N-Point Locality for Vertex Operators: Normal Ordered Products, Operator Product Expansions, Twisted Vertex Algebras

Journal of Pure and Applied Algebra 218 (2014) 2165–2203

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Journal of Pure and Applied Algebra

www.elsevier.com/locate/jpaa

N-point locality for vertex operators: Normal ordered products, operator product expansions, twisted vertex algebras

Iana I. Anguelova ∗, Ben Cox, Elizabeth Jurisich

Department of Mathematics, College of Charleston, 66 George Street, Charleston, SC 29424, United States article info abstract

Article history: In this paper we study fields satisfying N-point locality and their properties. We Received 24 July 2013 obtain residue formulae for N-point local fields in terms of derivatives of delta Received in revised form 29 January functions and Bell polynomials. We introduce the notion of the space of descendants 2014 of N-point local fields which includes normal ordered products and coefficients of Available online 20 March 2014 Communicated by D. Nakano operator product expansions. We show that examples of N-point local fields include the vertex operators generating the boson–fermion correspondences of types B, C MSC: and D-A. We apply the normal ordered products of these vertex operators to the 17B67; 81R10 setting of the representation theory of the double-infinite rank Lie algebras b∞, c∞, d∞. Finally, we show that the field theory generated by N-point local fields and their descendants has a structure of a twisted vertex algebra. © 2014 Elsevier B.V. All rights reserved.

1. Introduction

Vertex operators were introduced in string theory and now play an important role in many areas such as quantum field theory, integrable models, statistical physics, representation theory, random matrix theory, and many others. There are different vertex algebra theories, each designed to describe different sets of examples of collections of fields. The best known is the theory of super vertex algebras (see for instance [7,20,18,24,31,17]), which axiomatizes the properties of some, simplest, systems of vertex operators. Locality is a property that is of crucial importance for super vertex algebras [27] and the axioms of super vertex algebras are often given in terms of locality (see [24,17]). On the other hand, there are field theories which do not satisfy the usual locality property, but rather a generalization. Examples of these are the generalized vertex algebras [13,6], Γ -vertex algebras [29], deformed chiral algebras [21], quantum vertex algebras (see e.g. [16,8,28,30,1]). We consider λ ∈ C∗ to be a point of locality for two fields a(z), b(w)ifa(z)b(w) is singular at z = λw (for the precise definition see Section 4). The usual vertex algebra locality is then just locality at the single

* Corresponding author. E-mail addresses: [email protected] (I.I. Anguelova), [email protected] (B. Cox), [email protected] (E. Jurisich). http://dx.doi.org/10.1016/j.jpaa.2014.03.010 0022-4049/© 2014 Elsevier B.V. All rights reserved. 2166 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 point λ = 1. In this paper we study the following field theory problems: First, if we start with fields which are local at several, but finite number of points, then what are the properties that these fields and their descendants satisfy? Second, what is the algebraic structure that the system of descendants of such N-point local fields, in its entirety, satisfies? In Section 3 we prove the basic property of N-point local distributions: that they can be expressed in terms of derivatives of delta functions at the points of locality and Bell polynomials (Theorem 3.2). A question in any field theory is: if we start with some collection of (generating) fields, which fields do we consider to be their descendants? In this paper we take an approach motivated by quantum field theory and representation theory and (as was the case of usual 1-point locality) we consider derivatives of fields, operator product expansion coefficients and normal ordered products of fields to be among the descendant fields (for precise definitions see Section 4, and in particular Definition 4.21). For N-point locality it is also natural to consider substitutions at the points of locality to be among the descendant fields, i.e., if a(z)isa field and λ ∈ C∗ is a point of locality, then a(λw) is a descendant field (this of course trivially holds in the case of usual 1-point locality at λ = 1). In Section 4 we study the descendant fields and their properties. We show that the Operator Product Expansion (OPE) formula holds and we provide residue formulas for the OPE coefficients. We also prove properties for the normal order products of fields, such as the Taylor expansion property and the residue formulas. For two fields a(z), b(w) we give a general definition of products of fields a(z)(j,n)b(w), where for n 0 these new products coincide with the OPE coefficients and for n<0 they coincide with the normal ordered products (see Definition 4.18). This seemingly unjustified unification of these two types of descendants is in fact explained later by Lemma 7.9, which shows that both the OPE coefficients and the normal ordered products are just different residues of the same analytic continuation, see Lemmas 7.7 and 7.8. In Section 4 we also prove a generalization of Dong’s Lemma: if we start with fields that are N-point local at Nth roots of unity, then the entire system of descendants will consist of N-point local fields. We finish Section 4 by proving properties relating the OPE expansions of normal ordered products of fields. Among earlier studies of fields that satisfy properties closely related to N-point locality as defined here, our work is most closely related to that of [29]. But both our definition of products of fields (and thus descendant fields), and the twisted vertex algebra structure on the system of descendants generally differ from Li’s theory of Γ -vertex algebras (see Appendix A for a detailed comparison between the two constructions). Our definition is motivated by quantum field theory, and we specifically require that the OPE coefficients and the normal ordered products be elements in our twisted vertex algebras. In general, for a finite cyclic group Γ , Γ -vertex algebras are described by smaller collections of descendant fields, and are in fact subsets of the systems of fields that we consider. In Section 5 we detail three examples of N-point local fields, and calculate examples of their descendants fields. These three examples, although in some sense the simplest possible, are particularly important due to their connection to both representation theory and integrable systems. In Section 6 we show that the normal ordered products of the fields from Section 5 produce representations of the double-infinite rank Lie algebras b∞, c∞ and d∞. Although these representations have been known earlier (see [12,34] for b∞, [11] for c∞, [26,25] for d∞), the proofs we present are written in terms of normal ordered products and generating series, which is new for the cases of b∞ and c∞.Thecaseofd∞ is interesting because even though the operator product expansion is 1-point local with point of locality 1, it is necessary to consider N-points of locality in order to obtain the boson–fermion correspondence in this case (which was done in [3]). Finally, in Section 7 we state the definition of twisted vertex algebra [3],andshowthatitcanbe re-formulated in terms of N-point locality. This shows that twisted vertex algebras can be considered as a generalization of super vertex algebras. We finish with results establishing the strong generation theorem for a twisted vertex algebra and the existence of analytic continuations of arbitrary products of fields. I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2167

2. Notation and preliminary results

In this section we summarize the notation that will be used throughout the paper. Let U be an associative algebra with unit. We denote by U[[ z±1]] the doubly-inﬁnite series in the formal variable z: q y ±1 m U z = s(z) s(z)= smz ,sm ∈ U . m∈Z

An element of U[[ z±1]] is called a formal distribution. Similarly, U[[ z]] denotes the series in the formal variable z with only nonnegative powers in z: m U[[ z]] = s(z) s(z)= smz ,sm ∈ U . m0

±1 Let s(z) ∈ U[[ z ]]. The coeﬃcient s−1 is “the formal residue Reszs(z)”: Reszs(z):=s−1.Wedenoteby U[[ z±1,w±1]] the doubly-inﬁnite series in variables z and w: q y ±1 ±1 m n U z ,w = a(z,w) a(z,w)= am,nz w ,am,n ∈ U . m,n∈Z

Similarly, we will use the notations U[[ z,w]] , U[[ z±1,w]] , U[[ z,w±1]] . ±1 ±1 ±1 For any a(z,w) ∈ U[[ z ,w ]] the formal residue Resza(z,w) is a well-deﬁned series in U[[ w ]] : q y n ±1 Resza(z,w)= a−1,nw ∈ U w . n∈Z

A very important example of a doubly-inﬁnite series is given by the formal delta-function at z = w (recall e.g., [27,24,31]):

Deﬁnition 2.1. The formal delta-function at z = w is given by: δ(z,w):= znw−n−1. n∈Z

The formal delta-function at z = w is an element of U[[ z±1,w±1]] f o r a n y U—an associative algebra with unit. To emphasize its main property (see the ﬁrst part of Lemma 2.2), and in keeping with the common usage in physics we will denote this formal delta function by δ(z − w) (even though this is something of an abuse of notation). Let λ ∈ C be a ﬁxed complex number, λ = 0. In many cases (e.g., when U = End(V ) for a complex vector space V ) we can consider U to contain a copy of C. For the remainder of this paper, we assume C ⊂ U. The formal delta-function at z = λw is given by: δ(z − λw):=δ(z,λw)= λ−n−1znw−n−1. n∈Z

Again, by abuse of notation we write δ(z − λw) even though it depends on two formal variables z and w, and a parameter λ. As is well known for the formal delta-function at z = w (see e.g., [27,24,31]) we have the following properties for the formal delta-function at z = λw: 2168 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

Proposition 2.2. (See [27,24, Prop. 2.1].) For n ∈ Z,n 0, λ ∈ C\{0},

(1) Forany f(z) ∈ U[[ z±1]] , one has f(z)δ(z−λw)=f(λw)δ(z−λw) and in particular (z−λw)δ(z−λw)=0. ±1 (2) Forany f(z) ∈ U[[ z ]] we have Reszf(z)δ(z − λw)=f(λw). (3) δ(z − λw)=δ(z,λw)=λ−1δ(w, λ−1z)=λ−1δ(w − λ−1z). − (n) − (n−1) − (4) (z λw)∂λw δ(z λw)=∂λw δ(z λw) for n 1. − n+1 n − (5) (z λw) ∂z δ(z λw)=0.

∈ C (n) an Notation 2.3 (Divided powers). For any a A,whereA is an associative algebra, denote a := n! .

In this work we encounter formal delta functions at z = λiw,forλi ∈ C, λi =0, λi = λj,1 i, j N, which satisfy properties extending the properties above:

Lemma 2.4 (Factoring properties). For j, n ∈ Z, j, n 0,

(z − λiw)δ(z − λjw)=(λj − λi)wδ(z − λjw); (2.1) − (n) − (n−1) − − · (n) − (z λiw)∂λ wδ(z λjw)=∂λ w δ(z λjw)+(λj λi) w∂λ wδ(z λjw); (2.2) j j j (z − λ w)w−n−1 n ∂(n) δ(z − λ w)= i (−1)n−k(λ − λ )kwk∂(k) δ(z − λ w) . λj w j n+1 j i λj w j (λj − λi) k=0

For a rational function f(z,w)wedenotebyiz,wf(z,w) the expansion of f(z,w) in the region |z||w| (the region in the complex plane outside of all the points z = λiw, λi ∈ C,1 i n), and correspondingly for iw,zf(z,w).

Example 2.5.

1 1 i = λnz−n−1wn,i = − λ−n−1znw−n−1. (2.3) z,w z − λw w,z z − λw n0 n0

Moreover 1 1 δ(z − λw)=i − i , (2.4) z,w z − λw w,z z − λw 1 1 ∂(l) δ(z − λw)=i − i . (2.5) λw z,w (z − λw)l+1 w,z (z − λw)l+1 m ∈ ±1 For any a(z)= m0 amz U[[ z ]] we use the following notation: q y q y n −1 n 1 a(z)− := anz ∈ U z ,a(z)+ := anz ∈ U z . (2.6) n<0 n0 m n ∈ ±1 ±1 For any a(z,w)= m,n∈Z am,nz w U[[ z ,w ]] we use the following notation q y m n −1 ±1 a(z,w)z,− := am,nz w ∈ U z ,w , (2.7) m<0,n∈Z q y n ±1 a(z,w)z,+ := anz ∈ U z,w , (2.8) m0,n∈Z similarly for a(z,w)w,− and a(z,w)w,+. I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2169

Thus from (2.4) and (2.5) we see that

1 1 δ(z − λw) − = i ,δ(z − λw) = −i , (2.9) z, z,w z − λw z,+ w,z z − λw 1 1 ∂(l) δ(z − λw) = i , ∂(l) δ(z − λw) = −i . (2.10) λw z,− z,w (z − λw)l+1 λw z,+ w,z (z − λw)l+1 m ∈ ±1 ∈ C Lemma 2.6 (Cauchy formulas). For any a(z)= m0 amz U[[ z ]] and λ we have 1 Res a(z)i = ∂(n)a(λw) , (2.11) z z,w (z − λw)n+1 λw + 1 (n) Res a(z)i = −∂ a(λw)−. (2.12) z w,z (z − λw)n+1 λw

Lemma 2.7 (Linear independence property). Suppose λ1,...,λN are distinct nonzero complex numbers and

− N nk 1 c (w)∂(l) δ(z − λ w)=0. kl λkw k k=1 l=0

Then ckl(w)=0for all 1 k N, 0 l nk − 1.

Proof. Fix 1 j N. We multiply both sides of the equality above by nj −1 nr (z − λjw) (z − λrw) . r=1,r= j

2 Then using Lemma 2.2 and Lemma 2.4 and taking Resz gives us cj,nj −1(w)=0sincetheλk are distinct.

3. N-point locality of formal distributions

The notion of locality is fundamental to development of conformal ﬁeld theory and vertex algebra theory, as seen in [27,24,17]. Our notion of N-point locality is a direct generalization of these.

Deﬁnition 3.1 (N-point local). Given a(z,w) ∈ U[[ z±1,w±1]] w e s a y t h a t a(z,w)isN-point local at the ∗ points λi ∈ C , λi = λj,for1 i, j N if there exist exponents n1,n2,...,nN ∈ N such that

n1 n2 nN (z − λ1w) (z − λ2w) ···(z − λN w) a(z,w)=0.

∗ We will refer to the distribution as being N-point local, and refer to the points λi ∈ C as points of locality for a(z,w). We will refer to a formal distribution as satisfying N-point locality when we do not ∗ want to specify the points λi ∈ C . The following theorem is a necessary technical result which will allow us to deﬁne the operator product expansion and prove residue formulas for products of ﬁelds. This in turn allows us to generalize results from vertex algebra theory to our setting of twisted vertex algebras in the sections that follow.

±1 ±1 ∗ Theorem 3.2. A formal distribution a(z,w) ∈ U[[ z ,w ]] satisﬁes N-point locality at a set of points λi ∈ C ∗ and exponents ni ∈ N for i =1...N if and only

− N nk 1 a(z,w)= c (w)∂(l) δ(z − λ w) (3.1) kl λkw k k=1 l=0 2170 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

±1 for some ckl(w) ∈ U[[ w ]] .IfN-point locality holds then the ckl(w) are unique and deﬁned by r−1 N − − nj r+i − ni cj,nj −r(w)=Resz (z λjw) Pi,j(w) (z λiw) a(z,w) , (3.2) i=0 i=1,i= j for 1 r nj ;where

1 n (−1)kk! − − − − P0,j(w)= ,Pn,j(w)= k+1 Bn,k(p 1,j,p 2,j,...,pk n 1,j ),n 1; p0,j k=1 p0,j n! and Bn,k(x1,x2,...,xn−k+1) are the Bell polynomials in the variables N (i) − nr p−i,j(w)=∂z (z λrw) . r=1,r= j z=λj w

We will present a proof by multi-induction on n1,n2,...,nN . For the ﬁrst step, we need the following lemma:

±1 ±1 Lemma 3.3. Let a(z,w) ∈ U[[ z ,w ]] satisfy N-point locality for n1 = n2 = ···= nN =1, i.e.,

(z − λ1w)(z − λ2w) ···(z − λN w)a(z,w)=0.

Then

N q y ±1 a(z,w)= ck(w)δ(z − λkw), where ck(w) ∈ U w . (3.3) k=1

Proof. The proof is by induction on N.ForN = 1, suppose we have (for brevity denote λ1 just by λ):

(z − λw)a(z,w)=0, k l ∈ where a(z,w)= k,l∈Z ak,lz w , ak,l U.Wehave k+1 l k l+1 k l k l (z − λw)a(z,w)= ak,lz w − λak,lz w = ak−1,lz w − λak,l−1z w =0. k,l k,l k,l k,l

Hence we have the recurrence relation

ak−1,l = λak,l−1, for any k, l ∈ Z.

m Denote m := k + l,andAk := ak,m−k = ak,l, hence we have

m−1 m−1 ∈ Z Ak−1 = Ak , for any m, k .

Since this is true for any m ∈ Z, we will suppress the index m and just write

−1 Ak−1 = λAk, or Ak = λ Ak−1, I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2171

which is a linear recurrence relation for the doubly inﬁnite sequence Ak, k ∈ Z. As is well known, the solution to such a linear recurrence relation is (since the solution to the linear ﬁrst order characteristic equation is r = λ−1):

−k Ak = A0 · λ ,A0 can be chosen arbitrarily.

∈ Z m m −k Hence, for any m ,wehaveAk = A0 λ .Thuswehave k l m k m−k m −k k m−k a(z,w)= ak,lz w = Ak z w = A0 λ z w k,l∈Z m,k∈Z m,k∈Z m m+1 −k−1 k −k m m+1 − = A0 λw λ z w = A0 λw δ(z λw) m∈Z k∈Z m∈Z = c(w) · δ(z − λw).

Now assume the result holds for N, and consider

(z − λ1w)(z − λ2w) ···(z − λN w)(z − λN+1w)a(z,w)=0.

Regrouping (associativity applies here) and applying the induction hypothesis yields

N q y ±1 (z − λN+1w)a(z,w)= ck(w)δ(z − λkw), where ck(w) ∈ U w ,k=1,...,N. k=1

By Lemma 2.4(1):

N 1 (z − λN+1w)a(z,w)=(z − λN+1w) c˜k(w)δ(z − λkw), c˜k(w)= ck(w). (λN+1 − λk)w k=1

Hence we have N (z − λN+1w) a(z,w) − c˜k(w)δ(z − λkw) =0 k=1 which from the base case gives us

N a(z,w) − c˜k(w)δ(z − λkw)=˜cN+1(w)δ(z − λN+1w), k=1 which proves the induction step and thus the lemma. 2

Proof of Theorem 3.2. We will prove the theorem in two parts: ﬁrst we will prove that N-point locality implies

− N nk 1 a(z,w)= c (w)∂(l) δ(z − λ w), kl λkw k k=1 l=0

±1 where ckl(w) ∈ U[[ w ]]. In addition, we will obtain the particular form of the coeﬃcients ckl(w). For the ﬁrst part we will use multi-induction on n1,n2,...,nN . The base of the induction, n1 = n2 = ···= nN =1, is Lemma 3.3. Now assume the result is true for some N-tuple n1,n2,...,nm,...,nN , i.e., 2172 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

n1 nk nN (z − λ1w) ···(z − λmw) ···(z − λN w) a(z,w)=0 implies

− N nk 1 q y a(z,w)= c (w)∂(l) δ(z − λ w), for c (w) ∈ U w±1 . kl λkw k kl k=1 l=0

Suppose now we have for the tuple n1,n2,...,nm +1,...,nN

n1 nm+1 nN (z − λ1w) ···(z − λkw) ···(z − λN w) a(z,w)=0.

We can regroup (associativity applies here): n1 nk nN (z − λ1w) ···(z − λmw) ···(z − λN w) (z − λmw)a(z,w) =0.

Hence from the induction hypothesis we have

− N nk 1 (z − λ w)a(z,w)= c (w)∂(l) δ(z − λ w). (3.4) m kl λkw k k=1 l=0

Now from the factoring properties Lemma 2.4 wecanfactora(z −λ w)fromeachof∂(l) δ(z −λ w)—with m λkw k almost no expense for k = m, and with expense of raising the order of the derivative ∂(l) δ(z − λ w) for λmw m k = m: − − N nk 1 N nk 1 c (w)∂(l) δ(z − λ w)=(z − λ w) c˜ (w)∂(l) δ(z − λ w) kl λkw k m kl λkw k k=1 l=0 k= m, k=1 l=0 nm + c˜ (w)∂(l) δ(z − λ w) . ml λmw m l=1

We can rewrite (3.4) as − N nk 1 nm (z − λ w) a(z,w) − c˜ (w)∂(l) δ(z − λ w) − c˜ (w)∂(l) δ(z − λ w) =0 m kl λkw k ml λmw m k= m, k=1 l=0 l=1 which from Lemma 3.3 gives us

− N nk 1 nm a(z,w) − c˜ (w)∂(l) δ(z − λ w) − c˜ (w)∂(l) δ(z − λ w)=c δ(z − λ w). kl λkw k ml λmw m m0 m k= m, k=1 l=0 l=1

Now we proceed to calculating the precise form of the coeﬃcients from (3.2), which we will do in two steps. Suppose

− N nk 1 a(z,w)= c (w)∂(l) δ(z − λ w). kl λkw k k=1 l=0

First we write I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2173

N − − ni − ··· − nj 1 − nj (z λiw) = p0,j(w)+p−1,j(w)(z λjw)+ + p−nj +1,j(w)(z λjw) + O(z λjw) i=1,i= j (i) N n − − r | where p i,j(w)=∂z ( r=1,r= j (z λrw) ) z=λj w.Then

− N N nk 1 N (l) (z − λ w)ni a(z,w)= c (w) (z − λ w)ni ∂ δ(z − λ w) i kl i λkw k i=1,i= j k=1 l=0 i=1,i= j − nj 1 N (l) = c (w) (z − λ w)ni ∂ δ(z − λ w) jl i λj w j l=0 i=1,i= j − − nj 1 nj 1 l (l) = p− (w)(z − λ w) · c (w)∂ δ(z − λ w) . l,j j jl λj w j l=0 l=0

Then we get the linear system − nj 1 N r ni cjl(w)pr−l,j(w)=Resz (z − λjw) (z − λiw) a(z,w) . (3.5) l=r i=1,i= j

This linear system is upper-triangular, and has a unique solution for the coefficients cjl. Our formula (3.2) − ··· nj 1 for the coefficients follows from the following consideration: Let P (t)=p0,j + p−1,j t + + p1−nj ,jt be the polynomial in a formal variable t, where p−i,j = p−i,j(w) are the partial derivatives defined above. Let N nj −r−1 ni Ar := Ar(w)=Resz (z − λjw) (z − λiw) a(z,w) i=1,i= j

− ··· nj 1 and let A(t)=A0 +A1t+ +Anj −1t be the polynomial in the formal variable t. Consider the problem of ﬁnding a power series (or polynomial) Q(t) indexed as

− ··· nj 1 ··· Q(t)=qnj −1 + qnj −2t + + q0t + satisfying the equation

P (t) · Q(t)=A(t). (3.6)

The coeﬃcients ql for l nj −1 will satisfy the same linear system (3.5), and thus ql = cjl for l =0,...,nj −1. On the other hand we can solve (3.6) (and thus (3.5)) by just inverting the power series:

1 Q(t)=A(t) · . P (t)

The inversion 1/P (t) satisﬁes Faa di Bruno formula for f(P (t)), f(t)=1/t:

n − k 1 1 ( 1) k! n − − − − = + k+1 Bn,k(p 1,j ,p 2,j,...,pk n 1,j )t , P (t) p0,j n1 k=1 p0,j n! where Bn,k(x1,x2,...,xn−k+1) are the Bell polynomials (see e.g. [4,10]). We denote the coeﬃcients of this power series by 2174 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

1 n (−1)kk! − − − − P0,j(w)= ,Pn,j(w)= k+1 Bn,k(p 1,j,p 2,j,...,pk n 1,j),n 1. p0,j k=1 p0,j n!

Hence the solution to the power series problem (3.6) is given by

1 Q(t)=A(t) · = A(t) · P (w)tn; P (t) n,j n0 and thus r−1 r−1 N − − nj r+i − ni cj,nj −r = AiPi,j(w)=Resz (z λjw) Pi,j(w) (z λiw) a(z,w) . i=0 i=0 i=1 i= j

This ﬁnishes the proof of the “if” part of Theorem 3.2. The converse follows directly from the lemmas in Section 2. The uniqueness follows from Lemma 2.7. 2

Remark 3.4. If a distribution a(z,w)isN-point local with exponents n1,n2,...,nN as in Deﬁnition 3.1, then a(z,w)isalsoN-point local with exponents m1,m2,...,mN where mi ni for i =1,...,N.From the uniqueness of the coeﬃcients it follows that ckl(w)=0forl nk.

4. N-point local ﬁelds, operator product expansions (OPE’s), normal ordered products and their properties

±1 Deﬁnition 4.1. We say that a distribution a(z) ∈ U[[ z ]] i s even and N-point self-local at λ1,λ2,...,λN if there exist n1,n2,...,nN ∈ N such that n1 n2 nN (z − λ1w) (z − λ2w) ···(z − λN w) a(z),a(w) =0. (4.1)

In this case we set the parity p(a(z)) of a(z)tobe0. ±1 Web set {a, } = ab + ba. We say that a distribution a(z) ∈ U[[ z ]] i s N-point self-local at λ1,λ2,...,λN and odd if there exist n1,n2,...,nN ∈ N such that n1 n2 nN (z − λ1w) (z − λ2w) ···(z − λN w) a(z),a(w) =0. (4.2)

In this case we set the parity p(a(z)) to be 1. For brevity we will just write p(a) instead of p(a(z)). If a(z) ∈ U[[ z±1]] is even or odd, we say that a(z) is homogeneous.

Remark 4.2. In physics the notion of parity of a distribution is tied to its operator product expansion (see Lemma 4.10 below) which follows from its self-locality, and it was the approach we took here as well.

Remark 4.3. If two elements a(z),b(z) ∈ U[[ z±1]] commute (one of the elements has parity 0) or supercom- mute (both elements have parity 1), then they can be considered to be local at any λ ∈ C.Wecallsuch points of locality removable.

Consider the simplest cases of N-point self-local distributions starting with N =1.

Lemma 4.4. Let a(z) ∈ U[[ z±1]] be 1-point self-local at λ, of order 1, i.e.,

2 2 (z − λw) a(z)a(w) − (−1)p(a) a(w)a(z) =0, but a(z)a(w) − (−1)p(a) a(w)a(z) =0 .

Then λ2 =1. I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2175

Proof. From Theorem 3.2 we have that there exists c(w) ∈ U[[ w±1]] s u c h t h a t

2 a(z)a(w) − (−1)p(a) a(w)a(z)=c(w)δ(z − λw).

Now if we interchange the formal variables z and w we get

2 0=(w − λz) a(w)a(z) − (−1)p(a) a(z)a(w)

2 2 = − z − λ−1w (−1)p(a) λ a(z)a(w) − (−1)p(a) a(w)a(z) , which means that exists d(w) ∈ U[[ w±1]] s u c h t h a t

2 a(z)a(w) − (−1)p(a) a(w)a(z)=d(w)δ z − λ−1w .

Comparing the two, we get that c(w)δ(z − λw)=d(w)δ z − λ−1w .

2 Since c(w) =0from a(z)a(w) − (−1)p(a) a(w)a(z) =0, Lemma 2.7 then gives us λ2 =1. 2

There are of course a variety of cases even when λ2 = 1, depending on the coefficients c(w) ∈ U[[ w±1]] , with consequent restrictions on the coefficients c(w) placed by the parity. The simplest of these cases are as follows: the case of λ =1,c(w) = 1 is only possible for odd parity, and describes the neutral free fermion (the free fermion of type D), see e.g. [24,3], also Section 5.Thecaseofλ = −1, c(w) = 1 is only possible for even parity, and describes the free boson of type C (considered in detail in [32] as it gives rise to the CKP hierarchy). The case of λ = −1 and odd parity only allows for odd coefficients c(w), thus the easiest such case is c(w)=w; though for reasons of convenience we rescale and use λ = −1, c(w)=−2w.Thisisthe case of the free fermion of type B (see [2,3]), and is described in Section 5.

Remark 4.5. As a corollary to the proof of the lemma above it is easy to see that if a distribution a(z)hasa non-removable point of locality λ ∈ C, λ = ±1, then a(z)alsohasλ−1 as a non-removable point of locality.

Deﬁnition 4.6. Let a(z)andb(z) be homogeneous elements in U[[ z±1]] ( s e e Deﬁnition 4.1). We say that a(z) and b(z)areN-point mutually local at λ1,λ2,...,λN if there exist n1,n2,...,nN ∈ N such that n1 n2 nN p(a)p(b) (z − λ1w) (z − λ2w) ···(z − λN w) a(z)b(w) − (−1) b(w)a(z) =0. (4.3)

±1 Remark 4.7. If two elements a(z),b(z) ∈ U[[ z ]] are mutually local at λ1,λ2,...,λN , then if we choose M =max{n1,n2,...,nN } we have M p(a)p(b) (z − λ1w)(z − λ2w) ···(z − λN w) a(z)b(w) − (−1) b(w)a(z) =0. (4.4)

We will use the following notation: N − Deﬁnition 4.8. Πz,w = i=1(z λiw).

Then we can restate the N-point locality as follows: There exists M ∈ N, such that M − − p(a)p(b) Πz,w a(z)b(w) ( 1) b(w)a(z) =0. (4.5) 2176 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

∈ ±1 −n−1 Let a(z),b(z) U[[ z ]], we will use the indexing a(z)= n∈Z anz . Recall (notation (2.6))that with this indexing −n−1 −n−1 a(z)− := anz ,a(z)+ := anz . (4.6) n0 n<0

Definition 4.9 (Normal ordered product). Let a(z),b(z) be homogeneous distributions in U[[ z±1]] ( s e e Defi- nition 4.1). Define

p(a)p(b) :a(z)b(w): = a(z)+b(w)+(−1) b(w)a−(z). (4.7)

One calls this the normal ordered product of a(z)andb(w). We extend by linearity the notion of normal ordered product to any two distributions which are linear combinations of homogeneous distributions.

Lemma 4.10 (Operator Product Expansions (OPE’s)). Suppose a(z), b(w) are N-point mutually local. Then we have

− N nj 1 cjk(w) a(z)b(w)=iz,w k+1 +:a(z)b(w):, (4.8) (z − λjw) j=1 k=0

±1 where cjk(w) ∈ U[[ w ]] , j =1,...,N; k =0,...,nj − 1, are the coeﬃcients from Theorem 3.2.

Remark 4.11. Since the notion of normal ordered product is extended by linearity to any two distributions which are linear combinations of homogeneous distributions, the Operator Product Expansions formula above applies also to any two distributions which are linear combinations of homogeneous N-point mutually local distributions.

Proof. The N-point mutual locality of the distributions a(z), b(w) implies from Theorem 3.2 that there ±1 exists cjk(w) ∈ U[[ w ]] , j =1,...,N; k =0,...,nj − 1 such that

− N nj 1 a(z)b(w) − (−1)p(a)p(b)b(w)a(z)= c (w)∂(k) δ(z − λ w). (4.9) jk λj w j j=1 k=0

We can separate the nonnegative powers of z from the negative powers to get

− N nj 1 p(a)p(b) (k) a(z)±b(w) − (−1) b(w)a(z)± = c (w)∂ δ(z − λ w) ±, (4.10) jk λj w j z, j=1 k=0 and from (2.9) we have then ⎧ − ⎨ − N nj 1 cjk(w) iw,z − k+1 for subscript +, − − p(a)p(b) j=1 k=0 (z λj w) a(z)±b(w) ( 1) b(w)a(z)± = − (4.11) ⎩ N nj 1 cjk(w) − iz,w k+1 for subscript j=1 k=0 (z−λj w) which we can rewrite as

− N nj 1 − cjk(w) 2 a(z)b(w) :a(z)b(w): = iz,w k+1 . (4.12) (z − λjw) j=1 k=0 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2177

Notation 4.12 (Operator Product Expansion (OPE) and contraction). Since the commutation relations will only be aﬀected by the singular part of (4.8), we will write the above OPE as

− N nj 1 ∼ cjk(w) a(z)b(w) k+1 . (4.13) (z − λjw) j=1 k=0

The ∼ signiﬁes that we have only written the singular part, and also we have omitted writing explicitly the expansion iz,w, which we do acknowledge tacitly. Often also the following notation is used: ab = a(z)b(w) − :a(z)b(w): = a(z)−,b(w) , (4.14)

Remark 4.13 ((j, k)-products a(w)(j,k)b(w)). In the case of the usual 1 point locality (at z = w) the coeﬃ- cients cjk(w) require a single index, k, as the OPE expansion in that case is:

− n1 c (w) a(z)b(w)=i k +:a(z)b(w):. z,w (z − w)k+1 k=0

These coefficients ck(w) are used to define the k-products a(w)kb(w):=ck(w), for k =0, 1,...,n − 1. In the case of N-point locality we can define (j, k)-products a(w)(j,k)b(w):=cjk(w)forj =1,...,N; k =0,...,nj − 1. We can further define cjk(w):=0forj =1,...,N; k nj,seeRemarks 3.4 and 4.7.

We would like to define (j, k)-products a(w)(j,k)b(w)alsofork<0; j =1,...,N, and as in the case of 1-point locality we will use normal ordered products to define these new products. The definition of a normal ordered product :a(z)b(w): gives an element of U[[ z±1,w±1]] . I f a(z)andb(z) are arbitrary elements of U[[ z±1]], we cannot apply the substitution λz for w,(orλw for z), i.e., we cannot define elements :a(z)b(λz):. But, if we impose the following important restriction on the elements of U[[ z±1]] we consider, we can define :a(z)b(λz): and :a(λz)b(z): for any λ ∈ C∗:

Deﬁnition 4.14 (Field). Aﬁelda(z) on a vector space V is a series of the form −n−1 a(z)= a(n)z ,a(n) ∈ End(V ), (4.15) n∈Z such that for any v ∈ V there exists n 0witha(m)v =0form n.

Remark 4.15. The coeﬃcients a(n), n ∈ Z are usually called modes. The indexing above is generally used in vertex algebras, as in a vertex algebra the nonnegative modes (the modes in front of negative powers of z) annihilate a special vector called vacuum vector (hence are called annihilation operators), and the negative modes (the modes in front of the positive powers of z) are the creation operators.

Remark 4.16. Let a(z),b(z) ∈ End(V )[[z±1]] be fields on a vector space V .Then:a(z)b(λz): and :a(λz)b(z): are well defined elements of End(V )[[z±1]], and are also fields in V for any λ ∈ C∗.

Lemma 4.17 (Taylor expansion formula for normal ordered products). Let a(z),b(z) ∈ End(V )[[z±1]] be ﬁelds on a vector space V , λ ∈ C∗.Then (k) k iz,z0 :a(λz + z0)b(z): = : ∂λz a(λz) b(z): z0 . (4.16) k0 2178 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

Proof. From Proposition 2.4 in [24] (for instance) we have that (k) k iz,z0 a(λz + z0)= ∂λz a(λz) z0 ; k0 and we can separate the nonnegative powers of z from the negative powers of z on both sides to get (k) k iz,z0 a(λz + z0) z,± = ∂λz a(λz) z,±z0 . k0

Thus (k) k ± iz,z0 a(λz + z0)λz+z0, = ∂λz a(λz) z,±z0 , k0 and we have

iz,z0 :a(λz + z0)b(λz): − p(a)p(b) = iz,z0 a(λz + z0)λz+z0,+b(z)+( 1) b(z)a(λz + z0)λz+z0,− (k) k − p(a)p(b) (k) k = ∂λz a(λz) z,+b(z)z0 +( 1) b(z)∂λz a(λz) z,−z0 k0 k0 (k) k 2 = : ∂λz a(λz) b(z): z0 . k0

Definition 4.18 ((j, k)-products of fields). Let a(z),b(z) ∈ End(V )[[z±1]] be fields on a vector space V . Define

a(w)(j,−1)b(w):=:a(λjw)b(w): for j =1,...,N, (4.17) (n) a(w) − − b(w):=: ∂ a(λ w) b(w): for j =1,...,N, n>0, (4.18) (j, n 1) λj w j

a(w)(j,n)b(w)=cj,n(w)forj =1,...,N, n>0. (4.19)

It is easy to check that these products of fields are actually well defined and are in fact fields on V .

Remark 4.19. We chose to define products of fields a(w)(j,n)b(w) in two separate ways: by the OPE coefficients for n 0, and by the normal ordered products for n<0. We chose this definition because both the OPE’s and the normal ordered products are very important concepts in mathematical physics and quantum field theory. The OPE’s are used in quantum field theory to define (describe) the commutation relations and the parity of the particles (bosons vs fermions are defined by the symmetry/antisymmetry of the OPE’s). The normal ordered products give one of the most important operations used to produce new fields, and as such are used extensively both in quantum field theory and representation theory, as we will see in the Section 5. In the paper [29] the author defines products of fields for “compatible” pairs of vertex operators (see Definition 3.4 in [29]). The definition of compatible pairs in [29] is more general than our definition of N-point local fields. However, even when the two coincide our definition of products differs. In particular, in the case of a single pole in the OPE the two definitions may coincide for some of the products, but for multiple poles the definitions differ substantially (see Appendix A for examples). Our definition also incorporates parity, enabling us to apply it for the boson–fermion correspondences of type B, C and D-A. I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2179

Lemma 4.20 (Residue formulas for the products of ﬁelds). The following formulas hold for n 0,any j =1,...,N: M−n−1 N M (z − λiw) a(w) b(w)=Res P (w) i=1,i=j a(z)b(w) − (−1)p(a)p(b)b(w)a(z) ; (j,n) z i,j (z − λ w)−n−i i=0 j 1 p(a)p(b) 1 − − − − a(w)(j, n 1)b(w)=Resz a(z)b(w)iz,w n+1 ( 1) b(w)a(z)iw,z n+1 . (z − λjw) (z − λjw)

Proof. The formula for a(w)(j,n)b(w)andn 0 was established previously in Theorem 3.2. The formula for a(w)(j,−n−1)b(w)andn 0 follows directly from the deﬁnition of normal ordered products and the Cauchy formulas, Lemma 2.6. 2

The goal of this paper is to determine what type of structure is generated from a given number of fields which are mutually local at several points, if we incorporate all the descendants of those fields: descendants obtained by using normal order products, operator product expansions, derivatives and “substitutions” at the point of locality (i.e., at z = λiw). For the case of usual 1-point locality the structure that one obtains from allowing these operations (with imposing some additional auxiliary conditions) is a super vertex algebra structure (this is called the “Reconstruction Theorem”, see e.g. [19,24,31]). We will prove that if one allows locality at several, but finitely many points, one obtains instead the structure of a twisted vertex algebra (Theorem 7.12).

Definition 4.21 (The field descendants space FD{a0(z), a1(z),...,ap(z)}). Let a0(z),a1(z),...,ap(z)beho- mogeneous fields on a vector space W , which are self-local and pairwise local with points of locality 0 1 λ1,...,λN .DenotebyFD{a (z),a (z),...,ap(z)} the linear subspace of all fields on W obtained from the fields a0(z),a1(z),...,ap(z) as follows:

0 1 p 0 1 p (1) IdW ,a (z),a (z),...,a (z) ∈ FD{a (z),a (z),...,a (z)}; 0 1 p 0 p (2) If d(z) ∈ FD{a (z),a (z),...,a (z)},then∂z(d(z)) ∈ FD{a (z),...,a (z)}; 0 1 p 0 1 p (3) If d(z) ∈ FD{a (z),a (z),...,a (z)},thend(λiz) are also elements of FD{a (z),a (z),...,a (z)} for i =1,...,N; 0 1 p 0 (4) If d1(z),d2(z)arebothinFD{a (z),a (z),...,a (z)},then:d1(z)d2(z): is also an element of FD{a (z), 1 p a (z),...,a (z)}, as well as all products d1(z)(j,k)d2(z) for any j =1,...,N, k ∈ Z; (5) All ﬁnite linear combinations of ﬁelds in FD{a0(z),a1(z),...,ap(z)} are still in FD{a0(z),a1(z),..., ap(z)}.

0 1 Remark 4.22. The vector space FD{a (z),a (z),...,ap(z)} is given structure of a super vector space by the notion of parity of a ﬁeld (see Deﬁnition 4.1).

0 p 0 Remark 4.23. If a (z),...,a (z) have points of locality λ1,...,λN , then the fields in FD{a (z),...,ap(z)} ∗ will have points of locality λ where λ is in the multiplicative subgroup of C generated by λ1,...,λN .If we want the total number of points of locality to be finite, then this subgroup needs to be a finite cyclic subgroup of C∗.

For the following lemma, and the remainder of the paper, we will assume that the points of locality ∗ λ1,λ2,...,λN ∈ C form a multiplicative group (and hence are roots of unity). The proof of the lemma below uses only the group properties of the λis, and not that they are speciﬁcally roots of unity. We present the proof in this form in the interest of generality. 2180 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

Lemma 4.24 (Dong’s Lemma). Suppose a(z), b(z), c(z) are N-point mutually local fields on a vector space V , with points of locality λ1,λ2,...,λN ∈ C being the distinct elements of a finite multiplicative group. Let a(w)(j,k)b(w), j =1,...,N, k ∈ Z, be the products of fields defined in Definition 4.18. Then the field c(z) is N-point mutually local to a(w)(j,k)b(w) for any j =1,...,N, k ∈ Z.

Proof. Our proof is a generalization of the argument given in [27] (see also [24] and [31]). From Deﬁnition 4.8, by hypothesis there exists r ∈ N such that

r − p(a)p(b) r Πz1,z2 a(z1)b(z2)=( 1) Πz1,z2 b(z2)a(z1), (4.20) r − p(b)p(c) r Πz2,z3 b(z2)c(z3)=( 1) Πz2,z3 c(z3)b(z2), (4.21) r − p(a)p(c) r Πz1,z3 a(z1)c(z3)=( 1) Πz1,z3 c(z3)a(z1). (4.22)

We will prove the lemma separately for k 0andk<0, as we will use the Residue formulas from Lemma 4.20.Letk 0. Recalling Lemma 4.20, we need to show that there is an M 0 such that

M − p(c)(p(a)+p(b)) M Πz2,z3 A =( 1) Πz2,z3 B (4.23) where − k−n n − − p(a)p(b) A := (z1 λjz2) Πz ,z a(z1)b(z2)c(z3) ( 1) b(z2)a(z1)c(z3) , 1 2 − k−n n − − p(a)p(b) B := (z1 λjz2) Πz1,z2 c(z3)a(z1)b(z2) ( 1) c(z3)b(z2)a(z1) .

− k−n n − Note that in the above expressions when we write (z1 λjz2) Πz1,z2 it is short for iz1,z2 (z1 k−n n k−n n λjz2) Π , which equals iz ,z (z1 − λjz2) Π , as these are positive binomial powers. Indeed z1,z2 2 1 z1,z2 −(N−1)n N − −n taking Resz1 and multiplying (4.23) by z2 i=1,i= j (λj λi) we would get

M M Πz2,z3 a(z2)(j,k)b(z2)c(z3)=Πz2,z3 c(z3)a(z2)(j,k)b(z2) for k 0. To prove this one starts with the observation that for any i, j =1,...,N 2r 2r (z − λ z )2r = (z − λ z )2r−k(λ z − λ z )k 2 i 3 k 2 j 1 j 1 i 3 k=0 2r k 2r 4r−2k − −1 2r−k − λi = λj z1 λj z2 z1 z3 k λj k=0 so that for any i =1,...,N 2r N N kj 2r 2r − − λ − 2rN ··· 4r 2kj − −1 2r kj − i (z2 λiz3) = λj z1 λj z2 z1 z3 k1 kN λj kj =0 j=1 j=1 1jN and

2rN − 2rN ··· − 2rN Πz2,z3 =(z2 λ1z3) (z2 λN z3) 2r N N N ki,j 2r (4rN−2 N k ) − (2rN− N k ) λ · i=1 i,j − 1 i=1 i,j − i = λj z1 λj z2 z1 z3 . ki,j λj ki,j =0 i,j=1 j=1 i,j=1 1jN I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2181

From the group properties of the points of locality we have that

N N ki,j λ ki,j i =λ λi λ l z1 − z3 = (z1 − λlz3) j . λj i,j=1 l=1

In the above sum expansion of Π2rN then there two types of summands: the ones where for each l = z2,z3 ∈{ } 1,...,N the sum λi ki,j is at least r; and the summands where exists l 1, 2,...,N such that =λl λj N − r λi ki,j

r We interchanged b(z2)andc(z3) in the second summand due to the presence of Πz2,z3 . Further, the presence r of Πz1,z3 in the ﬁrst type of summands will allow us also to interchange a(z1)andc(z3)andtoget 2rN r − p(c)p(b) − − p(a)p(b) Πz ,z Πz ,z ( 1) a(z1)c(z3)b(z2) ( 1) b(z2)a(z1)c(z3) 2 3 2 3 2rN r − p(c)(p(a)+p(b)) − − p(a)(p(b)+p(c)) = Πz ,z Πz ,z ( 1) c(z3)a(z1)b(z2) ( 1) b(z2)c(z3)a(z1) 2 3 2 3 2rN r − p(c)(p(a)+p(b)) − − p(a)(p(b)+p(c))+p(b)p(c) = Πz2,z3 Πz2,z3 ( 1) c(z3)a(z1)b(z2) ( 1) c(z3)b(z2)a(z1) .

Thus on those ﬁrst type of summands we have r 2rN − − p(a)p(b) Πz ,z Πz ,z a(z1)b(z2)c(z3) ( 1) b(z2)a(z1)c(z3) 2 3 2 3 − p(c)(p(a)+p(b)) r 2rN − − p(a)p(b) =( 1) Πz2,z3 Πz2,z3 c(z3)a(z1)b(z2) ( 1) c(z3)b(z2)a(z1) . ∈{ } For the second type of summands there exists an l0 1,...,N such that the sum λi ki,j is less =λl λj 0 λi than r.Forany given l,andanygivenj is ﬁxed, we have a choice of i for which = λl as they lie in a λj group. Thus for any j ∈{1,...,N} exists an i such that k

N N − − ki,j = ki,j + kij ,j < (N 1)2r + r =(2N 1)r.

i=1 i=1,i= ij − N Hence 2Nr i=1 ki,j >rfor any j =1,...,N,andwecanfactor

N N − (2rN− N k ) − r r − 1 i=1 i,j (z1 λlz2) = Πz1,z2 from z1 λj z2 . l=1 j=1

2rN 2Nr Thus the second type of summands of the binomial sum expansion of Πz2,z3 will all have a factor of Πz1,z2 . Thus those summands will allow us to interchange a(z1)andb(z2) and will annihilate both r 2rN − − p(a)p(b) Πz2,z3 Πz2,z3 a(z1)b(z2)c(z3) ( 1) b(z2)a(z1)c(z3) and 2182 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

r 2rN − − p(a)p(b) Πz2,z3 Πz2,z3 c(z3)a(z1)b(z2) ( 1) c(z3)b(z2)a(z1) .

2rN As a result, for both type of summands of the binomial sum expansion of Πz2,z3 we have r 2rN − − p(a)p(b) Πz ,z Πz ,z a(z1)b(z2)c(z3) ( 1) b(z2)a(z1)c(z3) 2 3 2 3 r 2rN − p(c)(p(a)+p(b)) − − p(a)p(b) = Πz2,z3 Πz2,z3 ( 1) c(z3)a(z1)b(z2) ( 1) c(z3)b(z2)a(z1) .

− k−n n Hence we can multiply with the positive-powered binomial product (z1 λjz2) Πz1,z2 both sides and get

2Nr+r − p(c)(p(a)+p(b)) 2Nr+r Πz2,z3 A =( 1) Πz2,z3 B, and we have proved that the ﬁeld c(z)isN-point mutually local to a(w)(j,k)b(w) for any j =1,...,N, k 0.

Now for the products a(w)(j,k)b(w)whenk<0. In this case the proof is very similar, but since our 1 residue formula for a(w) − − b(w) contains the negative binomial power n+1 , we will need to pick (j, n 1) (z−λj w) a larger power M =2N(r + n +1)+r.Inthiscasewetake 1 − − p(a)p(b) 1 A := a(z1)b(z2)c(z3)iz1,z2 n+1 ( 1) b(z2)a(z1)c(z3)iz2,z1 n+1 , (z1 − λjz2) (z1 − λjz2) 1 − − p(a)p(b) 1 B := c(z3)a(z1)b(z2)iz1,z2 n+1 ( 1) c(z3)b(z2)a(z1)iz2,z1 n+1 . (z1 − λjz2) (z1 − λjz2)

The larger power M =2N(r + n +1)+r will ensure that again we will have 1 1 Πr Π2(r+n+1)N a(z )b(z )c(z )i − (−1)p(a)p(b)b(z )a(z )c(z )i z2,z3 z2,z3 1 2 3 z2,z3 n+1 2 1 3 z2,z1 n+1 (z1 − λj z2) (z1 − λjz2) 1 = Πr Π2(r+n+1)N (−1)p(c)(p(a)+p(b)) c(z )a(z )b(z )i z3,z3 z2,z3 3 1 2 z1,z2 n+1 (z1 − λjz2) − − p(a)p(b) 1 ( 1) c(z3)b(z2)a(z1)iz2,z1 n+1 . (z1 − λjz2)

The proof for the products a(w)(j,−n−1)b(w) is then ﬁnished in the same way as for the positive (OPE) products a(w)(j,n)b(w). 2

Corollary 4.25. Let a0(z),a1(z),...,ap(z) be given ﬁelds on a vector space W , which are self-local and pairwise local with points of locality i, i =1,...,N,where is a primitive root of unity. Then any two ﬁelds 0 1 i in FD{a (z),a (z),...,ap(z)} are self and mutually N-point local with points of locality , i =1,...,N.

Theorem 4.26 (Wick’s Theorem). (See [9,22] or [24].) Let ai(z) and bj(z) be N-point mutually local ﬁelds on a vector space V , satisfying

i j k i j k k k k k (1) [a (z)b (w),c (x)±]=[a b ,c (x)±]=0, for all i, j, k and c (x)=a (z) or c (x)=b (w). i j (2) [a (z)±,b (w)±]=0for all i and j.

Then

:a1(z) ···aM (z)::b1(w) ···bN (w): I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2183

min(M,N) ± i1 j1 ··· is js 1 ··· M 1 ··· N = a b a b :a (z) a (z)b (w) b (w):(i1,...,is;j1,...,js). s=0 i1<···

i j Here the subscript (i1,...,is; j1,...,js) means that those factors a (z), b (w) with indices i ∈{i1,...,is}, 1 M 1 N j ∈{j1,...,js} are to be omitted from the product :a ···a b ···b : and when s =0we do not omit any factors. The plus or minus sign is determined as follows: each permutation of adjacent odd ﬁelds changes the sign.

Lemma 4.27. Let a(z),b(z),c(z) be ﬁelds on a vector space W satisfying the hypothesis of Dong’s Lemma i and suppose {λ1,...,λN } = { | 1 i N}. We have N M :b(w)cac (w): N M cbac (w) a(z):b(w)c(w): ∼ (−1)p(a)p(b)i mp − (−1)p(a)p(b) i mp , z,w (z − mw)p+1 z,w (z − mw)p+1 m=1 p=0 m=1 p=0

bac where the triple coeﬃcient cmp(w) is deﬁned in the proof below.

Proof. First we note by Lemma 4.10

N M cba (z) b(w) a(z) − (−1)p(a)p(b)a(z)b(w) = −i mp . + + z,w (w − mz)p+1 m=1 p=0

We have p(b)p(c) a(z) b(w)+c(w)+(−1) c(w)b(w)− p(b)p(c) = a(z)b(w)+ c(w)+(−1) a(z)c(w) b(w)− N M cba (z) =(−1)p(a)p(b)b(w) a(z)c(w) − (−1)p(a)p(b) i mp c(w) + z,w (w − mz)p+1 m=1 p=0 N M ac p(b)p(c) cmp(w) p(b)p(c) +(−1) i b(w)− +(−1) :a(z)c(w): b(w)− z,w (z − mw)p+1 m=1 p=0 N M cac (w) =(−1)p(a)p(b)b(w) i mp + z,w (z − mw)p+1 m=1 p=0 N M ac p(b)p(c) cmp(w) +(−1) i b(w)− z,w (z − mw)p+1 m=1 p=0 N M cba (z) − (−1)p(a)p(b) i mp c(w) z,w (w − mz)p+1 m=1 p=0 p(a)p(b) p(b)p(c) +(−1) b(w)+ :a(z)c(w): +(−1) :a(z)c(w): b(w)−.

ba We use the OPE expansion of cmp(z)andc(w)

N M cba c c mp (w) cba (z)c(w)=i +:cba (z)c(w): mp z,w (z − mw)p+1 mp m=1 p=0 2184 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 so we have N M cba (z) i mp c(w) z,w (w − mz)p+1 m=1 p=0 N M N M cba N M ba 1 c m2p2 (w) :cmp(z)c(w): = iz,w iz,w + iz,w , (w − m1 z)p1+1 (z − m2 w)p2+1 (w − mz)p+1 m1=1 p1=0 m2=1 p2=0 m=1 p=0 which we can rearrange further to get N M cba (z) N M cb,ac(w) i mp c(w)=i mp + Reg bac (z,w); z,w (w − mz)p+1 z,w (z − mw)p+1 m=1 p=0 m=1 p=0

b,ac where we denote by cmp (w) the coeﬃcients in the combined singular part of the partial fraction expansions (after the re-arrangement), and the Reg b,ac(z,w) denotes the regular (nonsingular) part of the partial fraction expansion. If we use this, we have

a(z):b(w)c(w): N M :b(w)cac (w): N M cb,ac(w) =(−1)p(a)p(b)i mp − (−1)p(a)p(b)i mp z,w (z − mw)p+1 z,w (z − mw)p+1 m=1 p=0 m=1 p=0 p(a)p(b) (p(a)+p(c))p(b) b,ac +(−1) b(w)+:a(z)c(w): + (−1) :a(z)c(w):b(w)− + Reg (z,w).

ac b,ac 2 The only summands contributing to the OPE coeﬃcients are :b(w)cmp(w): and cmp (w).

b,ac We will use the triple OPE coeﬃcient cmp (w) later on in the paper, hence we formalize the following:

Notation 4.28 (Triple OPE coefficients). Let a(z),b(z),c(z) be fields on a vector space W satisfying the a,bc hypothesis of Dong’s Lemma. Denote by cmp (w) the coefficient in the double OPE expansion of N M cab (z) i mp c(w); z,w (w − mz)p+1 m=1 p=0

ab see above, where cmp is itself the coeﬃcient from the OPE expansion (4.8) of the ﬁelds a(w) and b(z),in particular

N M cab (z) a(w) b(z) − (−1)p(a)p(b)b(z)a(w) = −i mp . + + z,w (w − mz)p+1 m=1 p=0

ab,c Similarly, we denote by cmp (w) the coeﬃcient in the double OPE expansion of N M cab (w) i c(z) mp ; z,w (z − mw)p+1 m=1 p=0

bc where cmp is itself the coeﬃcient from the OPE expansion (4.8) of the ﬁelds a(z) and b(w), in particular

N M ab p(a)p(b) cmp(w) a(z)−b(w) − (−1) b(w)a(z)− = i . z,w (z − mw)p+1 m=1 p=0 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2185

a,bc From the lemma above we see that cmp (w) appears in the OPE expansion of b(z)(:a(w)c(w):), and ab,c cmp (w) appears in the OPE expansion of (:a(z)c(z):)b(w).

Corollary 4.29. Let a(z),b(z),c(z) be ﬁelds on a vector space W satisfying the hypothesis of Dong’s Lemma 4.24. We have

N M :b(iw)cac (w): a(z):b iw c(w): ∼ (−1)p(a)p(b)i mp z,w (z − mw)p+1 m=1 p=0 N M −(p+1)icbac (w) − (−1)p(a)p(b) i [m+i],p , z,w (z − mw)p+1 m=1 p=0 where in the index [m + i] is the residue of m + i modulo N.

5. Examples

We now consider the simplest examples of N-point local ﬁelds and give selected examples of elements of FD which can be used to obtain representations of certain Lie algebras. As in the preceding section, the points of locality in all examples are cyclic groups of roots of unity. Our ﬁrst three examples will have points j of locality {λ1,λ2} = {1, −1}, and the fourth has points of locality {λj = : j =0,...,N − 1} where is a primitive root of unity.

Example 5.1 (Free neutral fermion of type B and bosonization of type B). Let {λ1,λ2} = {1, −1}, we will { B } B consider elements of the space FD φ (z) , where φ (z)isanodd self-local ﬁeld, which we index in the B B n form φ (z)= n∈Z φn z .ByLemma 4.10 to describe its self locality, it is necessary and suﬃcient to give its OPE expansion:

−2w φB(z)φB(w) ∼ . (5.1) z + w

B ∈ This OPE expansion completely determines the commutation relations between the modes φn , n Z: B B − m φm,φn =2( 1) δm,−n1, (5.2) which show that the modes generate a Cliﬀord algebra ClB.

Remark 5.2. One often (especially in physics) writes the OPE expansion (5.1) in the following form

z − w φB(z)φB(w) ∼ , z + w

z−w as it illustrates at a glance the fact that this ﬁeld describes a fermion: the OPE z+w is antisymmetric with exchange of z and w.

With points of locality {1, −1} the following elements are in FD{φB(z)}:

B B B B φ (w)(2,0)φ (w)=−2w, φ (w)(2,n)φ (w)=0,n>0.

B B We have φ(w)(1,n)φ (w)=0forn 0. 2186 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

Remark 5.3. The OPE (5.1) of the ﬁeld φB(z) displays a property that is new to N-point local ﬁelds, as opposed to the usual 1-point locality. Due to the powers of w in the numerator (called “shifts”) we have a less symmetric OPEs on descendants, for example:

2w 2w 2 ∂ φB(z) φB(w) ∼ ,φB(z) ∂ φB(w) ∼ − . (5.3) z (z + w)2 w (z + w)2 z + w

We want to note that the shifts change with application of the derivative operator: we started with a shift w1 in the OPE of φB(z)andφB(w) for the pole at z = −w of order 1, we have no shift (shift of 1 = w0) B B in the OPE of φ (z)and∂wφ (w) for the pole at z = −w of order 1. This is important for the algebraic structure of twisted vertex algebra that we establish in Section 7,seeRemark 7.6.

| B| Let FB be the highest weight representation of ClB with the vacuum vector denoted 0 ,sothatφn 0 =0 B for n<0. We call FB the Fock space, and note that φ (z)aﬁeldonFB and thus we can form normal ordered products.

B B B B Corollary 5.4. :φ (z)φ (z): = φ (z)(1,−1)φ (z)=1.

Proof. By deﬁnition

B B B B p(φB )2 B B :φ (z)φ (z): = φ (z)+φ (z)+(−1) φ (z)φ−(z) B n B n − B n B n = φn z φn z φn z φn z n0 n0 n<0 n<0 B B m+n − B B m+n = φmφn z φmφn z . k0 m+n=k k<0 m+n=k

B B B B Note that the commutation relations (5.2) imply φmφn + φn φm =0ifbothm, n > 0, or if both m, n < 0; B B also φ0 φ0 = 1. Thus all but one of the terms above equals 0, B B B B m+n − B B m+n B B 2 :φ (z)φ (z): = φmφn z φmφn z = φ0 φ0 =1. k0 m+n=k k<0 m+n=k

The two-variable normal ordered product :φB(z)φB(w): is very important to the representation theory of inﬁnite rank Lie algebras, as it gives a representation of the algebra b∞ ([12,34], see also next section). Other elements of FD{φB(z)} are found by substituting at points of locality, in this case at λ = −1. Thus it is natural to consider the ﬁeld φB(−z), and all of its descendants. Of particular interest is then the normal ordered product :φB(z)φB(−z)::

Lemma 5.5. (See [12,2,3].) The ﬁeld hB(z) given by:

1 hB(z)= :φB(z)φB(−z): − 1 , (5.4) 4 B −2n−1 has only odd-indexed modes, h (z)= n∈Z h2n+1z and moreover it is a Heisenberg ﬁeld. I.e., it has OPE with itself given by:

zw(z2 + w2) 1 w2 1 w 1 w2 1 w hB(z)hB(w) ∼ ∼ + − + , (5.5) 2(z2 − w2)2 4 (z − w)2 4 (z − w) 4 (z + w)2 4 (z + w) and its modes, hn, n ∈ 2Z +1, generate a twisted Heisenberg algebra HZ+1/2 with relations [hm,hn]= m 2 δm+n,01, m, n are odd integers. I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2187

B B B We have the product φ (w)(2,−1)φ (w)=−1 − 4h (w), and the OPE yields the (j, n)-products for B B h (w)(j,n)h (w)forn 0:

1 1 hB(w) hB(w)= w, hB(w) hB(w)= w, (5.6) (1,0) 4 (2,0) 4 1 1 hB(w) hB(w)= w2,hB(w) hB(w)=− w2, (5.7) (1,1) 4 (2,1) 4 B B B B h (w)(1,n)h (w)=0,h(w)(2,n)h (w)=0 forn 2. (5.8)

Example 5.6 (Free twisted boson of type C). For {λ1,λ2} = {1, −1}, we now consider elements of the space FD{φC (z)}, where φC (z)isaneven self-local ﬁeld (see [11] and [32]) with OPE:

1 φC (z)φC (w) ∼ . (5.9) z + w C C n−1/2 We index the ﬁeld as follows: φ (z)= n∈Z+1/2 φn z (the half-integers are commonly used when indexing in this case, see [11] and [32]). In modes, the OPE leads to commutation relations: − 1 C C − n 2 φm,φn =( 1) δm,−n1. (5.10)

Despite the name of the field (φC (z)) the modes form a Lie algebra, not a Clifford algebra, which we denote C C by LC . The two-variable normal ordered product :φ (z)φ (w): is very important to the representation theory of infinite rank Lie algebras, as it gives the representation of the algebra c∞ (see [11] and Section 6 C C C C C C 1 below). In this case we have for the contraction φ φ of the fields φ (z),φ (w)that φ φ = iz,w z+w , which we will use for the application of Wick’s Theorem in Section 6. | C | Let FC be the highest weight module of LC with the vacuum vector 0 , such that φn 0 =0forn<0. C Then φ (z)isfieldonFC (which we call the Fock space) and thus we can define normal ordered products, and compute elements of FD{φC (z)}. First consider the normal ordered product :φC (z)φC (−z):. We have

Lemma 5.7. The ﬁeld hC (z) given by:

1 hC (z)= :φC (z)φC (−z): (5.11) 2 C C − C C −2n−1 satisﬁes h (z)=h ( z),thusweindexh (z) as h (z)= n∈Z+1/2 hnz (note the half-integers). Further, it is a Heisenberg ﬁeld, i.e., it has OPE with itself given by:

(z2 + w2) 1 1 1 1 hC (z)hC (w) ∼− ∼− − , (5.12) 2(z2 − w2)2 4 (z − w)2 4 (z + w)2 and its modes, hn, n ∈ Z +1/2, generate a twisted Heisenberg algebra HZ+1/2 with relations [hm,hn]= −mδm+n,01, m, n ∈ Z +1/2.

That the modes hn, n ∈ Z +1/2 generate a twisted Heisenberg algebra is ﬁrst seen in [11],afactalso mentioned in [32]. C C C We have the product φ (w)(2,−1)φ (w)=2h (w), and the lemma gives the (j, n)-products C C h (w)(j,n)h (w)forn 0:

C C C C h (w)(1,0)h (w)=0,h(w)(2,0)h (w)=0, (5.13) 1 1 hC (w) hC (w)=− ,hC (w) hC (w)=− , (5.14) (1,1) 4 (2,1) 4 C C C C h (w)(1,n)h (w)=0,h(w)(2,n)h (w)=0 forn 2. (5.15) 2188 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

The next pair of examples is based on a self-local generating ﬁeld we which is in fact well known as an example of a generating ﬁeld for a super vertex algebra, and thus is 1-point local at λ = 1. If we allow though substitutions at new points of locality, at λ = k, k =0, 1,...,N − 1, where is an N-th root of unity, we get a new and interesting structure which wasn’t otherwise possible—the boson–fermion correspondence of type D-A [3]. We start with N =2.

Example 5.8 (Free neutral fermion of type D and bosonization of type D-A). For this example, we again { } { − } D take points of locality λ1,λ2 = 1, 1 . We consider a single odd self-local ﬁeld φ (z), which we index D D −n−1/2 D in the form φ (z)= n∈Z+1/2 φn z .TheOPEofφ (z)isgivenby

1 φD(z)φD(w) ∼ . (5.16) z − w

D ∈ Z This OPE completely determines the commutation relations between the modes φn , n +1/2: D D φm,φn = δm,−n1, (5.17)

D D and so the modes generate a Clifford algebra ClD. The OPE yields φ (w)(1,0)φ (w) = 1, and D D D D φ (w)(1,n)φ (w)=0forn>0; φ (w)(2,n)φ (w)=0forn 0. In this example the Fock space is denoted FD and is the highest weight module of ClD with vacuum | D| vector 0 , such that φn 0 =0forn>0. (The space FD can be given a super-vertex algebra structure, as D is known from e.g. [24,25].) Note that φ (z)afieldonFD and thus normal ordered products are defined. The more general normal ordered product :φD(z)φD(w): is very important, this time to the representation theory the infinite rank Lie algebra d∞ ([25,33], also Section 6).

D D D D Corollary 5.9. :φ (z)φ (z): = 0 = φ (z)(1,−1)φ (z).

The space FD{φD(z)} contains the normal ordered product :φD(z)φD(−z): (for which the point of locality

λ2 = −1 is not removable).

Lemma 5.10. (See [2,3].) The ﬁeld hD(z) given by:

1 hD(z)= :φD(z)φD(−z): (5.18) 2 D −2n−1 has only odd-indexed modes, h (z)= n∈Z hnz and moreover it is a Heisenberg ﬁeld. I.e., it has OPE with itself given by:

zw 1 1 1 1 hD(z)hD(w) ∼ ∼ − (5.19) (z2 − w2)2 4 (z − w)2 4 (z + w)2 and its modes, hn, n ∈ Z, generate an ordinary, untwisted, Heisenberg algebra HZ with relations [hm,hn]= mδm+n,01, m, n integers.

D D D D D We have the product φ (w)(2,−1)φ (w)=−2h (w), and the products h (w)(j,n)h (w)forn 0:

D D D D h (w)(1,0)h (w)=0,h(w)(2,0)h (w)=0, (5.20) 1 1 hD(w) hD(w)= hD(w) hD(w)=− , (5.21) (1,1) 4 (2,1) 4 D D D D h (w)(1,n)h (w)=0,h(w)(2,n)h (w)=0 forn 2. (5.22) I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2189

We ﬁnish with the last example, which involves the primitive root of unity of order N ∈ N.

Example 5.11 (Bosonisation of type D-A (general order N)). In this example we extend the boson–fermion correspondence of type D-A from the previous example to a general order N as follows. Given a primitive { } { N−1} D N-th root of unity ,choose λ1,λ2,...,λN = 1,,..., . We again consider the free ﬁeld φ (z)= D −n−1/2 { D } n∈Z+1/2 φn z , with OPE’s with itself given by (5.16). Thus the descendant ﬁeld space FD φ (z) i D D i − contains all T φ (z)=φ ( z), for any 0 i N 1, with OPE’s

1 φD iz φD jw ∼ . iz − jw

j We note that for the above ﬁelds the points of locality λj = , j =1,...,N − 1 are not removable in the following normal ordered products, and in particular we have the following lemma:

Lemma 5.12. The ﬁeld hD(z) given by

− 1 N1 hD(z)= i−1:φD i−1z φD iz : (5.23) N i=0 D −Nn−1 can be indexed as h (z)= n∈Z hnz . Moreover, it is a Heisenberg ﬁeld, i.e., it has OPE with itself given by:

zN−1wN−1 hD(z)hD(w) ∼ , (5.24) (zN − wN )2

D D H and thus the commutation relations [hm,hn ]=mδm+n,01 for the Heisenberg algebra Z hold.

6. Applications to representation theory

In this section we consider the applications of the examples of normal ordered products from the previous section to the representation theory of the three double-infinite rank Lie algebras b∞, c∞ and d∞. The three representations given here also appear in the works of Date–Jimbo–Kashiwara–Miwa (see [12] and [11]), and the representation of d∞ is also considered in works of Wang (see [25,26,33]). We present the proofs here for completeness, and as an application of the N-point locality OPEs, the use of which generalizes the techniques already known for super vertex algebras. First, we recall the definitions and notations for the double-infinite rank Lie algebras a∞, b∞, c∞ and d∞ as in [23] The Lie algebraa ¯∞ is the Lie algebra of infinite matrices a¯∞ = (aij) i, j ∈ Z,aij =0for|i − j|0 . (6.1)

As usual denote the elementary matrices by Eij. The algebra a∞ is a central extension ofa ¯∞ by a central element c, a∞ =¯a∞ ⊕ Cc, with cocycle given by C(A, B)=Trace [J, A]B , (6.2) where the matrix J = i0 Eii.Inparticular 2190 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

C(Eij,Eji)=−C(Eji,Eij)=1, if i 0,j 1,

C(Eij,Ekl) = 0 in all other cases.

The three algebras ¯b∞,¯c∞ and d¯∞ are all deﬁned as subalgebras ofa ¯∞, each preserving diﬀerent bilinear form [23].

6.1. The inﬁnite dimensional Lie algebra b∞

The inﬁnite dimensional Lie algebra ¯b∞ is the subalgebra ofa ¯∞ consisting of the inﬁnite matrices pre- i serving the bilinear form B(vi,vj)=(−1) δi,−j , i.e., i+j−1 ¯b∞ = (aij) ∈ a¯∞ aij =(−1) a−j,−i . (6.3)

¯ ¯ ⊕ C 1 Denote by b∞ the central extension of b∞ by a central element c, b∞ = b∞ c, where we use 2 C (from (6.2)) as a cocycle for b∞,see[23]. Thus the commutation relations for the elementary matrices in b∞ are

1 [E ,E ]=δ E − δ E + C(E ,E )c. ij kl jk il li kj 2 ij kl

The generators for the algebra b∞ can be written in terms of these elementary matrices as: j i (−1) Ei,−j − (−1) Ej,−i,c i, j ∈ Z .

We can arrange the non-central generators in a generating series B j i i−1 −j E (z,w)= (−1) Ei,−j − (−1) Ej,−i z w . (6.4) i,j∈Z

The generating series EB(z,w)obeystherelations:EB(z,w)=−EB(w, z)and B B B B E (z1,w1),E (z2,w2) = −z2E (z1,w2)δ(w1 + z2)+w2E (z1,z2)δ(w1 + w2) B B + z2E (w1,w2)z1δ(z1 + z2) − w2E (w1,z2)δ(z1 + w2) − − − − c w1 z2 z1 w2 − c w2 z1 z2 w1 + iw1,z2 iz1,w2 iw2,z1 iz2,w1 4 w1 + z2 z1 + w2 4 z1 + w2 z2 + w1 − − − − − c w1 w2 z1 z2 c w2 w1 z2 z1 iw1,w2 iz1,z2 + iw2,w1 iz2,z1 . (6.5) 4 w1 + w2 z1 + z2 4 w1 + w2 z1 + z2 The proof of these is by standard calculations.

B Proposition 6.1. Let the ﬁeld φ (z) and the vector space FB be as in Example 5.1. Deﬁne

z − w Φ(z,w)=:φB(z)φB(w): − 1=φB(z)φB(w) − i . z,w z + w

B 1 → → ∞ The assignment E (z,w) 2 Φ(z,w), c IdFB gives a representation of the Lie algebra b .

Proof. −2z −2z −Φ(w, z)=−φB(w)φB(z)+i +1=φB(z)φB(w)+2wδ(z + w)+i +1 w,z z + w w,z z + w 2w 2w −2z = φB(z)φB(w)+i − i + i +1− Φ(z,w). z,w z + w w,z z + w w,z z + w I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2191

For the commutation relations of :φB(z)φB(w): one cannot apply Wick’s Theorem here, due to the fact B B B that φ0 φ0 =1(seeCorollary 5.4), i.e., φ (z)+ doesn’t anticommute with itself. We do direct computation using Φ(z,w): Φ(z1,w1),Φ(z2,w2) B B B B = φ (z1)φ (w1),φ (z2)φ (w2) B B B B B B B B = φ (z1) φ (w1),φ (z2) φ (w2) − φ (z1)φ (z2) φ (w1),φ (w2) B B B B B B B B + φ (z1),φ (z2) φ (w2)φ (w1) − φ (z2) φ (z1),φ (w2) φ (w1) B B B B = −2z2δ(w1 + z2)φ (z1)φ (w2)+2w2δ(w1 + w2)φ (z1)φ (z2) − 2z δ(z + z )φB(w )φB(w )+2w δ(z + w )φB(z )φB(w ) 2 1 2 2 1 2 1 2 2 1 z − w z − z = −2z δ(w + z ) Φ(z ,w )+i 1 2 +2w δ(w + w ) Φ(z ,z )+i 1 2 2 1 2 1 2 z1,w2 z + w 2 1 2 1 2 z1,z2 z + z 1 2 1 2 − − − w2 w1 z2 w1 2z2δ(z1 + z2) Φ(w2,w1)+iw2,w1 +2w2δ(z1 + w2) Φ(z2,w1)+iz2,w1 w2 + w1 z2 + w1

= −2z2δ(w1 + z2)Φ(z1,w2)+2w2δ(w1 + w2)Φ(z1,z2)

− 2w2δ(z1 + w2)Φ(w1,z2)+2z2δ(z1 + z2)Φ(w1,w2) − − − − w1 z2 z1 w2 − w2 z1 z2 w1 + iw1,z2 iz1,w2 iw2,z1 iz2,w1 w1 + z2 z1 + w2 z1 + w2 z2 + w1 − − − − − w1 w2 z1 z2 w2 w1 z2 z1 iw1,w2 iz1,z2 + iw2,w1 iz2,z1 . w1 + w2 z1 + z2 w1 + w2 z1 + z2

1 A direct comparison then shows that 2 Φ(z,w) indeed has the required commutation relations of EB(z,w). 2

6.2. The inﬁnite dimensional Lie algebra c∞

The inﬁnite dimensional Lie algebrac ¯∞ is the subalgebra ofa ¯∞ consisting of the inﬁnite matrices pre- i serving the bilinear form C(vi,vj)=(−1) δi,1−j , i.e., i+j−1 c¯∞ = (aij) ∈ a¯∞ aij =(−1) a1−j,1−i . (6.6)

The algebra c∞ is a central extension ofc ¯∞ by a central element c, c∞ =¯c∞ ⊕ Cc,withC the same cocycle as for a∞, (6.2) (see [23]). And the commutation relations for the elementary matrices in c∞ are

[Eij ,Ekl]=δjkEil − δliEkj + C(Eij,Ekl)c.

The generators for the algebra c∞ can be written in terms of these elementary matrices as: j i (−1) Ei,j − (−1) E1−j,1−i,i,j∈ Z;andc .

We can arrange the non-central generators in a generating series C j i i−1 −j E (z,w)= (−1) Eij − (−1) E1−j,1−i z w . (6.7) i,j∈Z

The generating series EC (z,w)obeystherelations:EC (z,w)=EC (w, z)and 2192 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

C C C C E (z1,w1),E (z2,w2) = E (z1,w2)δ(z2 + w1) − E (z2,w1)δ(z1 + w2) C C − E (w2,w1)δ(z1 + z2)+E (z1,z2)δ(w2 + w1) 1 1 − 1 1 +2ιz1,w2 ιw1,z2 c 2ιw2,z1 ιz2,w1 c z1 + w2 w1 + z2 w2 + z1 z2 + w1 1 1 − 1 1 +2ιz1,z2 ιw1,w2 c 2ιz2,z1 ιw2,w1 c. z1 + z2 w1 + w2 z2 + z1 w2 + w1

C Proposition 6.2. Recall the ﬁeld φ (z) on the highest weight module FC from Example 5.6. The assignment C C 1 → → ∞ E(z,w) i:φ (z)φ (w):, c 2 IdFC gives a representation of the Lie algebra c .(Herei is the imaginary unit.)

Proof. As a direct application of Wick’s Theorem (the conditions are easily checked to be satisﬁed) we have:

C C C C :φ (z1)φ (w1)::φ (z2)φ (w2): C C C C C C C C = φ (z1)φ (z2) :φ (w1)φ (w2): + φ (z1)φ (w2) :φ (z2)φ (w1): C C C C C C C C + φ (w1)φ (z2) :φ (z1)φ (w2): + φ (w1)φ (w2) :φ (z1)φ (z2): C C C C C C C C + φ (z1)φ (z2) φ (w1)φ (w2) + φ (z1)φ (w2) φ (w1)φ (z2) and hence C C C C i:φ (z1)φ (w1):,i: φ (z2)φ (w2): C C C C = −δ(z1 + z2):φ (w1)φ (w2): − δ(z1 + w2):φ (z1)φ (w2): + δ(z + w ):φC (z )φC (w ): + δ(w + w ):φC (z )φC (z ): 2 1 1 2 2 1 1 2 1 1 1 1 − ι ι − ι ι z1,z2 z + z w1,w2 w + w z1,w2 z + w w1,z2 w + z 1 2 1 2 1 2 1 2 1 1 1 1 + ιz2,z1 ιw2,w1 + ιz2,w1 ιw2,z1 . z2 + z1 w2 + w1 z2 + w1 w2 + z1

Note that we used (the rather peculiar) fact that for λ = −1wehaveδ(z1 +z2)=δ(z1, −z2)=−δ(z2 +z1)= −δ(z2, −z1). 2

6.3. The inﬁnite dimensional Lie algebra d∞

The inﬁnite dimensional Lie algebra d¯∞ is the subalgebra ofa ¯∞ consisting of the inﬁnite matrices preserving the bilinear form D(vi,vj)=δi,1−j , i.e., ¯ d∞ = (aij) ∈ a¯∞ aij = −a1−j,1−i . (6.8)

¯ ¯ ⊕ C 1 Denote by d∞ the central extension of d∞ by a central element c, d∞ = d∞ c, where we use 2 C (from (6.2)) as a cocycle for d∞,see[23]. The commutation relations for the elementary matrices in d∞ is

1 [E ,E ]=δ E − δ E + C(E ,E )c. ij kl jk il li kj 2 ij kl

The generators for the algebra d∞ can be written in terms of these elementary matrices as:

{Ei,j − E1−j,1−i,i,j∈ Z;andc}. I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2193

We can arrange the non-central generators in a generating series D i−1 −j E (z,w)= (Eij − E1−j,1−i)z w . (6.9) i,j∈Z

The generating series ED(z,w)obeystherelationsED(z,w)=−ED(w, z)and D D D D E (z1,w1),E (z2,w2) = E (z1,w2)δ(z2 − w1) − E (z2,w1)δ(z1 − w2) D D + E (w2,w1)δ(z1 − z2) − E (z1,z2)δ(w1 − w2) 1 1 − 1 1 + ιz1,w2 ιw1,z2 c ιz2,w1 ιw2,z1 c z1 − w2 w1 − z2 z2 − w1 w2 − z1 − 1 1 1 1 ιz1,z2 ιw1,w2 c + ιz2,z1 ιw2,w1 c. z1 − z2 w1 − w2 z1 − z2 w2 − w1

D Proposition 6.3. Recall the ﬁeld φ (z) on the highest weight module FD from Example 5.8. The assignment → D D → E(z,w) :φ (z)φ (w):, c IdFD gives a representation of the Lie algebra d∞.

7. Twisted vertex algebras

In this section we finally answer the following question: if we start with a given number of generating fields i a (z) (see a precise definition at 7.10), which are local at several but finitely many points λ1,λ2,...,λN (also i satisfying some auxiliary conditions), and we allow the operations of differentiation (∂za (z)), substitutions i at the locality points (a(λjz)), normal ordered products (:a (λkz)aj(λlz):)andOPEcoefficientsonall those fields and their descendants; and we further assume that all those descendants are still only local at λ1,λ2,...,λN ; then what is the resulting structure that we get? The first observation is, as we proved in Section 4, that in order for the descendants to stay local at finitely many points, then the points of locality have to form a multiplicative group. The only finite multiplicative subgroups of the complex plane C are the roots of unity cyclic groups. Hence if we want to have the total number of points of locality stay finite, the only possible choice for the N points of locality are the N-th roots of unity. Then the answer to our “generating descendants” problem is: the fields ai(z) will generate a twisted vertex algebra. Twisted vertex algebras of general order N were defined in [3] (of order 2 in [2]) in order to answer the question: what are the boson–fermion correspondences of type B, C and D-A? These correspondences must be isomorphisms between some type of structures, and we wanted to understand those structures. Although the boson–fermion correspondence of type A is an isomorphism of super vertex algebras (which are 1-point local), the boson–fermion correspondences of type B, C and D-A are isomorphisms of twisted vertex algebras (which are at least 2-point local). Before giving the definition of a twisted vertex algebra, we want to mention one other very important difference between super vertex algebras and twisted vertex algebras, and it is that the space of fields in a twisted vertex algebra is larger than the space of states (unlike in super vertex algebras where there is a state-field bijection); and by definition there is a strict surjective projection from the space of fields to the space of states in a twisted vertex algebra. In this, twisted vertex algebras resemble the concept of a deformed chiral algebras from [21], and differ from the concept of a Γ -vertex algebra of [29]. The Γ -vertex algebras were introduced in [29], and are indeed multilocal, but the number of fields included in a Γ -vertex algebra is not enough to describe the examples of the boson–fermion correspondences for which the concept of a twisted vertex algebra was introduced (see Appendix A).

We will work with the category of super vector spaces, i.e., Z2 graded vector spaces. The ﬂip map τ is deﬁned by

˜ τ(a ⊗ b)=(−1)a˜·b(b ⊗ a) (7.1) 2194 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 for any homogeneous elements a, b in the super vector space, wherea ˜ or p(a), ˜b or p(b) denote correspondingly the parity of a, b.

Deﬁnition 7.1 (The Hopf algebra HD = C[D]). The Hopf algebra HD = C[D] is the universal enveloping algebra of the commutative one-dimensional Lie algebra with even generator D, i.e., the polynomial algebra with a primitive generator D.Wehave i D(n) = D(k) ⊗ D(l). (7.2) k+l=n

Deﬁnition 7.2 (The Hopf algebra HN ). Let be a primitive root of unity of order N. The Hopf algebra HN T T is the Hopf algebra with a primitive generator D and a grouplike generator T subject to the relations:

n DT = TD, and (T) =1 (7.3)

HN has H as a Hopf subalgebra. Both H and HN are entirely even. T D D T

N Let be a primitive Nth root of unity. Denote by F (z,w) the space of rational functions in the formal variables z,w with only poles at z =0,w =0,z = iw, i =1,...,N. In other words N C −1 −1 − −1 − −1 − N−1 −1 F (z,w)= [z,w] z ,w , (z w) , (z w) ,..., z w . (7.4)

N +,w Also, denote F (z,w) the space of rational functions in the formal variables z,w with only poles at z =0, i N +,w ∈ N z = w, i =1,...,N. Note that we do not allow a pole at w =0inF (z,w) , i.e., if f(z,w) F (z,w), then f(z,0) is well deﬁned. If N is clear from the context, or the property doesn’t depend on the particular +,w value of N, we will just write F(z,w)orF(z,w) . N More generally, let F (z1,z2,...,zl) be the space of rational functions in variables z1,z2,...,zl with only k poles at zm =0,m =1,...,l,oratzi = zj,andi = j,1 k N. In other words N C | −1 − k −1 F (z1,z2,...,zl)= [zi 1 i l] zm , zi zj 1 m l, i = j . (7.5)

N +,zl F (z1,z2,...,zl) is the space of rational functions in variables z1,z2,...,zl with only poles at zm =0, k m =1,...,l− 1, or at zi = zj,fori = j.

Proposition 7.3. FN (z,w) and F (z,w)+,w are HN ⊗ HN (and consequently an H ⊗ H ) Hopf modules T T D D by

Dzf(z,w)=∂zf(z,w), (T)zf(z,w)=f(z, w), (7.6)

Dwf(z,w)=∂wf(z,w), (T)wf(z,w)=f(z,w). (7.7)

We will denote the action of elements h ⊗ 1 ∈ HN ⊗ HN on F (z,w) (and F (z,w)+,w)byh ·, correspond- T T z ingly h · will denote the action of the elements 1 ⊗ h ∈ HN ⊗ HN . w T T

Deﬁnition 7.4 (Twisted vertex algebra of order N). A twisted vertex algebra structure of order N is a collection of the following data (V,W,π,Y ):

• the space of ﬁelds V : a vector super space and an HN module graded as an H -module; T D • the space of states W : a vector super subspace of V ;

• a linear surjective projection map π : V → W , such that π|W = IdW I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2195

• a ﬁeld-state correspondence Y : a linear map from V to the space of ﬁelds on W ; • a vacuum vector: a vector |0 ∈W ⊆ V .

Satisfying the following set of axioms:

• vacuum axiom: Y (1,z)=IdW ; • modiﬁed creation axiom: Y (a, z)|0 |z=0 = π(a), for any a ∈ V ; • transfer of action: Y (ha, z)=h · Y (a, z) for any h ∈ HN ; z T • analytic continuation: For any a, b, c ∈ V exists Xz,w,0(a ⊗ b ⊗ c) ∈ W [[ z,w]] ⊗ F(z,w) such that

Y (a, z)Y (b, w)π(c)=iz,wXz,w,0(a ⊗ b ⊗ c);

• symmetry: Xz,w,0(a ⊗ b ⊗ c)=Xw,z,0(τ(a ⊗ b) ⊗ c); • Completeness with respect to Operator Product Expansions (OPE’s): For each i ∈ 0, 1,...,N−1, k ∈ Z, ∈ s ∈ Z and any a, b, c V , where a, b are homogeneous with respect to the grading by D, there exist lk,i , | s | − lk,i (N 1)(k + 1), such that

ﬁnite k s ⊗ ⊗ − i lk,i s Resz=iwXz,w,0(a b c) z w = w Y vk,i,w π(c) (7.8) s

s ∈ s ∈ Z for some homogeneous elements vk,i V , lk,i .

Remark 7.5. If V is an (ordinary) super vertex algebra, then the data (V,V,π = IdV ,Y) is a twisted vertex algebra of order 1. Moreover, it can be proved that a twisted vertex algebra (V,V,π = IdV ,Y) of order 1 is | s | − s a super vertex algebra, due to the restrictions lk,i (N 1)(k +1)intheshiftslk,i.

Remark 7.6 (Shift restriction). The axiom/property requiring completeness with respect to the Operator Product Expansions (OPE’s) is a weaker one than in the classical vertex algebra case. We can express this weaker axiom as follows. Any field v(w) is characterized by first, the doubly infinite sequence of the operators representing its modes (see Remark 4.15); and second, by the indexing of that doubly infinite sequence. One can shift the indexing of this sequence (i.e., place the 0 index at different modes), and each shift in the indexing of the sequence of modes corresponds to a multiplication by an integer power of w − − of the field v(w). For example, the OPE coefficient in (5.1) of Example 5.1 is 2w = 2wIdFB , (recall B B ∼ −2w φ (z)φ (w) z+w ). As a doubly-infinite sequence of modes, this is the sequence

(...,0,...,0, 0|index 0, −2, 0, 0, 0,...).

The distribution −2w is a ﬁeld, but is not a vertex operator in any vertex algebra. But if we shift the place of the 0-index, to

(...,0,...,0, 0, −2|index 0, 0, 0, 0,...),

− − − | this now represents the ﬁeld 2= 2IdFB , which is the vertex operator assigned to the element 2 0 .In other words, if we disregard the indexing place, but just consider the doubly-inﬁnite sequence of modes

(...,0,...,0, 0, −2, 0, 0, 0,...) it is represented in the twisted vertex algebra. The changing of the indexing place corresponds to multiplying − − −1 the original ﬁeld 2w = 2wIdFB by a factor of w in this case. This is then the role of the vertex map 2196 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

Y : in this case the map Y maps the field −2wIdW ∈ V to the vertex operator Y (−2|0 ,w)=−2IdW (here W = FB). The shift restriction that the completeness with respect to OPE’s axiom implies is that OPE’s ∼ −2w+w2 such as a(z)a(w) z+w , for example, are not allowed, as we wouldn’t be able to re-index the sequence of modes. In any physical model of quantum field theory in two dimensions it is very important that we are able to recover the doubly-infinite sequence of operators. In an (ordinary) super vertex algebra we have a stronger property, there the products of fields automatically are vertex operators in the same vertex algebra. This stronger property cannot hold in certain interesting examples, as one clearly sees in Example 5.1.The “modified completeness with respect to OPE’s” axiom is in some sense the weakest requirement one can impose.

Lemma 7.7 (Analytic continuation and symmetry imply N-point locality). Let (V,W,π,Y ) be a twisted vertex algebra of order N. Then for any a, b, c ∈ V there exists M such that M zN − wN Y (a, z)Y (b, w) − (−1)p(a)p(b)Y (b, w)Y (a, z) π(c)=0, (7.9) i.e., the ﬁelds Y (a, z), Y (b, w) are N-point local with points of locality 1,,...,N−1,where is a primitive N-th root of unity.

Proof. Since −1 −1 −1 N−1 −1 Xz,w,0(a ⊗ b ⊗ c) ∈ W [[ z,w]] ⊗ F(z,w)=W [[ z,w]] z ,w , (z − w) ,..., z − w , then exists M such that N N M −1 −1 z − w Xz,w,0(a ⊗ b ⊗ c) ∈ W [[ z,w]] z ,w .

Since in W [[ z,w]] [ z−1,w−1] the two expansions do not diﬀer, N N M N N M ιz,w z − w Xz,w,0(a ⊗ b ⊗ c) = ιw,z z − w Xz,w,0(a ⊗ b ⊗ c) .

But then from the symmetry axiom N N M N N M ιz,w z − w ιz,wXz,w,0(a ⊗ b ⊗ c)=ιw,z z − w ιw,zXz,w,0(a ⊗ b ⊗ c) N N M = ιw,z z − w ιw,zXw,z,0 τ(a ⊗ b) ⊗ c , from which follows that N N M N N M p(a)p(b) ιz,w z − w Y (a, z)Y (b, w)π(c)=ιw,z z − w (−1) Y (b, w)Y (a, z)π(c). 2

In fact, the converse lemma is also true:

Lemma 7.8 (N-point locality implies analytic continuation and symmetry). Let a(z), b(w) be ﬁelds on a vector space W which are N-point local with points of locality 1,,...,N−1,where is a primitive Nth root of unity. Then there exists Xz,w,0 : V ⊗ V ⊗ V → W [[ z,w]] ⊗ F(z,w) such that

a(z)b(w)π(c)=ιz,wXz,w,0(a ⊗ b ⊗ c); and Xz,w,0(a ⊗ b ⊗ c)=Xw,z,0(τ(a ⊗ b) ⊗ c). Moreover, Xz,w,0(a ⊗ b ⊗ c) is the unique such element in W [[ z,w]] ⊗ F(z,w) for any a, b ∈ V , π(c) ∈ W . I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2197

N N Proof. Let Πz,w = z − w .Let

⊗ ⊗ M − p(a)p(b) M Az,w(a b c)=ιz,wΠz,wY (a, z)Y (b, w)π(c)=( 1) ιw,zΠz,wY (b, w)Y (a, z)π(c), which holds from N-point locality. Thus q y −1 −1 Az,w(a ⊗ b ⊗ c) ∈ W (w) (z) ∩ W (z) (w) = W z,w z ,w .

Let then

⊗ ⊗ −M ⊗ ⊗ ∈ ⊗ Xz,w,0(a b c)=Πz,w Az,w(a b c) W [[ z,w]] F(z,w).

Then

⊗ ⊗ −M ⊗ ⊗ ιz,wXz,w,0(a b c)=ιz,wΠz,w Az,w(a b c) as required. Since we can also start from

⊗ ⊗ M Aw,z(b a c)=ιw,zΠz,wY (b, w)Y (a, z)π(c), we see that p(a)p(b) Az,w(a ⊗ b ⊗ c)=(−1) Aw,z(b ⊗ a ⊗ c)=Aw,z τ(a ⊗ b) ⊗ c , which symmetry transfers immediately to Xz,w,0(a ⊗ b ⊗ c). 2

These two lemmas imply that in the definition of twisted vertex algebra we can substitute the axioms of analytic continuation and symmetry with a single axiom of N-point locality. The axiom requiring completeness with respect to OPE’s is though much more easily expressible if we use the analytic continuation of fields (recall we have two different expressions for products of fields if we use the fields themselves and not the analytic continuation—see Lemma 4.20):

Lemma 7.9. Let (V,W,π,Y ) be a twisted vertex algebra of order N. Then for any a, b, c ∈ V . We have i k Resz=j wXz,w,0(a ⊗ b ⊗ c) z − w = Y (a, w)(j,k)Y (b, w)π(c), where the products Y (a, w)(j,k)Y (b, w) were the (i, k) products of the ﬁelds Y (a, z) and Y (b, w) we deﬁned in 4.18.

Proof. From the OPE of the ﬁelds Y (a, z)andY (b, w) (which follows from locality), we get from Lemma 4.10:

− N nj 1 Y (a, w)(j,k)Y (b, w) Y (a, z)Y (b, w)=iz,w k+1 +:a(z)b(w):. (z − λjw) j=1 k=0

From the analytic continuation property (speciﬁcally the uniqueness property of 7.8):

n −1 N j Y (a, w) Y (b, w)π(c) ⊗ ⊗ (j,k) Xz,w,0(a b c)= k+1 +:a(z)b(w):π(c). (z − λjw) j=1 k=0 2198 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

From 4.17 we have − (k) − k iw,(z−λj w):a λjw +(z λjw) b(w):= : ∂λw a(λw) b(w): (z λjw) . k0

i k Thus, the OPE part contains the residues Resz=j wXz,w,0(a ⊗ b ⊗ c)(z − w) for k 0, and the other i k residues Resz=j wXz,w,0(a ⊗ b ⊗ c)(z − w) for k<0 are contained in the normal ordered product part. Thus we clearly have i k Resz=j wXz,w,0(a ⊗ b ⊗ c) z − w = Y (a, w)(j,k)Y (b, w)π(c). 2

Definition 7.10 (Generating fields for a twisted vertex algebra of order N). We say that the fields {a0(z),a1(z),...,ag(z),...} are a set of generating fields for a twisted vertex algebra on a vector space of states W (where W contains the special vacuum vector |0 ) if the following conditions are satisfied:

0 0 • The field a (z)=IdW (i.e., the identity operator on W is always included, by convention as a (z)); i i i −k−1 i | | •Eachfielda (z), indexed by a (z)= n∈Z a(k)z , obeys the modified creation axiom: a (z) 0 z=0 = i | i i | ∈ a−1 0 ;wedenotea := a−1 0 W ; • The fields {a0(z),a1(z),...,ag(z),...} are homogeneous, self and mutually N-point local with points of locality 1,,...,N−1, where is a primitive Nth root of unity; • The set of fields {a0(z),a1(z),...,ag(z),...} is closed under OPE’s, which requires two conditions: (1) For any i, j =1,...,g,...

N M cij (w) ai(z)aj (w) ∼ mp , (7.10) (z − mw)p+1 m=1 p=0

where the OPE coefficients are uniformly shifted. We say that the OPE coefficients are uniformly g m,p m,p ∈ Z ij sp l ∈ C shifted if for any m, p,existssp , such that cmp(w)=w l=0 Ci,j,l a (w), where Ci,j,l are constants; but most importantly sp ∈ Z is independent of the choice of i, j, m (the shift sp is the same for any choice of the fields ai(z), aj(w) and any point of locality m). 1 i,jk ij,k (2) For any i, j =1,...,g,... the resulting OPE triple coefficients cmp (w)andcmp (w) are also uni- ij i,jk ij,k formly shifted and the shift sp is the same for each OPE coefficient cmp(w), cmp (w)andcmp (w)for any choice of the fields ai(z), aj(z), ak(z) and any point of locality m. i1 i2 ik • W = span{a a ...a |0 |i1,i2,...,ik ∈{1, 2,...,p}; ni ∈ N}. (−n1) (−n2) (−nk)

Remark 7.11. The indexing set for the set of the generating fields can be infinite, though the sum in the definition of uniformly shifted fields has to be finite.

Now we set up what are necessary preliminary results needed to prove the “Strong Generation Theorem” for a twisted vertex algebra. If we start with a set of generating ﬁelds {a0(z),a1(z),...,ag(z),...} on a vector space W , we will take as the super vector space V for the twisted vertex algebra the collection

FD{a0(z),a1(z),...ag(z),...} of the field descendants of the generating fields (see Definition 4.21). We need to define the vertex map Y and the projection map π from the definition of a twisted vertex algebra. We start with the vertex map Y . First, to the generating fields, which by the definition above are indexed as required, we associate the same vertex operator (no shift is necessary): Y : ai(z) → Y ai,z ,i=0, 1,...,g,...

1 Recall we deﬁned triple OPE coeﬃcients in Lemma 4.27. I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2199

For their descendants, we proceed as follows: i i i Y : ∂za (z) → Y Da ,z := ∂zY a ,z ,i=0, 1,...,g,...; m Y : ai mz → Y T ai ,z := Y ai,mz ,i=0, 1,...,g,..., m=0, 1,...,N − 1.

i j For the products of the generating ﬁelds: for the normal ordered products a (w)(m,k)a (w)(i, j = 0, 1,...,g,..., m =1,...,N, k<0), i.e., for the products with negative k, each normal ordered product maps to itself via the vertex map Y : i j i j Y a(m,k)a ,w := a (w)(m,k)a (w),i,j=0, 1,...,g, m=1,...,N, k <0.

i j i j (Here we write a(m,k)a as a shorthand notation for a (w)(m,k)a (w)). i j For the OPE coeﬃcients, i.e., for the products a (w)(m,k)a (w)withk 0, we have

g g i j m,k l m,k l Y a(m,k)a ,w := Ci,j,l Y a ,w = Ci,j,l a (w),m=1,...,N, k 0. l=0 l=0

Here we used the fact that for the generating fields {a0(z),a1(z),...,ag(z),...} the OPE coefficients are uniformly shifted,seeDefinition 7.10 above. We can proceed recursively as above for all the newly generated descendants in FD{a0(z),a1(z),...,ag(z)}, as the uniformly-shifted property holds for all the descendants by Lemma 4.27. Thus we can formulate an answer to the question that we started with in this paper:

Theorem 7.12 (Strong generation of a twisted vertex algebra). Let {a0(z),a1(z),...,ap(z)} be a set of generating ﬁelds for a twisted vertex algebra on a vector space of states W . Then the data (V,W,π,Y ) forms a twisted vertex algebra, where V = FD{a0(z),a1(z),...,ag(z)},themapY is deﬁned as above, and the projection map π is given by

π(v):=Y (v, z)|0 |z=0, for any v ∈ V.

Proof. Since the space of fields V is defined recursively by adding new descendants to V = FD{a0(z),a1(z), ...,ag(z)} (see Definition 4.21), we have to prove that in each recursive step the new descendant fields obey the axioms. By Dong’s Lemma all the new descendants are N-point local, from Lemma 7.8 it follows they satisfy the analytic continuation and the symmetry axioms. From Lemma 7.9 we know that the required residues are either OPE coefficients, or the normal ordered products defined in 4.20. The only property we need to prove is that the uniformity of the shift persists for the descendants. The uniformity follows from Lemma 4.27 and the definition of generating fields, as the OPE coefficients are finite linear combinations of the generating fields themselves. 2

There are a variety of different field theories, each of those theories is designed to describe different sets of examples of collections of fields. The best known is the theory of super vertex algebras (see e.g. [20,24,31, 7,17]). Twisted vertex algebras occupy intermediate step between super vertex algebras and the deformed chiral algebras of [21]. We choose here to define twisted vertex algebras with axioms closer to the deformed chiral algebra axioms, including the analytic continuation axiom. But twisted vertex algebras are in fact closer to super vertex algebras, in many ways. For one, they satisfy N-point locality (finitely many points of locality), unlike the deformed chiral algebras which have lattices of points of locality. Also, like in super vertex algebras, the axioms requiring existence of analytic continuation of product of two vertex operators plus the symmetry axiom do in fact enforce the property that analytic continuation of arbitrary product of fields exist; something that is not true for deformed chiral algebras (see [21]). We finish with that property: 2200 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

Proposition 7.13 (Analytic continuation for arbitrary products of ﬁelds). Let (V,W,π,Y ) be a twisted vertex algebra. For any ai ∈ V , i =1,...,k, there exist a rational vector valued function

⊗ k → ⊗ N +,zk Xz1,z2,...,zk : V W [[ z1,z2,...,zk]] F (z1,z2,...,zk) , such that

⊗ ⊗···⊗ Y (a1,z1)Y (a2,z2) ...Y(ak,zk)1 = iz1,z2,...,zk Xz1,z2,...,zk (a1 a2 ak).

Proof. The proof is by induction on the number of products k.Fork = 2 we have the axiom of analytic ∈ ⊗ ⊗ ∈ ⊗ continuation: for any a1,a2,c V exists Xz1,z2,0(a1 a2 c) W [[ z1,z2]] F(z1,z2) such that

⊗ ⊗ Y (a1,z1)Y (a2,z2)π(c)=iz1,z2 Xz1,z2,0(a1 a2 c).

If we take c =1=|0 we have the desired

⊗ ⊗ ⊗ ∈ ⊗ Xz1,z2 (a1 a2):=Xz,w,0(a1 a2 1) W [[ z1,z2]] F(z1,z2).

Suppose the analytic continuation property holds for a product of k−1 ﬁelds, and consider Y (a1,z1)Y (a2,z2) ...Y(ak,zk)1. From Lemma 7.7 we know that the ﬁelds Y (a1,z1),Y(a2,z2),...,Y(ak,zk)are N-point M k N − mutually local. Thus there exists a high enough power M so that if we multiply by Πz1,z2,...,zk := i

⊗···⊗ Y (a1,z1)Y (a2,z2) ...Y(ak,zk)1 = iz2,...,zk Y (a1,z1)Xz2,...,zk (a2 ak).

⊗ N +,zk Let Fz2,...,zk = W [[ z2,...,zk]] F (z2,...,zk) .Weseethat ⊗···⊗ ∈ Y (a1,z1)Xz2,...,zk (a2 ak) Fz2,...,zk (z1) .

But also

M Πz1,z2,...,zk Y (a1,z1)Y (a2,z2) ...Y(ak,zk)1 M − − = Πz1,z2,...,zk Y (a2,z2) ...Y(ak 1,zk 1)Y (ak,zk)Y (a1,z1)1.

But from the axioms of vertex algebra we know that Y (a1,z1)1 is regular in z1 (has no negative powers of z1). Thus we see that the last line has no pole in z1. But that forces

M ⊗···⊗ ∈ Πz1,z2,...,zk Y (a1,z1)Xz2,...,zk (a2 ak) Fz2,...,zk [[ z1]] ; and thus the required expansion

⊗ ⊗···⊗ Y (a1,z1)Y (a2,z2) ...Y(ak,zk)1 = iz1,z2,...,zk Xz1,z2,...,zk (a1 a2 ak) existsforanelement

⊗ ⊗···⊗ ∈ ⊗ N +,zk 2 Xz1,z2,...,zk (a1 a2 ak) W [[ z1,z2,...,zk]] F (z1,z2,...,zk) . I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203 2201

Appendix A. Comparison of twisted vertex algebras and Γ -vertex algebras

In Section 4 we defined products of fields a(w)(j,k)b(w) for any j =1, 2,...,N and k ∈ Z.Fork 0these products were the coefficients of the OPEs, and for k<0 they were the normal ordered products. As we mentioned in Section 4, in the paper [29] Li also defines products of fields for “compatible” pairs of vertex operators (see Definition 3.4 in [29]). We provide examples to demonstrate that the products of fields differ in our approach. We will use the products we calculated in Example 5.1 to compare them to the products according to B B Definition 3.4 in [29]. We will denote the products in [29] by φ (w)(α,k)φ (w). According to [29] the pair of B B fields (φ (x1),φ (x2)) is compatible by the polynomial f(x1,x2)=x1 + x2, since we have from the OPE (5.1)

B B B B (x1 + x2)φ (x1)φ (x2)=−2x2 +(x1 + x2):φ (x1)φ (x2):,

B B and :φ (x1)φ (x2): ∈ Hom(FB, FB((x1,x2))). Thus we calculate B B − B B (x1 + x)φ (x1)φ (x) x =x+x = 2x +(2x + x0) :φ (x + x0)φ (x): 1 0 − k (k) B B = 2x +(2x + x0) x0 : ∂x φ (x) φ (x):. k0

Next, −1 B B ix,x0 (x0 + x + x) (x1 + x)φ (x1)φ (x) x =x+x 1 0 (−1)lxl = −2x 0 + xk: ∂(k)φB(x) φB(x): (2x)l+1 0 x l0 k0 1 = −1+:φB(x)φB(x): + x +: ∂ φB(x) φB(x): + O x2 . 0 2x x 0

Thus, as we commented in Section 4, since we have a single pole in the OPE (at z = −w)hereourpositive (the OPE) products coincide with the [29] in that we have

B B B B φ (x)(1,k)φ (x)=φ (x)(1,k)φ (x)=0 fork 0.

On the other hand, our negative (normal ordered products) diﬀer from [29], for instance

B B B B B B − B B φ (x)(1,−1)φ (x)=:φ (x)φ (x): = 1 vs φ (x)(1,−1)φ (x)= 1+:φ (x)φ (x): = 0; B B B B B B 1 B B φ (x) − φ (x)=: ∂ φ (x) φ (x): vs φ (x) φ (x)= +: ∂ φ (x) φ (x):. (1, 2) x (1,−2) 2x x

For λ = −1(α = −1in[29]), a similar calculation shows that in this case our products coincide with [29]. The reason is the single pole in the OPE precisely at α = −1. We ﬁnish with a calculation of an example of products of [29] in the case of multiple poles in the OPE, D D and we will take as an example the OPE from (5.19). The pair of ﬁelds (h (x1),h (x2)) is compatible by 2 − 2 2 the polynomial f(x1,x2)=(x1 x2) ,thus 2 − 2 2 D D 2 − 2 2 D D x1 x h (x1)h (x) x =x+x =(x + x0)x + x1 x :h (x + x0)h (x): 1 0 2 2 k (k) D D =(x + x0)x + 2xx0 + x0 x0 : ∂x h (x) h (x):. k0 2202 I.I. Anguelova et al. / Journal of Pure and Applied Algebra 218 (2014) 2165–2203

D D For the ﬁrst three nontrivial products (h (x)(1,k)h (x)=0fork 2) we have

1 hD(x) hD(x)= = hD(x) hD(x), (1,1) 4 (1,1) 1 hD(x) hD(x)= vs hD(x) hD(x)=0, (1,0) 8x (1,0)

D D 1 D D 1 D D h (x) h (x)=− +:h (x)h (x): = − + h (x) − h (x). (1,−1) 16x2 16x2 (1, 1)

Hence products in [29] differ from ours, except at the highest order of the pole. As above, we can in general relate the products in [29] to ours, and vice versa, however the algebraic structures incorporating these fields are not equivalent. In [29] the author defines the notion of a Γ -vertex algebra, which is an algebraic structure incorporating certain collections of multi-local fields. Γ -vertex algebras however do not incorporate the examples we want to consider, namely the boson–fermion correspondences of types B, C and D-A, which motivated the definition of a twisted vertex algebra in [3]. Twisted vertex algebra is a related, but different structure from both the notion of a Γ -vertex algebra as defined in [29], and the concept of a twisted module of a super vertex algebra (as considered for instance in [20,14,15,5]). In general, for a finite cyclic group Γ , Γ -vertex algebras are described by smaller collections of descendant fields. The examples of Γ -vertex algebras in [29] can be incorporated as subspaces in the corresponding spaces of fields of the related twisted vertex algebras. In comparison to the notion of a (twisted) module of a super vertex algebra, the multi-local vertex operators in the twisted vertex algebra can be related to twisted vertex operators (as in [20]), after appropriate change-of-variables transformation changing twisted fields with fractional powers to fields with integer powers. But in the examples of twisted vertex algebras the vertex operators often act on a space of states that can be decomposed into two or more (twisted) modules of a suitable super vertex algebra; moreover such examples of twisted vertex algebras also incorporate (some of the) intertwining vertex operators between those modules (after appropriate change-of-variables). The twisted vertex algebras defined in [3] and studied here were motivated by the boson–fermion correspondences and were designed to describe the algebraic structure of such systems of vertex operators in its entirety.

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