} , the sheaf M ⊠ N ( ∞ Δ ) {\displaystyle {\mathcal {M}}\boxtimes {\mathcal {N}}(\infty \Delta )} is the sheaf on X 2 {\displaystyle X^{2}} whose sections are sections of the external tensor product M ⊠ N {\displaystyle {\mathcal {M}}\boxtimes {\mathcal {N}}} with arbitrary poles on the diagonal:
M ⊠ N ( ∞ Δ ) = lim → M ⊠ N ( n Δ ) , {\displaystyle {\mathcal {M}}\boxtimes {\mathcal {N}}(\infty \Delta )=\varinjlim {\mathcal {M}}\boxtimes {\mathcal {N}}(n\Delta ),}
Ω {\displaystyle \Omega } is the canonical bundle, and the 'diagonal extension by delta-functions' Δ ! {\displaystyle \Delta _{!}} is
Δ ! M = Ω ⊠ M ( ∞ Δ ) Ω ⊠ M . {\displaystyle \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 = A 1 {\displaystyle X=\mathbb {A} ^{1}} equivariant with respect to the group T {\displaystyle T} of translations.
Factorization algebra Chiral algebras can also be reformulated as factorization algebras.
See also Chiral homology Chiral Lie algebra
References
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.