Field Algebras

A field algebra is a “non-commutative” generalization of a vertex algebra. In this paper we develop foundations of the theory of field algebras. 0. Introduction Roughly speaking, the notion of a vertex algebra [B1] is a generalization of the notion of a unital commutative associative algebra where the multiplication depends on a parameter. (In fact, in [B2] vertex algebras are described as “singular” commutative associative rings in a certain category.) More precisely, an operator of left multiplication on a vertex algebra V by an element a ∈ V is a field Y (a, z) = ∑ n∈Z a(n)z , where a(n) ∈ EndV , and one requires that Y (a, z)b is a Laurent series in z for any two elements a, b ∈ V . The role of a unit element of an algebra is played in this context by a vacuum vector |0〉 ∈ V , satisfying (vacuum axioms) Y (|0〉, z) = IV , Y (a, z)|0〉 = e a, where IV ∈ EndV is the identity operator and T ∈ EndV . A linear map a 7→ Y (a, z) of a vector space V with a vacuum vector |0〉 to the space of EndV -valued fields satisfying (translation invariance) [T, Y (a, z)] = Y (Ta, z) = ∂zY (a, z), is called a state–field correspondence. This notion is an analogue of a unital algebra. For example, if V is an ordinary algebra with a unit element |0〉 and T is a derivation of V , then the formula Y (a, z)b = (ea)b, a, b ∈ V (0.1) defines a state–field correspondence (and it is easy to show that all of them with the property that Y (a, z) is a formal power series in z, are thus obtained). Furthermore, the associativity property of the algebra V is equivalent to the following property of the state–field correspondence (0.1): Y (Y (a, z)b,−w)c = Y (a, z − w)Y (b,−w)c , a, b, c ∈ V , (0.2) and the commutativity property of V is equivalent to: Y (a, z)b = e ( Y (b,−z)a ) , a, b ∈ V . (0.3) It turns out that for a general vertex algebra V identity (0.2) holds only after one multiplies both sides by (z − w) where N is sufficiently large (depending on Date: April 23, 2002. Revised May 6, 2002. The first author was supported by the Miller Institute for Basic Research in Science. The second author was supported in part by NSF grant DMS-9970007. 1 2 BOJKO BAKALOV AND VICTOR G. KAC a, b, c): (z − w)Y (Y (a, z)b,−w)c = (z − w)Y (a, z − w)Y (b,−w)c, N ≫ 0 . (0.4) One of the equivalent definitions of a vertex algebra is that it is a state–field correspondence satisfying (0.3) and (0.4) (see Theorem 6.3). A field algebra is a state–field correspondence satisfying only the associativity property (0.4). We believe that this is the right analogue of a unital associative algebra. In the present paper we are making the first steps towards a general theory of field algebras. The trivial examples of field algebras are provided by state–field correspondences (0.1): this is a field algebra if and only if the underlying algebra V is associative. The simplest examples of non-trivial field algebras are tensor products of field algebras (0.1) with vertex algebras. A special case of this is the algebra of matrices with entries in a vertex algebra. Other examples are provided by a smash product of a vertex algebra and a group of its automorphisms. Thus, many important ringtheoretic constructions with vertex algebras become possible in the framework of field algebras. One of our main results is the construction of a canonical structure of a field algebra in a tensor algebra T (R) over a Lie (even Leibniz) conformal algebra R (Theorem 5.1). Imposing relation (0.3) on T (R) gives the enveloping vertex algebra U(R) of R (cf. [K, GMS]). We also establish the field algebra analogues of the density and duality theorems in the representation theory of associative algebras (see Theorems 8.5 and 8.6) and discuss the Zhu algebra construction [Z] in the framework of field algebras. Note that the “field algebras” considered in [K, Sec. 4.11] are defined by a stronger than associativity axiom; we call them strong field algebras in the present paper. Surprisingly, it turns out that they are almost the same as vertex algebras (see Theorem 7.4), although the “trivial” field algebras (0.1) are automatically strong field algebras. In the present paper the results of [K, Sec. 4.11] on field algebras are corrected. The first examples of non-trivial field algebras (i.e., different from (0.1)) that are not vertex algebras were constructed in [EK]. The “quantum vertex algebras” of [EK] are field algebras satisfying in addition a certain “braided commutativity” generalizing (0.3). The relation of the present work to the paper [EK], and also to [B2] and [FR], will be discussed in a subsequent paper. 1. State–Field Correspondence Let V be a vector space (referred to as the space of states). Recall that a (End V -valued) field is an expression of the form a(z) = ∑ n∈Z a(n)z −n−1 , where z is an indeterminate, a(n) ∈ EndV , and for each v ∈ V one has: a(n)v = 0 for n ≫ 0 , i.e., a(z)v is a Laurent series in z. Denote by glf (V ) the space of all EndV -valued fields. For each n ∈ Z one defines the n-th product of fields a(z) and b(z) by the following formula: a(z)(n)b(z) = Resx ( a(x)b(z)ix,z(x− z) n − b(z)a(x)iz,x(x− z) n ) . (1.1)