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^[1] on a smooth algebraic curve ${\displaystyle X}$ is a right D-module ${\displaystyle {\mathcal {A}}}$, equipped with a D-module homomorphism $${\displaystyle \mu$$ :{\mathcal {A}}\boxtimes {\mathcal {A}}(\infty \Delta )\rightarrow \Delta _{!}{\mathcal {A}}} on ${\displaystyle X^{2}}$ and with an embedding ${\displaystyle \Omega \hookrightarrow {\mathcal {A}}}$, satisfying the following conditions

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

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

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 ${\displaystyle X=\mathbb {A} ^{1}}$ equivariant with respect to the group ${\displaystyle T}$ 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

1. Ben-Zvi, David; Frenkel, Edward (2004). Vertex algebras and algebraic curves (Second ed.). Providence, Rhode Island: American Mathematical Society. p. 339. ISBN 9781470413156.

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.