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Frenkel. Vertex algebras (Bourbaki seminar)(41s).pdf |
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875-03 suitable vertex algebra. We mention below two such constructions (both due to Beilinson and Drinfeld): in one of them the relevant vertex algebra is associated to an affine KacMoody algebra of critical level (see §6.5), and in the other it is the chiral Hecke algebra (see §7). Another application of vertex algebras to algebraic geometry is the recent construction by Malikov, Schechtman and Vaintrob [MSV] of a sheaf of vertex superalgebras on an arbitrary smooth algebraic variety, called the chiral deRham complex, which is reviewed in §6.6. Vertex algebras form a vast and rapidly growing sub ject, and it is impossible to cover all ma jor results (or even give a comprehensive bibliography) in one survey. For example, because of lack of space, I have not discussed such important topics as the theory of conformal algebras [K2, K3] and their chiral counterpart, Lie∗ algebras [BD2]; quantum deformations of vertex algebras [B3, EK, FR]; and the connection between vertex algebras and integrable systems. Most of the material presented below (note that §§4–6 contain previously unpublished results) is developed in the forthcoming book [FB]. I thank A. Beilinson for answering my questions about chiral algebras and D. Ben-Zvi for helpful comments on the draft of this paper. The support from the Packard Foundation and the NSF is gratefully acknowledged....
875-04 where each φj is a homogeneous linear endomorphism of V of degree −j . Two fields φ(z ) and ψ (z ) are called mutually local if there exists N ∈ Z+ , such that (1) (z − w)N [φ(z ), ψ (w)] = 0...
Here Res denotes the (−1)st Fourier coefficient of a Laurent series. Thus, H is a onedimensional central extension of the commutative Lie algebra C((t)). Note that the relations (2) are independent of the choice of local coordinate t. Thus we may define a...
The Reconstruction Theorem stated below implies that once we have defined Y (b−1 , z ), there is (at most) a unique way to extend this definition to other vectors of π . This is not very surprising, since as a commutative algebra with derivation T , π is freely generated by b−1 . Explicitly, the fields corresponding to other elements of π are constructed by the formula 1 n (4) Y (b−n1 b−n2 . . . b−nk , z ) = :∂ n1 −1 b(z ) . . . ∂z k −1 b(z ):. (n1 − 1)! . . . (nk − 1)! z The columns in the right hand side of the formula stand for the normal ly ordered product, which is defined as follows. First, let : bn1 . . . bnk : be the monomial obtained from bn1 . . . bnk by moving all bni with ni < 0 to the left of all bnj with nj ≥ 0 (in other words, moving all “creation operators” bn , n < 0, to the left and all “annihilation operators” bn , n ≤ 0, to the right). The important fact that makes this definition correct is that the operators bn with n < 0 (resp., n ≥ 0) commute with each other, hence it does not matter how we order the creation (resp., annihilation) operators among themselves (this...
875-08 (including |0 ), where j1 ≥ j2 ≥ . . . ≥ jm ≥ 0, and if ji = ji+1 , then αi ≥ αi+1 with respect to the order on the set S . Theorem 2.5 ([FKRW]). — Under the above assumptions, the above data together with the assignment 1 j :∂ j1 aα1 (z ) . . . ∂zm aαm (z ):, (5) Y (aα1 α −j1 . . . aαm αm −jm |0 , z ) = −∆ 1 −∆ j 1 ! . . . jm ! z define a vertex algebra structure on V . Here we use the following general definition of the normally ordered product of fields. Let φ(z ), ψ (w) be two fields of respective conformal dimensions ∆φ , ∆ψ and Fourier coefficients φn , ψn . The normally ordered product of φ(z ) and ψ (z ) is by definition the formal power series n m m :φ(z )ψ (z ): = φm ψn−m + ψn−m φm z −n−∆φ −∆ψ .
∈Z ≤−∆φ >− ∆ φ...
in w and z − w. By degree reasons, this is an element of V ((w))((z − w)). By locality, Y (A, z )Y (B , w)C is the expansion in V ((z ))((w)) of an element of V [[z , w]][z −1 , w−1 , (z − w)−1 ], which we can map to V ((w))((z − w)) sending z −1 to (w + (z − w))−1 considered as a power series in positive powers of (z − w)/w. Proposition 2.6 ([FHL, K2]). — Any vertex algebra V satisfies the fol lowing associativity property: for any A, B , C ∈ V we have the equality in V ((w))((z − w)) (7) Y (A, z )Y (B , w)C = Y (Y (A, z − w)B , w)C....
875-11 can simplify this functional realization by considering only the n–point functions of the generating fields. For example, in the case of the Heisenberg vertex algebra we consider for each ϕ ∈ π ∗ the n–point functions (11) ωn (z1 , . . . , zn ) = ϕ(b(z1 ) . . . b(zn )|0 )....
The fields corresponding to other monomials are obtained by formula (5). Explicit computation shows that the fields J a (z ) (and hence all other fields) are mutually local....
This theorem is proved in [FF3, FF4] for generic k and in [dBT] for all k . The vertex 0 algebra Hk (g) is called the W–algebra associated to g and denoted by Wk (g). We have: Wk (sl2 ) = Virc(k) , where c(k ) = 1 − 6(k + 1)2 /(k + 2). For g = sl3 , the W–algebra was first constructed by Zamolodchikov, and for g = slN by Fateev and Lukyanov [FL]. Since Vk (g) is conformal for k = −h∨ (see §3.2), Wk (g) is also conformal, with W1 playing the role of conformal vector. On the other hand, W−h∨ (g) is a commutative vertex algebra, which is isomorphic to the center of V−h∨ (g) [FF3] (see §6.5). The simple quotient of Wk (g) for k = −h∨ + p/q , where p, q are relatively prime integers greater than or equal to h∨ , is believed to be a rational vertex algebra. Moreover, if g is simply-laced...
875-18 the commutation relations of the Virasoro algebra with central charge c. The operators Ln , n ≥ 0, then define an action of the Lie algebra Der+ O (Ln corresponds to −tn+1 ∂t ). It follows from the axioms of vertex algebra that this action can be exponentiated to an action of Aut O. For f (z ) ∈ Aut O, denote by R(f ) : V → V the corresponding operator. n n Given a vector field v = vn z n+1 ∂z , we assign to it the operator v = − v n Ln .
≥−1 ≥− 1...
875-20 π≤1 = C1 ⊕ Cb−1 , gives rise to a rank two subbundle of Π, which we denote by B. It dual bundle is an extension (17) 0 → Ω → B∗ → OX → 0....
4.4. General twisting prop erty. We have seen in §4.2 that the vertex operation Y is in some sense invariant under the Aut O–action (see formula (16)). Therefore Y gives rise to a well-defined operation on the twist of V by any Aut O–torsor. This “twisting property” follows from the fact that Aut O acts on V by “internal symmetries”, that is by exponentiation of Fourier coefficients of the vertex operator Y (ω , z ). It turns out that vertex algebras exhibit a similar twisting property with respect to any group G of internal symmetry, i.e., a group (or more generally, an ind-group) obtained by exponentiation of Fourier coefficients of vertex operators. Using associativity, one can obtain an analogue of formula (16) for the transformations of Y under the action of G. This formula means that we get a well-defined operation on the twist of V by any G–torsor. For example, let V be a vertex algebra with a g–structure of level k = −h∨ , i.e., one equipped with a homomorphism Vk (g) → V , whose image is not contained in C|0 ....
875-22 fn Now observe that any f = ∈ g ⊗ Kx gives rise to an operator on Vk (g)x , nz f = Resx (f , J (z ))dz = fn bn , which does not depend on the choice of the coordinate z . Since by definition ϕ(J (z ) · A)dz extends to a regular one-form on X \x, we obtain from the residue theorem that ϕ(f · A) = 0 for all f ∈ gx,out = g ⊗ C[X \x]. Hence we obtain a map from C (X, x, Vk (g)) to the space of gx,out –invariant functionals on Vk (g)x . Lemma 5.3. — The space of conformal blocks C (X, x, Vk (g)) is isomorphic to the space of gx,out –invariant functionals on Vk (g)x . Thus we recover the common definition of conformal blocks as gx,out –invariants. They have been extensively studied recently because of the relation to the moduli spaces of G–bundles on curves (see §6.4 below). More generally, we have: C P (X, x, Vk (g)) = HomgP,out (Vk (g)P , C), where gP,out is the Lie algebra of sections of P × g over X \x. x x x
G...
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