An approach to Quantum Conformal Algebra
aa r X i v : . [ m a t h . QA ] D ec An approach to Quantum Conformal Algebra
Carina Boyallian and Vanesa Meinardi ∗ Abstract
We aim to explore if inside a quantum vertex algebras, we can find theright notion of a quantum conformal algebra.
Since the pioneering papers [BPZ, Bo1], there has been a great deal of worktowards understanding of the algebraic structure underlying the notion of theoperator product expansion (OPE) of chiral fields of a conformal field theory.The singular part of the OPE encodes the commutation relations of fields, whichleads to the notion of a conformal algebra [K1].In [BK], they develop foundation of the theory of field algebras, which are a“non-commutative version” of a vertex algebra. Among other results they showthat inside certain field algebras, more precisely strong field algebras ( where the n -product axiom holds) we have a conformal algebra and a diferential algebratoghehter with certain compatibility equations, and conversely, having this twostructures plus those equations we can recover a strong field algebra. One ofthese equations is the conformal analog of the Jacobi Identiy. They call aconformal algebra satisfying this equation Leibnitz conformal algebra.A definition of a quantum vertex algebra, which is a deformation of a ver-tex algebra, was introduced by Etingof and Kazhdan in 1998,[EK]. Roughlyspeaking, a quantum vertex algebra is a braided state-field correspondencewhich satisfies associativity and braided locality axioms. Such braiding is aone-parameter braiding with coefficients in Laurent series.Recently in [DGK], they developed a structure theory of quantum vertex al-gebras, parallel to that of vertex algebras. In particular, they introduce braidedn-products for a braided state-field correspondence and prove for quantum ver-tex algebras a version of the Borcherds identity.Following [BK], in this article, we try to determine the quantum analog ofthe notion of conformal algebra inside a quantum vertex algebra V . For thispurpose, we introduced new products parametrized by Laurent polinomials f ,and we showed that all this products are determined by those corresponding ∗ Ciem - FAMAF, Universidad Nacional de C´ordoba - (5000) C´ordoba, Argentina < [email protected] - [email protected] > . = 1 and f = z − . The case f = 1 coincides with the λ -product defininga conformal algebra([K1],[BK]). This allows us to deal with the coefficientsof the braiding in V . An important remark is that V together with the λ -product is no longer a Leibnitz conformal algebra, since due to the braiding,the analog of the Jacoby identity involves not only the products correspondingto f = 1 (as in [BK]), but those of f = z − . We translate to this language thehexagon axiom, quasi-associativity and associativity relations, and the braidedskew-symmetry in a quantum vertex algebra, and all this allows us to give anequivalent definition of quantum vertex algebra and present a candidate of aquantum conformal algebra.The article is organized as follows. In Section 2 we review all the definitionsand basic notion of field algebras and braided field algebras. In Section 3 weintroduce the ( λ, f )-product and prove some of its properties and we finishthe section proving in Theorem 3 that shows that having a strong braidedfield algebra is the same of having a conformal algebra, a differential algebrawith unit with some compatibility equations. In Section 4, we translate thehexagon axiom, quasi-associativity, and associativity relations, and the braidedskew-symmetry in a quantum vertex algebra, we give an equivalent definitionof quantum vertex algebra and present a candidate of a quantum conformalalgebra. In this section review some basic definitions followig [BK],[DGK]. Throughoutthe paper all vector spaces, tensor products,etc are over a field K of characteristiczero, unless otherwise specified. Given a vector space V , we let V [[ z, z − ]] be the space of formal power serieswith coefficients in V ; they are called formal distributions . A qauntum field over V is a formal distribution a ( z ) ∈ (End V )[[ z, z − ]] with coefficients in End V, suchthat a ( z ) v ∈ V (( z )) for every v ∈ V. Hereafter V (( z )) = V [[ z ]][ z − ] stands forthe space of Laurent series with coefficients in V. Throughout the article ι z,w (resp ι w,z ) denotes the geometric series expansionin the domain | z | > | w | (resp | w | > | z | ), namely we set for n ∈ Z ,ι z,w ( z + w ) n = X l ∈ Z + (cid:18) nl (cid:19) z n − l w l where (cid:18) nl (cid:19) = n ( n − · · · ( n − l + 1) l ! . For an arbitrary formal distribution a ( z ) , we haveRes z ( a ( z )) = a − , (1)2hich is the coefficient of z − . Denote by gl f ( V ) the space of all End V -valuedfields. We also need the Taylor’s Formula (cf. Proposition 2.4,[K1]), namely, ι z,w a ( z + w ) = X j ∈ Z + ∂ jz j ! a ( z ) w j = e w∂ z a ( w ) . (2)For each n ∈ Z one defines the n -th product of fields a ( z ) and b ( z ) by thefollowing formula: a ( z ) ( n ) b ( z ) = Res x ( a ( x ) b ( z ) ι x,z ( x − z ) n − b ( z ) a ( x ) ι z,x ( x − z ) n ) . (3)Denote by a ( z ) + = X j ≤− a ( j ) z − j − , a ( z ) − = X j ≥ a ( j ) z − j − . In this subsection we recall the definition of a field algebra, conformal algebrasand its properties following [BK]A state-field correspondence on a pointed vector space ( V, | i ) is a linear map Y : V ⊗ V → V (( z )) , a ⊗ b → Y ( z )( a ⊗ b ) satisfying(i) (vacuum axioms ) Y ( z )( | i ⊗ a ) = a, Y ( z )( a ⊗ | i ) ∈ a + V [[ z ]] z ;(ii) (translation covariance)[ T, Y ( z )]( a ⊗ b ) = ∂ z Y ( z )( a ⊗ b ),(iii) Y ( z )( T a ⊗ b ) = ∂ z Y ( z )( a ⊗ b ),where T ( a ) := ∂ z ( Y ( z )( a ⊗ | i )) | z =0 = a ( − | i , is called the translation opera-tor.Note that we will also denote by Y the map Y : V → End V [[ z, z − ]] , a Y ( a, z ) = P k ∈ Z a ( k ) z − k − , such that Y ( a, z ) b = Y ( z )( a ⊗ b ) . Note that Y ( a, z ) is a quantum field, i.e Y ( a, z ) b ∈ V (( z )) for any b ∈ V. The following results, proved in [BK], will be usefull in the sequel.
Proposition 1. (cf. [BK], Prop.2.7). Given Y : V ⊗ V → V (( z )) satisfayingconditions (i) and (ii) above, we have:(a) Y ( z )( a ⊗ | i )) = e zT a ; (b) e wT Y ( z )(1 ⊗ e − wT ) = ι z,w Y ( z + w ) . If, moreover, Y is a state-field correspondence, then(c) Y ( z )( e wT ⊗
1) = ι z,w Y ( z + w ) . Y, define Y op ( z )( u ⊗ v ) = e zT Y ( − z )( v ⊗ u ) . (4)Then Y op is also a state-field correspondence, called the opposite to Y. (cf. [BK],Prop 2.8).Let ( V, | i ) be a pointed vector space and let Y be a state-field correspon-dence. Recall that Y satisfies the n -th product axiom if for all a, b ∈ V and n ∈ Z Y ( z )( a ( n ) b, z ) = Y ( z ) ( n ) Y ( z )( a ⊗ b ) . (5)We say that Y satisfies the associativity axiom if for all a, b, c ∈ V , thereexists N ≫ z − w ) N Y ( − w )(( Y ( z ) ⊗ a ⊗ b ⊗ c )= ( z − w ) N ι z,w Y ( z − w )(1 ⊗ Y ( − w ))( a ⊗ b ⊗ c ) . (6)Let ( V, | i ) be a pointed vector space. As in [BK], a field algebra ( V, | i , Y )is a state-field correspondence Y for ( V, | i ) satisfying the associativity axiom(6). A strong field algebra ( V, | i , Y ) is a state-field correspondence Y satisfyingthe n -th product axiom (5).Let ( V, | i ) be a pointed vector space and let Y be a state-field correspon-cence. For a, b ∈ V, [BK] defined the λ - product given by a λ b = Res z e λz Y ( z )( a ⊗ b ) = X n ≥ λ n n ! a ( n ) b. (7)and the · - product on V , which is denote as a · b = Res z z − Y ( z )( a ⊗ b ) = a ( − b. (8)The vacuum axioms for Y implies | i · a = a = a · | i , (9)while the translation invariance axioms imply T ( a · b ) = T ( a ) · b + a · T ( b ) , (10)and T ( a λ b ) = ( T a ) λ b + a λ ( T b ) , ( T a ) λ b = − λa λ b (11)for all a, b ∈ V. Notice that from these equations we can derive that T ( | i ) = 0and | i λ a = 0 = a λ | i for a ∈ V. Conversely, if we are given a linear operator T, a λ -product and a · -producton ( V, | i ), satisfying the above properties (9)-(11), we can reconstruct the state-field correspondence Y by the formulas Y ( a, z ) + b = ( e zT a ) · b, Y ( a, z ) − b = ( a − ∂ z b )( z − ) , (12)4here Y ( a, z ) = Y ( a, z ) + + Y ( a, z ) − . A K [ T ]-module V, equipped with a linear map V ⊗ V → K ⊗ V, a ⊗ b → a λ b, satisfying (11) is called a ( K [ T ])- conformal algebra . On the other hand withrespect to the · -product, V is a ( K [ T ])- differential algebra (i.e an algebra withderivation T ) with a unit | i . Summarizing, (Cf. [BK], Lemma 4.1), we have that, giving a state-fieldcorrespondence on a pointed vector space ( V, | i ) is equivalent to provide V witha structure of a K [T]-conformal algebra and a structure of a K [ T ]-differentialalgebra with a unit | i . Now, recall the following results. Later on, we will prove some analogousresult for the braided environment.
Lemma 1. ([BK], Lemma 4.2) Let ( V, | i ) be a pointed vector space and let Y be a state-field correspondence. Fix a, b, c ∈ V. Then the collection of n -thproduct identities Y ( z )( a ( n ) b ⊗ c, z ) = ( Y ( z ) ( n ) Y ( z ))( a ⊗ b ⊗ c ) (for n ≥ )implies ( a λ b ) λ + µ = a λ ( b µ c ) − b µ ( a λ c ) , (13) a λ ( b · c ) = ( a λ b ) · c + b · ( a λ c ) + Z λ ( a λ b ) µ c dµ. (14) The ( − -st product identity Y ( z )( a ( − b ⊗ c ) = ( Y ( z ) ( − Y ( z ))( a ⊗ b ⊗ c ) implies ( a · b ) λ c = ( e T ∂ λ a ) · ( b λ c ) + ( e T ∂ λ b ) · ( a λ c ) + Z λ b µ ( a λ − µ c ) dµ, (15)( a · b ) · c − a · ( b · c ) = Z T dλ a ! · ( b λ c ) + Z T dλ b ! · ( a λ c ) . (16)Identity (13) is called the (left) Jacobi identity . A conformal algebra satis-fying this identity for all a, b, c ∈ V is called a (left) Leibnitz conformal algebra .Equation (14) is known as the “non-commutative” Wick formula, while (16) iscalled the quasi-associativity formula.Finally, we also recall the following result.
Theorem 1. ([BK], Theorem 4.4) Giving a strong field algebra structure ona pointed vector space ( V, | i ) is the same as providing V with a structure ofLeibnitz K [ T ] -conformal algebra and a structure of a K [ T ] -differential algebrawith a unit | i , satisfying (14)-(16). Recall also the following result.
Theorem 2. ([BK], Theorem 6.3) A vertex algebra is the same as a field algebra ( V, | i , Y ) for which Y = Y op . Therefore we may assume this as a definition of vertex algebra.5 .3 Braided Field Algebras
We will follow the notation and presentation introduced in [DGK].Throughout the rest of the paper we shall work over the algebra K [[ h ]] offormal series in the variable h , and all the algebraic structures that we willconsider are modules over K [[ h ]].A topologically free K [[ h ]] -module is isomorphic to W [[ h ]] for some K -vectorspace W. Note that W [[ h ]] ≇ W ⊗ K [[ h ]] , unless W is finite-dimensional over K , andthat the tensor product U [[ h ]] ⊗ K [[ h ]] W [[ h ]] of topologically free K [[ h ]]-modulesis not topologically free, unless one of U and W are finite dimensional. For anyvector space U and W, the completed tensor product by U [[ h ]] ˆ ⊗ K [[ h ]] W [[ h ]] := ( U ⊗ W )[[ h ]] (17)This is a completion in h -adic topology of U [[ h ]] ⊗ K [[ h ]] W [[ h ]] . Given a topologically free K [[ h ]]-module V, we let V h (( z )) = (cid:8) a ( z ) ∈ V [[ z, z − ]] | a ( z ) ∈ V (( z )) mod h M for every M ∈ Z ≥ (cid:9) . (18)Namely, expanding a ( z ) = P n ∈ Z a ( n ) z − n − , we ask thatlim n → + ∞ a ( n ) = 0in h -adic topology.Let V be a topologically free K [[ h ]]-module. Following [DGK], we callEnd K [[ h ]] V - valued quantum field an End K [[ h ]] V -valued formal distribution a ( z )such that a ( z ) b ∈ V h (( z )) for any b ∈ V. Later on, we will need the following lemmas, proved in [DGK](cf. Lemma 3.2and 3.3).
Lemma 2.
Let | i ∈ V and T : V → V be a K [[ h ]] -linear map such that T ( | i ) = 0 . Then for any End K [[ h ]] V -valued quantum field a ( z ) such that [ T, a ( z )] = ∂ z a ( z ) (translation covariance), we have a ( z ) | i = e zT a = X k ≥ T k ak ! z k , (19) where a = Res z z − a ( z ) | i . Lemma 3.
Let T : V → V be a K [[ h ]] -linear map and let a ( z ) be an End K [[ h ]] V -valued quantum field such that [ T, a ( z )] = ∂ z a ( z ) . We have e wT a ( z ) e − wT = ι z,w a ( z + w ) . (20)6et V be a topologically free K [[ h ]]-module, with a given non-zero vector | i ∈ V ( vacuum vector) and a K [[ h ]]-linear map T : V → V such that T ( | i ) =0 (translation operator). Again, following [DGK],(a) A topological state-field correspondence on V is a linear map Y : V ˆ ⊗ V → V h (( z )) , (21)satisfying(i) (vacuum axioms) Y ( z )( | i ⊗ v ) = v and Y ( z )( v ⊗ | i ) ∈ v + V [[ z ]] z, for all z ∈ V ;(ii) ( translation covariance ) ∂ z Y ( z ) = T Y ( z ) − Y ( z )(1 ⊗ T ) = Y ( z )( T ⊗ , (22)(b) A braiding on V is a K [[ h ]] − linear map S ( z ) : V ˆ ⊗ V → V ˆ ⊗ V ˆ ⊗ ( K (( z ))[[ h ]]) (23)such that S = 1 + O ( h ) . A braided state-field correspondance is a quintuple ( V, | i , T, Y, S ) where Y is a topological state-field correspondance and S is a braiding as above.We will use the following standard notation: given n ≥ i, j ∈ { , · · · , n } , we let S i,j ( z ) : V b ⊗ n → V b ⊗ n ˆ ⊗ ( K (( z ))[[ h ]]) , (24)act in the i -th and j -th factors (in this order) of V b ⊗ n , leaving the other factorsunchanged.A braided vertex algebra is a quintuple ( V, | i , T, Y, S ) where Y is a topologi-cal state-field correspondance and S is a braiding as above, satisfying the follow-ing S - locality : for every a, b ∈ V and M ∈ Z ≥ , there exists N = N ( a, b, M ) ≥ z − w ) N Y ( z )(1 ⊗ Y ( w )) S ( z − w )( a ⊗ b ⊗ c )= ( z − w ) N Y ( w )(1 ⊗ Y ( z ))( b ⊗ a ⊗ c ) , (25)where this equality holds mod h M , for all c ∈ V. Y , set Y op ( z )( u ⊗ v ) = e zT Y ( − z )( v ⊗ u ) . (26)It was shown in [DGK], Lemma 3.6, that in a braided vertex algebra V wehave Y ( z ) S ( z )( a ⊗ b ) = Y op ( z )( a ⊗ b ) (27)for all a, b ∈ V. After the proof of this result,(cf. Remark 3.7, [DGK]) they point out thatit is enough to have the S -locality (25) holding just for c = | i , to prove that Y S = Y op in a braided vertex algebra. We will use this remark later.We recall at this point two important Propositions for our sequel. Proposition 2. ([EK], Prop. 1.1) Let V be a braided vertex algebra. for every a, b, c ∈ V and M ∈ Z ≥ , there exists N ≥ such that ι z,w (( z + w ) N Y ( z + w )(1 ⊗ Y ( w )) S ( w ) S ( z + w )( a ⊗ b ⊗ c ))= ( z + w ) N Y ( w ) S ( w )( Y ( z ) ⊗ a ⊗ b ⊗ c ) mod h M . (28) Proposition 3. ([DGK], Proposition 3.9) Let ( V, | i , T, Y, S ) be a braided ver-tex algebra. Extend Y ( z ) to a map V ˆ ⊗ V ˆ ⊗ ( K (( z ))[[ h ]]) in the obvious way.Then, modulo Ker Y ( z ) , we have(a) S ( | i ⊗ a ) ≡ | i , and S ( z )( | i ⊗ a ) ≡ | i ⊗ a ; (b) [ T ⊗ , S ( z )] ≡ − ∂ z S ( z ) (left shift condition);(c) [1 ⊗ T, S ( z )] ≡ ∂ z S ( z ) (right shift condition);(d) [ T ⊗ ⊗ T, S ( z )] ≡ ;(e) S ( z ) S ( − z ) = 1 (unitary).Moreover, we have the quantum Yang-Baxter equation:(f ) S ( z − z ) S ( z − z ) S ( z − z ) ≡ S ( z − z ) S ( z − z ) S ( z − z ) , modulo Ker ( Y ( z )(1 ⊗ Y ( z ))(1 ⊗ ⊗ Y ( z )( − ⊗ − ⊗ − ⊗ | i ))) . On the structure of braided state-field corres-pondence
As in [BK], we aim to show that there are, inside certain braided vertex algebras,a “braided conformal algebra” and a “differential algebra” satisfying some familyof equation. Conversely, we will show that given such structures under somenice conditions, we can give some reconstruction theorem.Let ( V, | i , T, Y, S ) be a braided-state field correspondence. For n ∈ Z , the quantum n-product Y ( z ) S ( n ) Y ( z ) is defined as( Y ( z ) S ( n ) Y ( z ))( a ⊗ b ⊗ c ) = Res x ( ι x,z ( x − z ) n Y ( x )(1 ⊗ Y ( z ))( a ⊗ b ⊗ c ) − ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) S ( z − x )( b ⊗ a ⊗ c )) . (29)Now, we have the following result. Lemma 4.
Given ( V, | i , T, Y, S ) a braided state-field correspondence satisfyingthe equations [ T ⊗ , S ( z )] = − ∂ z S ( z ) , (30)[1 ⊗ T, S ( z )] = ∂ z S ( z ) . (31) The quantum n-product (29) satisfies the following equation ∂ z ( Y ( a, z ) S n Y ( b, z )) = ( ∂ z Y ( a, z )) S n Y ( b, z ) + Y ( a, z ) S n ( ∂ z Y ( b, z )) . (32) Proof.
Applying the definition of quantum n -product (29), using integration byparts and translation covariance (22), the LHS becomes9 es x ι x,z ∂ z (( x − z ) n Y ( x )(1 ⊗ Y ( z )))( a ⊗ b ⊗ c ) − Res x ι z,x ∂ z (( x − z ) n Y ( z )(1 ⊗ Y ( x )) S ( z − x ))( b ⊗ a ⊗ c )= Res x ι x,z ∂ z ( x − z ) n Y ( x )(1 ⊗ Y ( z ))( a ⊗ b ⊗ c )+ Res x ι x,z ( x − z ) n Y ( x )(1 ⊗ ∂ z )(1 ⊗ Y ( z ))( a ⊗ b ⊗ c ) − Res x ι z,x ∂ z ( x − z ) n Y ( z )(1 ⊗ Y ( x )) S ( z − x )( b ⊗ a ⊗ c ) − Res x ι z,x ( x − z ) n ∂ z Y ( z )(1 ⊗ Y ( x )) S ( z − x )( b ⊗ a ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) ∂ z S ( z − x )( b ⊗ a ⊗ c )= − Res x ι x,z ∂ x ( x − z ) n Y ( x )(1 ⊗ Y ( z ))( a ⊗ b ⊗ c )+ Res x ι x,z ( x − z ) n Y ( x )(1 ⊗ Y ( z ))(1 ⊗ T ⊗ a ⊗ b ⊗ c )+ Res x ι z,x ∂ x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) S ( z − x )( b ⊗ a ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )( T ⊗ ⊗ Y ( x )) S ( z − x )( b ⊗ a ⊗ c )+ Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) ∂ x S ( z − x )( b ⊗ a ⊗ c )= Res x ι x,z ( x − z ) n Y ( x )( T ⊗ ⊗ Y ( z ))( a ⊗ b ⊗ c )+ Res x ι x,z ( x − z ) n Y ( x )(1 ⊗ Y ( z )(1 ⊗ T ⊗ a ⊗ b ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x ))(1 ⊗ T ⊗ S ( z − x )( b ⊗ a ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x ))( T ⊗ ⊗ S ( z − x )( b ⊗ a ⊗ c ) . (33)On the other hand using translation covariance, RHS becomes Res x ι x,z ( x − z ) n Y ( x )( T ⊗ ⊗ Y ( z ))( a ⊗ b ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) S ( z − x )(1 ⊗ T ⊗ b ⊗ a ⊗ c )+ Res x ι x,z ( x − z ) n Y ( x )(1 ⊗ Y ( z ))(1 ⊗ T ⊗ a ⊗ b ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) S ( z − x )( T ⊗ ⊗ b ⊗ a ⊗ c ) . (34)Due to equations (30) and (31) we get( T ⊗ ⊗ S ( z − x )( b ⊗ a ⊗ c ) = S ( z − x )( T ⊗ ⊗ b ⊗ a ⊗ c ) + ∂ x S ( z − x )( b ⊗ a ⊗ c ) , (35)and(1 ⊗ T ⊗ S ( z − x )( b ⊗ a ⊗ c ) = S ( z − x )(1 ⊗ T ⊗ b ⊗ a ⊗ c ) − ∂ x S ( z − x )( b ⊗ a ⊗ c ) . (36)10pplying equations (35) and (36) to RHS, we get Res x ι x,z ( x − z ) n Y ( x )( T ⊗ ⊗ Y ( z ))( a ⊗ b ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x ))(1 ⊗ T ⊗ S ( z − x )( b ⊗ a ⊗ c )+ Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) ∂ x S ( z − x )( b ⊗ a ⊗ c )+ Res x ι x,z ( x − z ) n Y ( x )(1 ⊗ Y ( z ))(1 ⊗ T ⊗ a ⊗ b ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x ))( T ⊗ ⊗ S ( z − x )( b ⊗ a ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) ∂ x S ( z − x )( b ⊗ a ⊗ c )= Res x ι x,z ( x − z ) n Y ( x )( T ⊗ ⊗ Y ( z ))( a ⊗ b ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x ))(1 ⊗ T ⊗ S ( z − x )( b ⊗ a ⊗ c )+ Res x ι x,z ( x − z ) n Y ( x )(1 ⊗ Y ( z ))(1 ⊗ T ⊗ a ⊗ b ⊗ c ) − Res x ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x ))( T ⊗ ⊗ S ( z − x )( b ⊗ a ⊗ c ) . (37)Then equations (33) and (37) are equal, therefore the claim follows. Remark . Recall that, as we quote in Proposition 3, it was shown by [DGK]that in a braided vertex algebra, equations (30) and (31) hold mod Ker Y . In[EK], condition (30) is asked as part of the definition of a braided vertex operatoralgebra. In this context, asking (31) is equivalent to ask T to be a derivation ofa braided vertex operator algebra. It is shown in [Li], that if in addition we askthe undelying field algebra to be non-degenerate (cf. definition 5.12, [Li]), wehave that (31) holds in a braided vertex algebra where the associativity relation(6) holds (cf. [EK]).Let ( V, | i , T, Y, S ) be a braided-state field correspondence. Y satisfies the quantum n -th product identities if for all a, b, c ∈ V and n ∈ Z Y ( z ) S ( n ) Y ( z )( a ⊗ b ⊗ c ) = Y ( z )( a S ( n ) b ⊗ c ) , (38)where a S ( n ) b = Res( z n Y ( z ) S ( z )( a ⊗ b )) . (39) Y satisfies the associativity relation if for any a, b, c ∈ V and M ∈ Z ≥ thereexists N ∈ Z ≥ such that ι z,w ( z + w ) N Y ( z + w )((1 ⊗ Y ( w )))( a ⊗ b ⊗ c )= ( z + w ) N Y ( w )( Y ( z ) ⊗ a ⊗ b ⊗ c ) mod h M , (40)Let ( V, | i ) be a pointed vector space. We define a braided field algebra ( V, | i , Y, T, S ) is a braided state-field correspondence Y satisfying the associa-tivity relation (6). We also introduce a strong braided field algebra ( V, | i , Y, T, S )11s a state-field correspondence Y satisfying the quantum n -th product identi-ties (38). This are the braided versions of field algebra and strong field algebraintroduced by [BK].Let ( V, | i , T, S ) be a braided-state field correspondence. For a, b ∈ V, f ∈ K (( z ))[[ h ]] , we define the ( λ, f )- product by the a ( λ,f ) b = Res z e λz f ( z ) Y ( z )( a ⊗ b ) = X n ∈ Z ≥ , finite X i ∈ Z f i ( h ) a ( n + i ) b λ n n ! ∈ V ⊗ K [ λ ][[ h ]](41)where f ( z ) = P i ∈ Z f i ( h ) z i , f i ( h ) ∈ K [[ h ]] . Note that f i ( h ) = 0 for i << Remark . If in adition we ask V to have a structure of K (( z ))-module structure,more precisely z k ( a ( n ) b ) = a ( n + k ) b , this ( λ, f )-product resembles the operationsintroduced in [GKK]. Instead, we are asking V to have a braiding that involvessome elements of K (( z ))[[ h ]].We have the following useful Lemma. Lemma 5.
Given ( V, | i , T, S ) be a braided-state field correspondence, we have(a) a ( λ,z m f ) b = ∂ mλ a ( λ,f ) b for m ≥ , and f ∈ K (( z ))[[ h ]] . In particular, a ( λ,z m ) b = ∂ mλ a ( λ, b for m ≥ ,(b) a ( λ,z − k ) b = (( λ + T ) ( k − a ) ( λ,z − ) b, for k ≥ ,Proof. Let f ( z ) = P i f i ( h ) z i , item ( a ) follows from the definition of ( λ, f )-product: a ( λ,z m f ) b = Res z e λz z m f ( z ) Y ( z )( a ⊗ b )= X i f i ( h )Res z X k ≥ λ k /k ! X j ∈ Z a ( j ) b z − j − k + m + i = X i f i ( h ) X k ≥ λ k − m / ( k − m )! a ( k + i ) b = ∂ mλ a ( λ,f ) b. Applying definition of ( λ, f )-product and using integration by parts and trans-lation covariance we get item ( b ), namely: a ( λ,z − k ) b = a ( λ, ( − ∂ ) ( k − z z − ) b = Res z e λz ( − ∂ ) ( k − z z − Y ( z )( a ⊗ b )= Res z z − ∂ ( k − z e λz Y ( z )( a ⊗ b )= Res z z − e λz k − X r =0 λ ( r ) Y ( z )( T ( k − − r ) a ⊗ b )= Res z z − e λz Y ( z )(( λ + T ) ( k − a ⊗ b )= (( λ + T ) ( k − a ) ( λ,z − ) b. f = 1 in (41), we recover the λ -product introduced in (7) fora state-field correspondence. We will denote a ( λ, = a λ b . Observe also that,due to the Lemma above, any ( λ, f )-product can be written in terms of the λ -product and the ( λ, z − )-product.The vacuum axioms for Y imply that, | i ( λ,z − ) a = a = a ( λ,z − ) | i , (42)while the translation invariance axioms show that, T ( a ( λ,f ) b ) = T ( a ) ( λ,f ) b + a ( λ,f ) T ( b ) (43)and T ( a ) ( λ,f ) b = − λa ( λ,f ) b − a ( λ,f ′ ) b (44)for all a, b ∈ V and f ∈ K (( z )). Note that, when f = 1 in (43) and (44), werecover equation (11).Conversely, if we are given a pointed topologically free K [[ h ]]-module ( V, | i ) , togheter with a K [[ h ]]- linear map T , a braiding S , a ( λ, λ, z − )-product on V satisfying the properties (42)-(44), we can reconstructthe braided state-field correspondence Y by the formulas: Y ( a, z ) + b = ( e zT a ) ( λ,z − ) b | λ =0 , Y ( a, z ) − b = ( a ( − ∂ z , b )( z − )) , (45)where Y ( a, z ) = Y ( a, z ) + + Y ( a, z ) − . We will need the following Lemma.
Lemma 6.
We have that a ( λ,f ( l ) ) b = (( − λ − T ) l a ) ( λ,f ) b, for all a and b ∈ V and l ≥ . Here and further f ( l ) ( z ) = ∂ lz f ( z ) .Proof. Straightforward using (44).For the following Proposition it will be useful to introduce the followingnotation: a ( · ,f ) b := a ( λ,z − f ) b | λ =0 = Res z z − f ( z ) Y ( z )( a ⊗ b ) = X i ∈ Z f i ( h ) a ( i − b, (46)for a, b ∈ V, f ∈ K (( z ))[[ h ]] , f ( z ) = P i ∈ Z f i ( h ) z i , f i ( h ) ∈ K [[ h ]] . Note that inthe case f = 1 we obtain the ·− product in [BK], ( cf. (8)), namely13 ( · , b = a ( λ,z − ) b | λ =0 = a · b, since it is easy to show that a ( λ,z − ) b = a · b + Z λ a µ b dµ (47). Whith all this, we can state the following result. Proposition 4.
Let ( V, | i , T, Y, S ) be a braided state field correspondence such S -locality holds for c = | i . Then the collection of the n -th quantum productidentities (29) for n ≥ − implies: ( a − α − T b ) α + β c = − b α ( a ( β ) c ) + r X i =1 X l ≥ ( − l a i ( β, ( f i ( z )) ( l ) ) ( b i ( α,x l ) c ) , (48)( a − λ b ) · c = − b ( λ − T ) ( a · c ) + r X i =1 X l ≥ ( − l a i ( · , ( f i ( z )) ( l ) ) ( b i ( λ − T,x l ) c )+ Z T − λ ( a − λ b ) µ c dµ, (49)( a · b ) λ c = ( e T ∂ λ b ) · ( a λ c ) − Z − T ( a − µ − T b ) λ c dµ + r X i =1 [( e T ∂ λ a i ) ( · ,f i ( z )) ( b iλ c ) − X l ≥ Z λ a i ( µ, ( f i ( z )) ( l ) ) ( b i ( λ − µ,x l ) c ) dµ ] , (50)( a · b ) · c = b · ( a · c ) + Res z Z T d λ b ! · ( b λ c ) − Z − T ( a µ − T b ) · c dµ + r X i =1 X l ≥ Z T d λ a i ! · ( b iλ, D l f i ( z ) c )+ r X i =1 X m, l ≥ ( − l a i ( · ,z m +1 ) ( b i ( ., D l ( f i ( z )) z − m ) c ) , (51) where D l = z l ∂ ( l ) z and S ( x )( a ⊗ b ) = P ri =0 f i ( z ) a i ⊗ b i . roof. Recall that the fact that the S -locality holds for c = | i , implies that Y ( z ) S ( z ) = Y op ( z ). Applying definitions of λ -product and the definition of Y op , due to Lemma 27 we get a λ b = Res z e λz Y ( z )( a ⊗ b )= Res z e ( λ + T ) z Y op ( − z )( b ⊗ a )= − Res z e − ( λ + T ) z Y ( z ) S ( z )( b ⊗ a ):= − b S− ( λ + T ) a. (52)The collection of n -th product identities (38) together (52) are equivalent to: Y ( a − λ b, z ) c = − Y ( b S ( λ − T ) a, z ) c = − X n ≥ Y ( z )( b S ( n ) a ⊗ c ) ( λ − T ) n n != − X n ≥ Y ( z ) S ( n ) Y ( z )( b ⊗ a ⊗ c ) ( λ − T ) n n != − X n ≥ [Res x ι x,z ( x − z ) n Y ( x )(1 ⊗ Y ( z ))( b ⊗ a ⊗ c )+ ι z,x ( x − z ) n Y ( z )(1 ⊗ Y ( x )) S ( z − x )( a ⊗ b ⊗ c )] ( λ − T ) n n != − Res x e ( λ − T )( x − z ) Y ( x )(1 ⊗ Y ( z ))( b ⊗ a ⊗ c )+ Res x e ( λ − T )( x − z ) r X i =1 e − x∂ z ( f i ( z )) Y ( z )(1 ⊗ Y ( x ))( a i ⊗ b i ⊗ c )= − e ( − λ + T ) z [ b ( λ − T ) ( Y ( a, z ) c )+ r X i =1 X l ≥ ( − ∂ z ) ( l ) ( f i ( z )) Y ( a i , z )( b i ( λ − T,x l ) c )] . (53)Taking Res z e ( λ + µ ) z , and changing λ − T by α and µ + T by β , we obtain (48).Taking Res z z − in (53) and using e ( − λ + T ) z z − = z − + R − λ + T e µz dµ, we get( a − λ b ) · c = − b ( λ − T ) ( a · c ) − Z T − λ b ( λ − T ) ( a µ c ) dµ + r X i =1 X l ≥ ( − l [ a i ( · , ( f i ( z )) ( l ) ) ( b i ( λ − T,x l ) c )+ Z T − λ a i ( µ, ( f i ( z )) ( l ) ) ( b i ( λ − T,x l ) c ) dµ ] . (54)This, together with (48), implies (49) (after the substitution µ ′ = λ + µ − T ).Applying definitions of ( − Y op , due to (27) we get15 · b = Res z z − Y ( z )( a ⊗ b )= Res z z − e zT Y op ( − z )( b ⊗ a )= Res z z − e − zT Y ( z ) S ( z )( b ⊗ a )= Res z z − Y ( z ) S ( z )( b ⊗ a ) + Z − T e µz Y ( z ) S ( z )( b ⊗ a ) dµ = b S· a + Z − T b S µ a dµ. (55)Then this equation together the quantum ( − Y ( a · b, z ) c = Y ( z )( b S ( − a ⊗ c ) + Z − T Y ( z )( b S µ a ⊗ c ) dµ = ( Y ( z ) S ( − Y ( z ))( b ⊗ a ⊗ c ) + Z − T Y ( z )( b S µ a ⊗ c ) dµ = Y ( b, z ) + Y ( a, z ) c + r X i =1 Y ( a i , z )( b i − ∂ z c )( z − f i ( z ))+ Z − T Y ( b S µ a, z ) c dµ. (56)Taking Res z e λz and using integration by parts, we get:( a · b ) λ = Res z ( e T ∂ λ e λz b ) · ( a λ c ) + Res z r X i =1 Y ( a i , z )( b iλ − ∂ z c ( e λz z − f i ( z )))+ Z − T ( b S µ a ) λ c dµ = ( e T ∂ λ b ) · ( a λ c ) + Res z r X i =1 Y ( a i , z )( b iλ − ∂ z c ( z − f i ( z ) + Z λ f i ( z ) e µz dµ ) − Z − T ( a − µ − T b ) λ c dµ = ( e T ∂ λ b ) · ( a λ c ) − Z − T ( a − µ − T b ) λ c dµ + r X i =1 [( e T ∂ λ a i ) ( · ,f i ( z )) ( b iλ c )+ X l ≥ Z λ a i ( µ, ( f i ( z )) ( l ) ) ( b i ( λ − µ,x l ) c ) dµ ] . (57)16ue to (56) and taking Res z z − , we get( a · b ) · c = Res z z − Y ( b, z ) + Y ( a, z ) + c + Res z z − Y ( b, z ) + Y ( a, z ) − c + Res z z − Z − T Y ( b S µ a, z ) c dµ + r X i =1 X l ≥ ( − l Res z z − Y ( a i , z )( f i ( z )) ( l ) ( ∂ z ) l Y ( b i , z ) − c = b · ( a · c ) + Res z z − )(( e zT − b ) · ( Y ( a, z ) − c ) + Z − T ( b S µ a ) · c dµ + r X i =1 X l ≥ Res z z − (( e zT − a i ) · (( f i ( z )) ( l ) ( ∂ z ) l Y ( b i , z ) − c )+ r X i =1 X l ≥ Res z z − Y ( a i , z ) − (( f i ( z )) ( l ) ( ∂ z ) l Y ( b i , z ) − c )= b · ( a · c ) + Res z Z T e λz d λ b ! · ( Y ( b, z ) − c ) − Z − T ( a µ − T b ) · c dµ + r X i =1 X l ≥ Res z Z T e λz d λ a i ! · ( f i ( z )) ( l ) ( ∂ z ) l Y ( b i , z ) − c )+ r X i =1 X m,l ≥ ( − l a i ( · ,z m +1 ) ( b i ( ., D l ( f i ( z )) z − m ) c ) , (58)which proves (51).Note that V together with the λ -product is what [BK] called conformalalgebra , and V with ( λ, z − )-product is also a K [ T ]-differential algebra withunit due to (42) and (43).An important remark is that V togheter with the λ -product is no longer aLeibnitz conformal algebra, since due to the braiding, the analog of the Jacobyidentity involves ( λ, z − )-products.With this in mind, we can prove the following Theorem 3.
Giving a braided state field correspondence ( V, | i , Y, S ) , satisfyingthe S -locality for | i and the axiom of quantum ( n ) -product (29) implies toprovide V with a structure of a conformal algebra and a structure of a K [ T ] -differential algebra with a unit | i , satisfying (48)-(51).Conversely, given V a topologically free K [[ h ]] -module, a K [[ h ]] - linear map T and a braiding S . Assume that V has a structure of conformal algebra anda structure of a K [ T ] -differential algebra with a unit | i , satisfying (48)-(51)and S satisfies (30)- (31), then ( V, | i , Y, S ) , is a braided state field correspon-dence satisfying the axiom of quantum ( n ) -product, namely a strong braided fieldalgebra. roof. If ( V, | i , Y, S ) is a braided state field algebra satisfying the axiom ofquantum n -product, then by the above discussion we can define a ( λ, f )-producton V satisfying all the requirement. Conversely, given a ( λ, f )-product we definea braided state field correspondence Y by (45). In the proof of Lemma 4, wehave seen that the equations (48)-(49) are equivalent to the identitiesRes z ( Y ( b S n a, z ) − Y ( b, z ) S ( n ) Y ( a, z )) F = 0 , a, b ∈ V, n ≥ , F = e λz or z − , (59)while the equations (50)-(51) are equivalent to the identitiesRes z X k ≥ [( − ∂ z ) ( k ) Y ( b S ( k − a, z ) − k X j =0 ( − k ∂ ( k − j ) z Y ( b, z ) S ( k − ∂ ( j ) z Y ( a, z )] F = 0 , (60)for a, b ∈ V and F = e λz or z − .Due to Lemma 4 and using translation invariance of Y , this identity is equiv-alent to Res z X k ≥ [ Y ( b S ( k − a, z ) − Y ( b, z ) S ( k − Y ( a, z )]( ∂ z ) ( k ) F = 0 , (61) a, b ∈ V, F = e λz or z − . Using the translation invariance of Y and integration by parts, we see thatidentity (59) holds also with F replaced with ∂ z F. Hence equations (59) and(61) hold for all F = z l , l < . For F = e λz , taking coefficients at power of λ shows that they are satisfied also for F = z l , l ≤ . This implies the n -thquantum product axioms for n ≥ − . The the proof remains the same thatproof of Theorem 4.4 [BK].
In this section, based on what we have seen in Section 3, we aim to give adefinition of braided conformal algebra. Until now, we didn’t ask any furhterstructure for the braiding S besides (30) and (31). We have the following results. Proposition 5.
If the hexagon relation S ( x )( Y ( z ) ⊗
1) = ( Y ( z ) ⊗ S ( x ) ι x,z S ( x + z ) (62) holds in a braided state field correspondence, then we have that: S ( x )( a ( λ,f ) b ⊗ c ) = X l ≥ ∂ lλ (( · ( λ,f ) · ) ⊗ S ( x ) ∂ ( l ) x S ( x )( a ⊗ b ⊗ c ) (63)= e ∂ λ ∂ x (( · ( λ,f ) · ) ⊗ S ( x ) S ( x )( a ⊗ b ⊗ c ) | x = x (64) for all a, b, c ∈ V . roof. Applying the definition of ( λ, f )-product and using the hexagon relation(62), definition of S, Taylor expansion and change of variables we get, S ( x )( a ( λ,f ) b ⊗ c ) = S ( x )Res z e λz f ( z )( Y ( z ) ⊗ a ⊗ b ⊗ c )= Res z e λz f ( z ) S ( x )( Y ( z ) ⊗ a ⊗ b ⊗ c )= Res z e λz f ( z )( Y ( z ) ⊗ S ( x ) ι x,z S ( x + z )( a ⊗ b ⊗ c )= X i,j ∈ Z Res z e λz f ( z )( Y ( z ) ⊗ h i ( x ) ι x,z g j ( x + z )( a ( j ) ⊗ b ( i ) ⊗ ( c ( j ) ) ( i ) )= X i,j ∈ Z h i ( x )Res z e λz f ( z )( e z∂ x g j ( x ))( Y ( a ( j ) , z ) b ( i ) ⊗ ( c ( j ) ) ( i ) )= X i,j,m,r ∈ Z X k,l ∈ Z ≥ h i ( x ) λ ( k ) f r Res z g ( l ) j ( x ) a ( j )( m ) b ( i ) ⊗ ( c ( j ) ) ( i ) z − m − k + l + r = X i,j,r ∈ Z X k,l ∈ Z ≥ h i ( x ) λ ( k ) f r g ( l ) j ( x ) a ( j )( k + l + r ) b ( i ) ⊗ ( c ( j ) ) ( i ) = X i,j,r ∈ Z X k ≥ l ∈ Z ≥ h i ( x ) λ ( k − l ) f r g ( l ) j ( x ) a ( j )( k + r ) b ( i ) ⊗ ( c ( j ) ) ( i ) = X l ≥ ∂ lλ ( · ( λ,f ) · ) S ( x ) ∂ ( l ) x S ( x )( a ⊗ b ⊗ c ) , (65)where S ( x )( a ⊗ b ⊗ c ) = P i h i ( x ) a ⊗ b i ⊗ c i and S ( x )( a ⊗ b ⊗ c ) = P j g j ( x ) a j ⊗ b ⊗ c j .Similarly, we have the following results. Proposition 6.
If the associativity relation holds, namely, there exists N ∈ Z ≥ such that ι z,w ( z + w ) N Y ( z + w )((1 ⊗ Y ( w )))( a ⊗ b ⊗ c ) = ( z + w ) N Y ( w )( Y ( z ) ⊗ a ⊗ b ⊗ c ) mod h M , for any a, b, c ∈ V and M ∈ Z ≥ in a (braided) state field correspon-dence, then ∂ Nλ a λ ( b µ c ) = ∂ Nλ ( a λ b ) λ + µ c, (66) mod h M , for all a, b, c ∈ V. Proof.
Changing z + w by x in the associativity relation we have x N Y ( x )(1 ⊗ Y ( w ))( a ⊗ b ⊗ c ) = x N Y ( w ) ι x,w ( Y ( x − w ) ⊗ a ⊗ b ⊗ c ) . (67)Taking Res x Res w e λx e µw to the LHS of (67) and using Lemma 5 (a), we haveRes x Res w e λx e µw x N Y ( x )((1 ⊗ Y ( w )))( a ⊗ b ⊗ c )= ( a ( λ,x N ) ( b µ c )) = ∂ Nλ ( a λ ( b µ c )) . (68)Now, taking Res x Res w e λx e µw to the RHS of (67), using Taylor’s formula(2), translation covariance and Lemma 5 (a),19es x Res w e λx e µw x N Y ( w ) ι x,w ( Y ( x − w ) ⊗ a ⊗ b ⊗ c )= Res x Res w e λx e µw x N Y ( w ) e − w∂ x ( Y ( x ) ⊗ a ⊗ b ⊗ c )= Res x Res w e λx e µw x N Y ( w )( Y ( x ) ⊗ e − wT a ) ⊗ b ⊗ c )= Res w e µw Y (( e − wT a ) ( λ,x N ) b, w ) c = Res w e µw ∂ Nλ Y (( e − wT a ) λ b, w ) c = ∂ Nλ Res w e µw e wλ Y ( a λ b, w ) c = ∂ Nλ ( a λ b ) λ + µ c. (69)Equating (68) and (69), we finish the proof.Now, we will show a similar resut but for the quasi-associativity (70). Proposition 7.
Let V be a braided state field correspondance. Suppose that forevery a, b, c ∈ V and M ∈ Z ≥ there exists N ≥ such that ι z,w (( z + w ) N Y ( z + w )(1 ⊗ Y ( w )) S ( w ) S ( z + w )( a ⊗ b ⊗ c ))= ( z + w ) N Y ( w ) S ( w )( Y ( z ) ⊗ a ⊗ b ⊗ c ) mod h M . (70) holds mod h M . Then X i,j ∂ Nσ a j ( σ,g j ) ( b i ( − λ + µ,h i ) ( c j ) i ) | σ = λ = ( ∂ λ + ∂ µ ) N X r ( a λ b ) r ( µ,f r ) c r mod h M , (71) where S ( x )( a λ b ) ⊗ c = X r f r ( x )( a λ b ) r ⊗ c r ) ,S ( x )( a ⊗ b ⊗ c ) = X j g j ( x )( a j ⊗ b ⊗ c j ) ,S ( x )( a j ⊗ b ⊗ c j ) = X i h i ( x )( a j ⊗ b i ⊗ ( c j ) i ) . Proof.
Taking Res x Res w e λz e µw to the LHS of (70), using Taylor’s formula (2)and integration by parts, we haveRes z Res w e λz e µw ι z,w (( z + w ) N Y ( z + w )(1 ⊗ Y ( w )) S ( w ) S ( z + w )( a ⊗ b ⊗ c )) == Res z Res w e λz e µw ( z + w ) N e w∂ z ( Y ( z ))(1 ⊗ Y ( w )) S ( w ) e w∂ z ( S ( z ))( a ⊗ b ⊗ c ))= X i,j Res z Res w e − w∂ z ( e λz ( z + w ) N ) e µw Y ( z ))(1 ⊗ Y ( w )) g j ( z ) h i ( w )( a j ⊗ b i ⊗ ( c j ) i ) . (72)It is straightforward that e − w∂ z ( z + w ) N = z N and e − w∂ z e λz = e − wλ e λz ,thus 20 i,j Res z Res w e − w∂ z (cid:0) e λz ( z + w ) N (cid:1) e µw Y ( z )(1 ⊗ Y ( w )) g j ( z ) h i ( w )( a j ⊗ b i ⊗ ( c j ) i )= X i,j Res z Res w e − wλ e λz ( z ) N e µw Y ( z )(1 ⊗ Y ( w )) g j ( z ) h i ( w )( a j ⊗ b i ⊗ ( c j ) i )= X i,j a j ( λ,x N g j ) ( b i ( − λ + µ,h i ) ( c j ) i )= X i,j ∂ Nσ a j ( σ,g j ) ( b i ( − λ + µ,h i ) ( c j ) i ) | σ = λ . (73)In the last equality we used Lemma 5(a). Now, lets take residues in the RHSof (70) and use Lemma 5(a)again. ThusRes z Res w e λz e µw ( z + w ) N Y ( w ) S ( w )( Y ( z ) ⊗ a ⊗ b ⊗ c )= X r N X k =0 (cid:18) Nk (cid:19) Res w e µw f r ( w ) w N − k ( a ( λ,z k ) b ) r ⊗ c r = X r N X k =0 (cid:18) Nk (cid:19) ( a ( λ,z k ) b ) r ( µ,f r ( w ) w N − k ) ⊗ c r = ( ∂ λ + ∂ µ ) N X r ( a λ b ) r ( µ,f r ) c r . (74)Equating mod h M , we have the desired result.Finaly, let us translate the condition Y ( z ) S ( z ) = Y op ( z ) to the ( λ, f )-product. Proposition 8.
Suppose we have a state-field correspondance V where Y ( z ) S ( z ) = Y op ( z ) holds. Then, for a and b ∈ V , − b − λ − T a = X i a i ( λ,f i ) b i , (75) where S ( z )( a ⊗ b ) = P i f i ( z ) a i ⊗ b i Proof.
We have thatRes z e λz Y ( z ) S ( z )( a ⊗ b ) = X i Res z e λz f i ( z ) Y ( a i , z ) b i = X i a i ( λ,f i ) b i . (76)21n the other hand, using Y ( z ) S ( z ) = Y op ( z ),Res z e λz Y ( z ) S ( z )( a ⊗ b ) = Res z e λz Y op ( z )( a ⊗ b )= Res z e λz e T z Y ( − z )( b ⊗ a )= − Res z e ( − λ − T ) z Y ( z )( b ⊗ a )= − b − λ − T a, (77)finishing the proof.A braided vertex algebra where the associativity relation holds, is called quantum vertex algebra. (Cf. Definition 3.12, [DGK]). In the CharacterizationTheorem (cf. Theorem 5.13,[DGK]) they proved, among other equivalences,that a quantum vertex algebra is a braided state field correspondence suchthat the associativity relation and Y S = Y op holds. We have shown in thediscussion before Lemma 6, combined with the fact that all ( λ, f ) products canbe rewritten in terms of λ -products and ( λ, z − )-products, that having a braidedstate field correspondence is the same of having topologically free K [[ h ]]-module V, together with a K [[ h ]]-linear map T : V → V , a distinguished vector | i , abraiding S on V and linear maps ( λ, f ) : V ⊗ V → K [ λ ][[ h ]] , a ⊗ b → a ( λ,f ) b for f ∈ K (( Z ))[[ h ]], such that | i ( λ,z − ) a = a = a ( λ,z − ) | i , (78) T ( a ( λ,f ) b ) = T ( a ) ( λ,f ) b + a ( λ,f ) T ( b ) (79)and T ( a ) ( λ,f ) b = − λa ( λ,f ) b − a ( λ,f ′ ) b (80)for all a, b ∈ V . Combining this with Porposition 6 and Proposition 8 we havethe following. Theorem 4.
Let V be topologically free K [[ h ]] -module, together with a K [[ h ]] -linear map T : V → V , a distinguished vector | i , a braiding S on V . Define in V linear maps ( λ, f ) : V ⊗ V → K [ λ ][[ h ]] , a ⊗ b → a ( λ,f ) b for f ∈ K (( z ))[[ h ]] , suchthat the equation above hold. Let Y be a topological state-field correspondence.The following statements are equivalent:(i) ( V, T, | i , Y, S ) is a quantum vertex algebra.(ii) ( V, T, | i , ( · ( λ,f ) · ) , S ) satisfies the equations: | i ( λ,z − ) a = a = a ( λ,z − ) | i , (81) T ( a ( λ,f ) b ) = T ( a ) ( λ,f ) b + a ( λ,f ) T ( b ) (82) and T ( a ) ( λ,f ) b = − λa ( λ,f ) b − a ( λ,f ′ ) b (83) for all a, b ∈ V , and − b − λ − T a = X i a i ( λ,f i ) b i , (84)22 here S ( z )( a ⊗ b ) = P i f i ( z ) a i ⊗ b i , and there exists N >> such that ∂ Nλ a λ ( b µ c ) = ∂ Nλ ( a λ b ) λ + µ c, (85) mod h M , for all a, b, c ∈ V. If was proved in Proposition 3.13 in [DGK] that if a braided vertex algebrasatisfies the hexagon relation then the associativity relation holds.Assume that we have a braided vertex algebra V and the hexagon relationholds, thus we have a quantum vertex algebra. I we also ask in V the condition[ T ⊗ , S ( x )] = − ∂ x S ( x ) and [1 ⊗ T, S ( x )] = ∂ x S ( x ) , (which hold, for instance, in what [EK] called non-degenerate quantum vertexalgebra), and consider here the λ -product above, we showed that ( V, T, S ) to-gether with the λ -product is a conformal algebra ( in the sense of [BK]), sittinginside our quantum vertex algebra such that (64) holds. All these, leads us tothe following definition. Definition 1. A quantum conformal algebra is a topologically free K [[ h ]]-module V, together with a K [[ h ]]-linear map T : V → V , a braiding S on V and a linear map λ : V ⊗ V → K [ λ ] , a ⊗ b → a λ b such that:( a, b, c ∈ V )(i) [ T ⊗ , S ( x )] = − ∂ x S ( x ) (left shift condition);(ii) [1 ⊗ T, S ( x )] = ∂ x S ( x ) (right shift condition);(iii) T ( a λ b ) = ( T a ) λ b + a λ ( T b ) , ( T a ) λ b = − λa λ b ;(iv) S ( x )( a λ b ⊗ c ) = e ∂ λ ∂ x (( · λ · ) ⊗ S ( x ) S ( x )( a ⊗ b ⊗ c ) | x = x , (hexagonrelation).Moreover if we ask(iv) a λ ( b µ c ) = ( a λ b ) λ + µ c ,we call V associative quantum conformal algebra. References [BPZ] A.Belavin, A. Polyakov y A. Zamolodchikov, Infinite conformal symme-try in two dimensional quantum field theory, Nuclear Phys. B 241, N º Internat.Mat.Res.Notices (2003),123-159. 23K1] V. Kac, Vertex Algebra for Begginers, ed., AMS (1998).[DGK] A. De Sole, M. Gardini and V. Kac, On the structure of quantum vertexalgebras, J. Math. Phys. , J. Math.Phys. (1998), 2290-2305.[EK] P. Etingof, D.Kazhdan, Quantization of Lie bialgebras, Part V: QuantumVertex operator algebras, em Selecta Math(New Series) , Selecta.Math. (New series)11