Generalized Vertex Algebras

arXiv:math/0602072v1 [math.QA] 4 Feb 2006

Generalized Vertex Algebras Bojko Bakalov1 , Victor G. Kac2 , 1

Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA; bojko [email protected] 2 Department of Mathematics, MIT, Cambridge, MA 02139, USA; [email protected] Abstract We give a short introduction to generalized vertex algebras, using the notion of polylocal fields. We construct a generalized vertex algebra associated to a vector space h with a symmetric bilinear form. It contains as subalgebras all lattice vertex algebras of rank equal to dim h and all irreducible representations of these vertex algebras.

1 Introduction A vertex algebra is essentially the same as a chiral algebra in two-dimensional conformal field theory [BPZ, G, DMS]. In mathematics, vertex algebras arose naturally in the representation theory of infinite-dimensional Lie algebras and in the construction of the “moonshine module” for the Monster simple finite group [B, FLM]. Some of the most important vertex (super)algebras are the vertex (super)algebras VQ associated to integral lattices Q [FK, B, FLM, K, LL]. If the lattice Q is not necessarily integral, one gets on VQ the structure of a generalized vertex algebra as introduced in [FFR, DL, M] (under the name “vertex operator para-algebra” in [FFR]). This notion includes as special cases the notion of a “vertex superalgebra” (called just a “vertex algebra” in [K]) and the notion of a “colored vertex algebra” from [X]. The “parafermion algebras” of Zamolodchikov and Fateev [ZF1, ZF2] are also closely related to generalized vertex algebras (see [DL, Chapter 14]). The theory of generalized vertex algebras (and further generalizations) is developed in detail in the monograph [DL], and important examples of generalized vertex algebras are constructed in [FFR, DL, M, GL]. The treatment of [FFR, DL, M] is centered around a “Jacobi identity,” which generalizes the Jacobi identity from [FLM] (the latter is equivalent to the Borcherds identity from [K]). In the present paper we give a short introduction to generalized vertex algebras, utilizing the approach of [BN] (for D = 1) which is based on the notion 1

2

Generalized Vertex Algebras

of polylocal fields. The notion of locality plays an important role in the theory of (generalized) vertex algebras (see [G, DL, K, Li1, GL, LL]). It is natural both from a physical and from a mathematical point of view to extend it to polylocality of fields in several variables. Our definition of a generalized vertex algebra is slightly more general than the ones from [FFR, DL, M] in that we allow a more general grading condition, and we do not assume an action of the Virasoro algebra (cf. [GL]). In particular, in [DL, M] only rational lattices Q give rise to generalized vertex algebras VQ . In our setting, Q is allowed to be the whole vector space, which leads to a new generalized vertex algebra Vh associated to any vector space h with a symmetric bilinear form. It is remarkable that Vh contains as subalgebras the lattice vertex (super)algebras VQ for all integral lattices Q ⊆ h, as well as all of their irreducible representations. The paper is organized as follows. In Sect. 2 we recall the notion of a quantum field and its generalization that corresponds to taking non-integral powers of the formal variable. We also introduce fields in several variables and (generalized) polylocality, following [BN]. In the presence of a grading by an abelian group Q, we define parafermion fields in Sect. 3.1. Then in Sect. 3.2 we show that a translation covariant, local and complete system of parafermion fields can be extended uniquely to a statefield correspondence. This result implies Uniqueness and Existence Theorems that generalize those of [G, FKRW, K] (see also [Li1, GL, LL]). The definition of a generalized vertex algebra is given in Sect. 3.3 in terms of (generalized) locality. In Sect. 3.4 we introduce a natural action of the cohomology group H2 (Q, C× ) on the isomorphism classes of generalized vertex algebras. We also show how in a special case the notion of a generalized vertex algebra reduces to that of a Q-graded vertex superalgebra. In Sect. 3.5 we discuss the operator product expansion of local parafermion fields, and we prove formal associativity and commutativity relations generalizing those of [FB, LL]. In Sect. 3.6 we derive the (generalized) Jacobi identity (= Borcherds identity), thus showing that our definition is a generalization of the ones from [FFR, DL, M]. The exposition of Sections 2, 3.2, 3.5 and 3.6 follows closely [BN]. In Sect. 4 we introduce the notion of a module over a generalized vertex algebra, following [DL]. We show that the notion of a twisted module over a vertex (super)algebra [FFR, D2] (cf. [DK]) is a special case of the notion of a module over a generalized vertex algebra, as observed in [Li3]. In Sect. 5 we construct the generalized vertex algebra Vh associated to a vector space h with a symmetric bilinear form, and we discuss its subalgebras VQ associated to integral lattices Q ⊆ h. 2 Quantum fields in several variables and polylocality 2.1 Spaces of formal series We first fix some notation to be used throughout the paper. Let N = {1, 2, . . . } and Z+ = {0, 1, 2, . . . }. By z, w, etc., we will denote formal commuting variables. All vector spaces are over the field C of complex numbers. We will denote

Bojko Bakalov, Victor G. Kac

3

by V a vector space, and by V [z] (respectively, V [[z]]) the space of polynomials (respectively, formal power series) in z with coefficients in V . We will identify the subsets Γ ⊆ C/Z with subsets Γ ⊆ C that are Zinvariant, i.e., that satisfy Γ + Z ⊆ Γ. Given such P a subset Γ, we denote by V [[z]]z Γ the space of all finite sums of the form i ψi (z)z di , where di ∈ Γ Γ and ψP i (z) ∈ V [[z]]. Another way to write the elements of V [[z]]z is as infinite n sums n fn z , where fn ∈ V and n runs over the union of finitely many sets of the form {di + Z+ } with di ∈ Γ. In particular, V [[z]]z Z is exactly the space V ((z)) ≡ V [[z]][z −1 ] of formal Laurent series. P We denote by V [[z, z Γ ]] the space of all formal infinite series n∈Γ fn z n with fn ∈ V . For Γ = Z this coincides with the space V [[z, z −1 ]] of formal power series in z, z −1 . Note that V [[z]]z Γ is a C((z))-module, while V [[z, z Γ ]] is only a C[z, z −1]-module, V [[z]]z Γ being its submodule. In addition, both V [[z]]z Γ and V [[z, z Γ ]] are equipped with the usual action of the derivative ∂z . In the same way, we introduce spaces of formal series in several variables V [[z1 , . . . , zs ]]z1Γ1 · · · zsΓs and V [[z1 , z1Γ1 , . . . , zs , zsΓs ]]. The latter consists of all infinite sums of the form X fn1 ,...,ns ∈ V , fn1 ,...,ns z1n1 · · · zsns , nj ∈Γj

while the former is its subspace consisting of such sums with each nj running over the union of finitely many sets of the form {di,j + Z+ } with di,j ∈ Γj . Let us point out that V [[z1 , z2 ]]z1Γ1 z2Γ2 consists of series in which the powers of z1 and z2 are uniformly bounded from below. In contrast, elements of the space V [[z1 ]]z1Γ1 [[z2 ]]z2Γ2 have powers of z2 that are bounded from below but the powers of z1 are possibly unbounded. For N ∈ C, we define the formal expansions ιz1 ,z2 (z1 − z2 )N := e−z2 ∂z1 z1N X N N −j z (−z2 )j ∈ C[[z1 ]]z1Γ [[z2 ]] , = j 1

(1)

j∈Z+

:= eπiN e−z1 ∂z2 z2N X N = eπiN (−z1 )j z2N −j ∈ C[[z2 ]]z2Γ [[z1 ]] , j N

ιz2 ,z1 (z1 − z2 )

(2)

j∈Z+

where Γ = N + Z. Note that the spaces in the right-hand sides of Eqs. (1) and (2) are modules over the localized ring C[[z1 , z2 ]]z1 ,z2 . More generally, it makes sense to multiply Eq. (1) by an element of V [[z1 , z2 ]]z1Γ1 z2Γ2 thus producing an element of V [[z1 ]]z1N +Γ1 [[z2 ]]z2Γ2 , and similarly for Eq. (2). In this way, we extend the above expansions by linearity to maps N ιz1 ,z2 : V [[z1 , z2 ]]z1Γ1 z2Γ2 z12 → V [[z1 ]]z1N +Γ1 [[z2 ]]z2Γ2 , N ιz2 ,z1 : V [[z1 , z2 ]]z1Γ1 z2Γ2 z12 → V [[z2 ]]z2N +Γ2 [[z1 ]]z1Γ1 ,


4

where z12 := z1 − z2 . Obviously, when N ∈ Z+ , expansions (1) and (2) are equal to each other and coincide with the binomial expansion of (z1 − z2 )N . Similarly, we define expansions of (z1 + z2 )N for N ∈ C. One has the following analog of Taylor’s formula: ιz1 ,z2 ιz12 ,z2 f (z12 + z2 , z2 ) = ιz1 ,z2 f (z1 , z2 ) ,

z12 := z1 − z2 ,

(3)

N for every localized series f (z1 , z2 ) = g(z1 , z2 ) z12 with N ∈ C and g(z1 , z2 ) ∈ C C C[[z1 , z2 ]]z1 z2 . Indeed, it is enough to prove Eq. (3) for f (z1 , z2 ) = z1M with M ∈ C, in which case it follows from (1) (see e.g. Proposition 2.2 from [BN] for more details).

2.2 Polylocal quantum fields We define a quantum field in m variables z1 , . . . , zm (or just an m-field for C short) to be a linear map from V to the space V [[z1 , . . . , zm ]]z1C · · · zm . Alternatively, an m-field A(z1 , . . . , zm ) can be viewed as a formal series from C (End V )[[z1 , z1C , . . . , zm , zm ]] with the property that for every v ∈ V one has: X d di,m z1 i,1 · · · zm ψi (z1 , . . . , zm ) , A(z1 , . . . , zm )v = i

where the sum is finite, di,j ∈ C and ψi ∈ V [[z1 , . . . , zm ]]. If A is an m-field, then for every partition {1, . . . , m} = J1 ⊔ · · · ⊔ Jr

(disjoint union),

the restriction e 1 , . . . , ur )v := A(z1 , . . . , zm )v A(u z

j :=us

for j∈Js

(4)

makes sense and defines an r-field. We will assume that V is endowed with an endomorphism T (called translation operator) and with a vector |0i (called vacuum vector), such that T |0i = 0. An m-field A is called translation covariant if T A(z1 , . . . , zm ) − A(z1 , . . . , zm ) T =

m X

∂zk A(z1 , . . . , zm ) .

k=1

Let us point out that a product A(z1 , . . . , zm )B(zm+1 , . . . , zm+n ) of two fields is not a field in general. Indeed, by the above definition, for every v ∈ V we have C A(z1 , . . . , zm )v ∈ V [[z1 , . . . , zm ]]z1C · · · zm , C C B(zm+1 , . . . , zm+n )v ∈ V [[zm+1 , . . . , zm+n ]]zm+1 · · · zm+n ,

which implies that A(z1 , . . . , zm )B(zm+1 , . . . , zm+n )v belongs to the space C C C [[zm+1 , . . . , zm+n ]]zm+1 · · · zm+n . V [[z1 , . . . , zm ]]z1C · · · zm


5

In general, elements of the latter space have unbounded powers of z1 , . . . , zm . As a consequence, the restriction of the above product for coinciding arguments is not well defined in general. We will show below that one can “regularize” this product to make a field if the following definition is satisfied (see [BN]). Definition 2.2. An m-field A and an n-field B are called mutually local if there exist complex numbers Ni,j (i = 1, . . . , m, j = 1, . . . , n) and η ∈ C× such that m m+n Y Y

ιzi ,zj (zi − zj )Ni,j A(z1 , . . . , zm )B(zm+1 , . . . , zm+n )

(5)

i=1 j=m+1

=η

m m+n Y Y

ιzj ,zi (zi − zj )Ni,j B(zm+1 , . . . , zm+n )A(z1 , . . . , zm ) .

i=1 j=m+1

A 1-field that is local with respect to itself is usually called a local field; a 2-field that is local with respect to itself is called a bilocal field. An m-field, for general m, which is local with respect to itself, is called a polylocal field. The following important result is a straightforward variation of Theorem 4.1 from [BN]. Proposition 2.2. Let A(z1 , . . . , zm ) and B(zm+1 , . . . , zm+n ) be an m-field and an n-field, respectively, which are mutually local as above. (a) Every restriction of A for coinciding arguments is also a field and is mutually local with respect to B (see Eq. (4)). (b) If the field A is translation covariant, then its restrictions for coinciding arguments are also translation covariant fields. (c) If A is translation covariant, then A(z1 , . . . , zm )|0i ∈ V [[z1 , . . . , zm ]]. (d) Every partial derivative ∂zk A is a field and is mutually local with respect to B. If the field A is translation covariant, then ∂zk A is also translation covariant. (e) The left-hand side FA,B of Eq. (5) is an (m + n)-field. If the fields A and B are local with respect to a p-field C, then FA,B is also local with respect to C. If both fields A and B are translation covariant, then FA,B is also translation covariant. Remark 2.2. The above Proposition remains valid if one considers a more genQ eral notion of polylocality, where in Eq. (5) the product (zi − zj )Ni,j is replaced with some function of the differences zi − zj . One can replace that product with an even more general function if the translation covariance condition is modified accordingly. See also [Li4], where related ideas (in the case m = n = 1) were applied in the investigation of “quantum vertex algebras.”


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2.3 Operator product expansion As a corollary of Proposition 2.2, every m-field A(z1 , . . . , zm ) can be expanded in 1-fields as follows (see [BN]). Consider for v ∈ V the formal expansion ιz,w1 · · · ιz,wm−1 A(z + w1 , . . . , z + wm−1 , z)v

:= exp(w1 ∂z1 + · · · + wm−1 ∂zm−1 ) A(z1 , . . . , zm )v z1 =···=zm =z ∈ V [[z]]z C [[w1 , . . . , wm−1 ]] .

(6)

This is a formal power series in w1 , . . . , wm−1 with coefficients of the form ψi (z)v ∈ V [[z]]z C for some uniquely defined fields ψi (z) (i running over some index set). All ψi (z) are fields because they are obtained from A(z1 , . . . , zm ) by the operations of differentiation and restriction. If, in addition, A is translation covariant and is local with respect to some other fields B, C, etc., then all the fields ψi (z) are also translation covariant and local with respect to B, C, etc. Formal expansion (6) is called the operator expansion of A(z1 , . . . , zm ). Applying this expansion to the field FA,B given by the left-hand side of Eq. (5), we get what is called the operator product expansion (OPE) of two mutually local fields A and B. The following simple observation will be useful in the sequel. Remark 2.3. It follows from Proposition 2.2(c) that for v = |0i the right-hand side of Eq. (6) is just the Taylor series expansion of A(z + w1 , . . . , z + wm−1 , z)|0i ∈ V [[z, w1 , . . . , wm−1 ]] . This implies that the linear span of all coefficients of A(z1 , . . . , zm )|0i coincides with the linear span of all coefficients of all ψi (z)|0i, where {ψi (z)} is the collection of fields appearing in operator expansion (6). 3 Generalized vertex algebras 3.1 Parafermion fields Now we will introduce a grading on the vector space V and on the space of fields, which will allow us to make the notion of locality more concrete. Let us fix a (not necessarily finite or discrete) abelian L group Q, and let us assume that our vector space V is Q-graded: V = α∈Q Vα . In addition, assume we are given a symmetric bilinear map ∆ : Q × Q → C/Z. As before, we will identify cosets Γ ∈ C/Z with subsets of C of the form d + Z for some d ∈ C. Note that, although ∆(α, β) is defined mod Z, e−2πi∆(α,β) is a welldefined complex number. Definition 3.1. (a) A parafermion field (or just a field for short) of degree α ∈ Q is a formal series a(z) ∈ (End V )[[z, z C ]] with the property that a(z)b ∈ Vα+β [[z]]z −∆(α,β)

for b ∈ Vβ .

(7)


7

We denote by Fα (V ) the vector space of all fields of degree α, and by F (V ) := L α∈Q Fα (V ) the vector space of all fields on V .

(b) Two parafermion fields a(z), b(z) of degrees α and β, respectively, are called mutually local if (see Eqs. (1), (2)) ιz,w (z − w)N a(z)b(w) = η(α, β) ιw,z (z − w)N b(w)a(z)

(8)

for some N ∈ ∆(α, β) and η(α, β) ∈ C× . It is easy to see that Eq. (8) forces N ∈ ∆(α, β) and the following relation: η(α, β) η(β, α) = e−2πi∆(α,β) ,

α, β ∈ Q .

(9)

Then the above definition of locality is symmetric with respect to switching a(z) and b(z), due to Eqs. (1), (2), (9). The notion of locality can be extended to not necessarily homogeneous fields (i.e., of fixed degree) by requiring that all their homogeneous components be mutually local. ¯ 1}, ¯ we can think of V = V¯0 ⊕ V¯1 as a Example 3.1. For Q = Z/2Z = {0, superspace. Then the choice ∆(α, β) = Z, η(α, β) = (−1)αβ corresponds to the usual locality for fields on V (see [G, DL, K, Li1]). Remark 3.1. Our notation coincides with that in [FFR, Chapter 0], except that in [FFR] the abelian group Q is denoted by Γ and is assumed finite. Let us compare our notation to that in [DL]. What is denoted by (α, β) in [DL] corresponds to our ∆(α, β); however, in [DL] all (α, β) belong to a finite subgroup of C/2Z, while ∆(α, β) take values in C/Z. The function c(α, β) from [DL] coincides with η(α, β) eπi∆(α,β) in our notation; then the condition c(α, β) = c(β, α)−1 is equivalent to Eq. (9) above. We prefer to work with η(α, β) instead of c(α, β) because eπi∆(α,β) is not well defined when ∆(α, β) is defined mod Z. We are going to write fields in the conventional form X a(z) = a(n) z −n−1 , a(n) ∈ End V .

(10)

n∈C

Then Eq. (7) is equivalent to the following conditions: a(n) b ∈ Vα+β

for a(z) ∈ Fα (V ) , b ∈ Vβ

a(n) b = 0 if

n∈ / ∆(α, β)

and or Re n ≫ 0 .

(11)

The coefficients a(n) in Eq. (10) are called modes of a(z). Assume, in addition, that we are given the vacuum vector |0i ∈ V0 and translation operator T ∈ End V preserving the Q-grading of V and such that T |0i = 0. As before, a field a(z) is called translation covariant if T a(z) − a(z) T = ∂z a(z) . In this case, Proposition 2.2(c) implies that a(z)|0i ∈ V [[z]]. On the other hand, the bilinearity of ∆ implies ∆(α, 0) = Z for all α ∈ Q. Thus, every translation covariant parafermion field a(z) of degree α satisfies a(z)|0i ∈ Vα [[z]].

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3.2 Completeness and state-field correspondence As in the previous subsection, let V be a Q-graded vector space endowed with a translation operator T and a vacuum vector |0i. Definition 3.2. A system of fields {φi (z)}i∈I is called local iff φi (z) and φj (z) are mutually local for every i, j ∈ I. The system {φi (z)} is called translation covariant iff every φi (z) is translation covariant. Finally, the system {φi (z)} is called complete iff the coefficients of all formal series φi1 (z1 ) · · · φin (zn )|0i (n ∈ N) together with |0i span the whole vector space V . The next result shows that, given a translation covariant, local and complete system of fields on V , one can extend it uniquely to a state-field correspondence. Theorem 3.2. Let {φi (z)}i∈I be a translation covariant, local and complete system of fields on V . Then for every a ∈ V there exists a unique field, denoted as Y (a, z), which is translation covariant, local with respect to all φi (z), and such that Y (a, z)|0i|z=0 = a. The proof follows closely that of Theorem 4.2 from [BN]. Let us consider the vector space F of all translation covariant fields that are local with respect to φi (z) for all i ∈ I. By Proposition 2.2(c), there is a well-defined linear map Φ: F → V , χ(z) 7→ χ(z)|0i z=0 .

Our goal is to show that this map is an isomorphism of vector spaces. Without loss of generality, we will assume that the system of fields {φi (z)} is homogeneous, i.e., that every field φi (z) has a certain degree αi . Consider for every fixed m ∈ N and i1 , . . . , im ∈ I the m-field Y A(z1 , . . . , zm ) := ιzk ,zl (zk − zl )Nkl φi1 (z1 ) · · · φim (zm ) , 1≤k