Chiral Algebras

Vertex Algebras and Algebraic Curves

“…In addition, it is required that F 1 has a global section (unit) satisfying natural properties. It is proved in [BD3] that under these conditions each F n is automatically a left D-module. Moreover, the right D X -module F r 1 = F 1 ⊗ Ω X acquires a canonical structure of chiral algebra, with µ given by the composition j * j * F r 1 ⊠ F r 1 = j * j * F r 2 → ∆ !…”

“…In fact, more is true: both of these commutative vertex algebras carry what can be called Poisson vertex algebra (or coisson algebra, in the terminology of [BD3]) structures, which are preserved by this isomorphism.…”

“…As shown in [BD3], the space of coinvariants of a commutative vertex algebra A is canonically isomorphic to the ring of functions on Γ ∇ (X, Spec A). In our case we obtain: Γ ∇ (X, Spec AL g ) ≃ OpL g (X), and so H(X, x, AL g (D)) ≃ AL g (X).…”

“…Beilinson and Drinfeld define a chiral algebra on a curve X as a right D-module A on X equipped with a homomorphism j * j * A ⊠ A → ∆ ! A satisfying the conditions of Corollary 7.2 (see [BD3,G]). …”

“…They have also found applications in such fields as algebraic geometry, theory of finite groups, modular functions and topology. Recently Beilinson and Drinfeld have introduced a remarkable geometric version of vertex algebras which they called chiral algebras [BD3]. Chiral algebras give rise to some novel concepts and techniques which are likely to have a profound impact on algebraic geometry.…”