| Home / lib / M_Mathematics / MP_Mathematical physics / | ||
Frenkel. Vertex algebras (Bourbaki seminar)(41s).pdf |
|
Size 0.4Mb Date Jun 5, 2003 |
2. DEFINITION AND FIRST PROPERTIES OF VERTEX ALGEBRAS 2.1. Let R be a C–algebra. An R–valued formal power series (or formal distribution) in variables z1 , z2 , . . . , zn is an arbitrary infinite series i i i i A(z1 , . . . , zn ) = ··· ai1 ,...,in z11 · · · znn ,
1 ∈Z n ∈Z...
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...
This is a field of conformal dimension ∆φ + ∆ψ . The normal ordering of more than two fields is defined recursively from right to left, so that by definition :A(z )B (z )C (z ): = :A(z )(:B (z )C (z ):):. It is easy to see that in the case of the Heisenberg vertex algebra π this definition of normal ordering coincides with the one given in §2.3. 2.5. The meaning of lo cality. The product φ(z )ψ (w) of two fields is a well–defined End V –valued formal power series in z ±1 and w±1 . Given v ∈ V and ϕ ∈ V ∗ , consider the matrix coefficient ϕ, φ(z )ψ (w)v ∈ C[[z ±1 , w±1 ]]. Since Vn = 0 for n < 0, we find by degree considerations that it belongs to C((z ))((w)), the space of formal Laurent series in w, whose coefficients are formal Laurent series in z . Likewise, ϕ, ψ (w)φ(z )v belongs to C((w))((z )). As we have seen in §2.2, the condition that the fields φ(z ) and ψ (w) literally commute is too strong, and it essentially keeps us in the realm of commutative algebra. However, there is a natural way to relax this condition, which leads to the more general notion of locality. Let C((z , w)) be the field of fractions of the ring C[[z , w]]; its elements may be viewed as meromorphic functions in two formal variables. We have natural embeddings (6) C((z ))((w)) ←− C((z , w)) −→ C((w))((z ))....
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 )....
Note that the translation operator is equal to L−1 . The Lie algebra Der O generates infinitesimal changes of coordinates. As we will see in §4, it is important to have this Lie algebra (and even better, the whole Virasoro algebra) act on a given vertex algebra by “internal symmetries”. This property is formalized by the following definition. Definition 3.1. — A vertex algebra V is cal led conformal, of central charge c ∈ C, if V contains a non-zero conformal vector ω ∈ V2 , such that the corresponding vertex operator n −n−2 satisfies: L−1 = T , L0 |Vn = n Id, and L2 ω = 1 c|0 . Y (ω , z ) = ∈ Z Ln z 2 These conditions imply that the operators Ln , n ∈ Z, satisfy the commutation relations of the Virasoro algebra with central charge c. We also obtain a non-trivial homomorphism Virc → V , sending L−2 vc to ω . For example, the vector 1 b2 1 + λb−2 is a conformal vector in π for any λ ∈ C. The 2− corresponding central charge equals 1 − 12λ2 . The Kac-Moody vertex algebra Vk (g) is conformal if k = −h∨ , where h∨ is the dual Coxeter number of g. The conformal vector is aa2 1 given by the Sugawara formula 2(k+h∨ ) (J−1 ) vk , where {J a } is an orthonormal basis of g. Thus, each g–module from the category O of level k = −h∨ is automatically a module over the Virasoro algebra. 3.3. Boson–fermion corresp ondence. Let C be the Clifford algebra associated to the vector space C((t)) ⊕ C((t))dt equipped with the inner product induced by the residue pairing. It has topological generators ∗ ψn = tn , ψn = tn−1 dt, n ∈ Z, satisfying the anti-commutation relations (14)
∗ ∗ [ψn , ψm ]+ = [ψn , ψm ]+ = 0, ∗ [ψn , ψm ]+ = δn,−m ....
where Sλ : πµ → πµ+λ is the shift operator, Sλ · 1µ = 1µ+λ , [Sλ , bn ] = 0, n = 0. V The boson–fermion correspondence is an isomorphism of vertex superalgebras Z, ∗ which maps ψ (z ) to Y (1−1 , z ) and ψ (z ) to Y (11 , z ). For more details, see, e.g., [K2]. 3.4. Rational vertex algebras. Rational vertex algebras constitute an important class of vertex algebras, which are particularly relevant to conformal field theory [dFMS]. In order to define them, we first need to give the definition of a module over a vertex algebra. Let V be a vertex algebra. A vector space M is called a V –module if it is equipped with the following data: • gradation: M = ⊕n∈Z+ Mn ; • operation YM : V → End M [[z , z −1 ]], which assigns to each A ∈ Vn a field YM (A, z ) of conformal dimension n on M ; sub ject to the following conditions: • YM (|0 , z ) = IdM ; • For all A, B ∈ V and C ∈ M , the series YM (A, z )YM (B , w)C is the expansion of an element of M [[z , w]][z −1 , w−1 , (z − w)−1 ] in M ((z ))((w)), and YM (A, z )YM (B , w)C = YM (Y (A, z − w)B , w)C , in the sense of Proposition 2.6....
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...
| © 2007 eKnigu | ||