Full vertex algebra and bootstrap -- consistency of four point functions in 2d CFT
aa r X i v : . [ m a t h . QA ] J un Full vertex algebra and bootstrap- consistency of four point functions in 2d CFT
Yuto Moriwaki ∗ Kavli Institute for the Physics and Mathematics of the Universe, Chiba, Japan
Abstract.
In physics, it is believed that the consistency of two dimensional conformalfield theory follows from the bootstrap equation. In this paper, we introduce the notionof a full vertex algebra by analyzing the bootstrap equation, which is a “real analytic”generalization of a Z -graded vertex algebra. We also give a mathematical formulationof the consistency of four point correlation functions in two dimensional conformal fieldtheory and prove it for a full vertex algebra with additional assumptions on the conformalsymmetry. In particular, we show that the bootstrap equation together with the conformalsymmetry implies the consistency of four point correlation functions. As an application, adeformable family of full vertex algebras parametrized by the Grassmanian is constructed,which appears in the toroidal compactification of string theory. This give us examplessatisfying the above assumptions. I ntroduction Quantum field theory has a long history in physics (see e.g. [S]) and has attracted math-ematicians in the past several decades due to its unexpected mathematical consequences orconjectures, for example [W, Ma]. One aim of quantum field theory is to calculate n pointcorrelation functions , that is, the vacuum expectation value of an interaction of n particles.An interaction of n particles decomposes into subsequent interactions of three particles.Thus, an n point correlation function can be expressed in terms of three point correlationfunctions, depending on a choice of decompositions. A quantum field theory requiresthat the resulting n point correlation functions are independent of the choice of decom-positions. This principle is known as the consistency of quantum field theory . Althoughit is known to be di ffi cult to construct mathematically rigorous quantum field theories,surprisingly many such examples, especially conformal field theories , (i.e., quantum fieldtheories with “conformal symmetry”), have been found in two dimension, see [FMS].In (not necessarily two dimensional) conformal field theories, it is believed in physics,that the whole consistency of n point correlation functions follows from the bootstrapequations (or hypothesis) , which are distinguished consistencies of four point correlationfunctions [FGG, Poly2]. This hypothesis was used successively by Belavin, Polyakov ∗ email: [email protected] and Zamolodchikov in [BPZ] where the modern study of two dimensional conformalfield theories was initiated.In two dimensional conformal field theory, fields are operator valued real analytic func-tions. It is noteworthy that the subalgebra consisting of holomorphic fields satisfies apurely algebraic axiom, which was introduced by Borcherds [B], see also [G]. It is calleda vertex algebra or a vertex operator algebra [FLM] and has been studied intensively bymany authors, see e.g., [LL, FHL, FB, K]. We note that in the language of a vertex alge-bra, three point correlation functions correspond to the vertex operator and the bootstrapequations essentially correspond to the Borcherds identity, which is an axiom of vertexalgebras. In contrast, the axiom of the (non-holomorphic) whole algebra of a conformalfield theory needs analytic properties and seems impossible to describe in an algebraicway.Moore and Seiberg constructed a conformal field theory as an extension of holomor-phic and anti-holomorphic vertex operator algebras by their modules [MS1, MS2]. Thebootstrap equations in this case are translated as a monodromy invariant property of thefour point correlation functions. In the physics literature, this property was reformulatedlater by Fuchs, Runkel and Schweigert in [FRS], which says that the algebra describ-ing the conformal field theory is a Frobenius algebra object in a braided tensor categoryconstructed from holomorphic and anti-holomorphic vertex operator algebras.A mathematical approach in this direction is due to Huang and Kong [HK] based on theabstract representation theory of vertex algebras developed by Huang and Lepowsky in aseries of papers [HL1, HL2, HL3, H1, H2]. One of the prominent results is obtained byHuang [H3, H4], which states that the representation category of a regular vertex operatoralgebra (of strong CFT type) inherits a modular tensor category structure. Note that avertex operator algebra is called regular if every (weak) module is completely reducible.Then, Huang and Kong [HK] introduced a notion of full field algebras, which is amathematical axiomatization of the algebras describing two dimensional conformal fieldtheories. Their definition is based upon a part of the consistencies of n point correlationfunctions. In [HK], they constructed conformal field theories, called diagonal theories in physics, as finite module extensions of the tensor product of regular vertex operatoralgebras in two variables “( z , ¯ z )”.In this paper, we study an axiom of conformal field theory on C P and introduce anotion of full vertex algebras starting with the bootstrap equations, which are expected tobe su ffi cient to derive the whole consistency of the theory.The definition of full vertex algebras is independent of the theory of vertex algebras,which is an essential building block of the approach by Huang and Kong. Rather, theexistence of vertex subalgebras as holomorphic and anti-holomorphic part of the algebrais naturally derived from our definition as expected in physics, see below for details.The notion of full vertex algebras essentially appears in [HK]. There they showedthat as long as a full field algebra is an extension of a tensor product of regular vertex algebras, the notions of full field algebras and full vertex algebras are equivalent [HK,Theorem 2.11].Before going into the details, let us summarize the key ingredients of the paper: • to introduce a reasonable space Cor of real analytic functions serving as the fourpoint correlation functions (Section 0.1), • to define and analyze parenthesized correlation functions S A (Section 0.2), • to formulate and prove the consistency of four point functions in the language offull vertex algebras (Section 0.3).As a byproduct, we give an example of a deformable family of full vertex algebras cor-responding to the irrational conformal field theory appearing in the toroidal compactifi-cation of string theory [Polc1]. Mathematically, this is a generalization of a lattice vertexalgebra [B, FLM]. We remark that a deformation of a theory is one of the most impor-tant ingredient of quantum field theory and conformal field theory [IZ]. This kind ofdeformations will be studied in more general setting in our next paper [Mo2]. The authorhopes that our approach will be beneficial in the future investigation on the conformalfield theories in higher dimensions.0.1. Space of four point correlation function.
In order to formulate the consistency ofconformal field theory on the projective space C P , we need to fix a space of functions“consisting of four point correlation functions”. Let X = { ( z , . . . , z ) ∈ ( C P ) | z i , z j for any i , j } be a space of ordered configurations of four points in C P . Then, afour point correlation function is, roughly, a C -valued real analytic function on X withpossible singularities along the diagonals, { ( z , . . . , z ) ∈ ( C P ) | z i = z j } for 1 ≤ i < j ≤
4. Recall that the automorphism group of C P is PSL C , whose action is called the linearfractional transformation and which is the global conformal symmetry in two dimension.Since the real analytic function ξ : X → C P \ { , , ∞} , ( z , z , z , z ) ( z − z )( z − z )( z − z )( z − z )is invariant under the diagonal action of PSL C on X ⊂ ( C P ) , it gives a homeomor-phism from the coset space PSL C \ X to C P \ { , , ∞} . Any four point function ofquasi-primary states in 2d conformal field theory possesses the global conformal symme-try. Thus, by the homeomorphism, a four point function can be regarded as a real analyticfunction on C P \ { , , ∞} with possible singularities at { , , ∞} , which we call confor-mal singularities. We remark that | z | r is not a real analytic function at 0, where | z | = z ¯ z ,the square of the absolute value, and r ∈ R . A function with a conformal singularity at 0has an expansion X r , s ∈ R a r , s z r ¯ z s , (0.1)where a r , s ∈ C and a r , s = r − s < Z . This series is absolutely convergent in anannulus 0 < | z | < R (for the precise definition, see Section 1.1). By the assumption, a r , s z r ¯ z s = a r , s z r − s | z | s is a single-valued function around 0. Denote by F , , ∞ the space of real analytic functions on C P \{ , , ∞} with possible conformal singularities at { , , ∞} . Thespace F , , ∞ contains a monodromy invariant linear combination of solutions of di ff eren-tial equations with possible regular singularities at { , , ∞} . For example, a combinationof hypergeometric functions, f Ising ( z ) = | − √ − z | / + | + √ − z | / , (0.2)has conformal singularities at { , , ∞} , which appears as a four point function of the twodimensional Ising model. The expansion of f Ising at z = + | z | / / − z / − ¯ z / + | z | / ( z + ¯ z ) / + z ¯ z / − z / − z / + . . . . We introduce a space Cor spanned by real analytic functions on X of the form: Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij f ◦ ξ ( z , z , z , z ) , (0.3)where f ∈ F , , ∞ and α i j , β i j ∈ R such that α i j − β i j ∈ Z for any 1 ≤ i < j ≤
4. Here,we allow these functions to have singularities around { z i = ∞} i = , , , . By the condition α i j − β i j ∈ Z , we have ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij = | z i − z j | α ij (¯ z i − ¯ z j ) β ij − α ij , which implies that Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij is a single valued real analytic functionon X . Physically, a four point correlation function of quasi-primary states in compactconformal field theory in two dimensions is of the form (0.3). Thus, we call Cor aspace of four point correlation functions. We remark that non-compact conformal fieldtheories, e.g., the Liouville field theory and non-compact WZW conformal field theoryare excluded from our consideration, where we need to consider some measure on thespace of irreducible representations and integral over there.0.2. Consistency and parenthesized correlation functions.
Roughly speaking, a con-formal field theory on C P , considered in this paper, consists of an R -graded C -vectorspace F = L h , ¯ h ∈ R F h , ¯ h with a distinguished vector ∈ F , a symmetric bilinear form( − , − ) : F × F → C , which is normalized as ( , ) =
1, and a linear map Y ( − , z ) : F → End( F )[[ z ± , ¯ z ± , | z | R ]] , a Y ( a , z ) = X r , s ∈ R a ( r , s ) z − r − ¯ z − s − , where z means the pair of the formal variables z and ¯ z . Physically, a vector in F h , ¯ h is astate with the spin h − ¯ h and the energy h + ¯ h , corresponds to the vacuum state and Y ( a , z ) is the field located at z associated with a state a ∈ F , called a vertex operator .It is convenient to write the vertex operator Y ( a , z ) as a ( z ). For example, ( a ( x ) a )( x )means Y ( Y ( a , x ) a , x ) and a ( x ) a ( x ) means Y ( a , x ) Y ( a , x ). A four point correla-tion function is a linear map S : F ⊗ → Cor , ( a , a , a , a ) S ( a , a , a , a )( z , z , z , z ) which physically calculates the vacuum expectation value of states a , a , a , a ∈ F at( z , z , z , z ) ∈ X . A four point correlation can be calculated in many ways by using thevertex operator, e.g., the formal power series of the variables ( z , ¯ z , . . . , z , ¯ z ),( , a ( z ) a ( z ) a ( z ) a ( z ) ) , is absolutely convergent to S ( a , a , a , a )( z , z , z , z ) ∈ Cor in the region {| z | >> | z | >> | z | >> | z |} . Another choice of compositions of vertex operators is (cid:16) , (cid:18)(cid:16) a ( x ) a (cid:17) ( x ) a (cid:19) ( x )( a ( x ) ) (cid:17) . (0.4)Such compositions are described by parenthesized products of symbols { , , , , ⋆ } , inthis case, ((42)1)(3 ⋆ ). To be more precise, let n ∈ Z > and Q n be the set of parenthesizedproducts of n + , , . . . , n , ⋆ with ⋆ at right most, e.g., (((31)6)(24))(5 ⋆ ) ∈ Q . We can naturally associate a formal power series to each element in Q n , which wecall a parenthesized n point correlation function . To give a simple example, an element(((31)6)(24))(5 ⋆ ) ∈ Q defines the following parenthesized 6 point correlation function: S (((31)6)(24))(5 ⋆ ) ( a , . . . , a ; x , . . . , x ) : = (cid:16) , "(cid:18)(cid:16) a ( x ) a (cid:17) ( x ) a (cid:19) ( x ) a ( x ) a ( x ) a ( x ) (cid:17) , where a , . . . , a ∈ F . For A ∈ Q n , we combinatorially give a space T A of formal variables x , ¯ x , . . . , x n , ¯ x n which contains the image of all the parenthesized n point functions. Thus,the parenthesized correlation function defines the map, S A : F ⊗ n → T A for each A ∈ Q n .The consistency of quantum field theory says that all parenthesized four point correla-tion functions associated with all the elements in Q and a , a , a , a ∈ F (or more gener-ally n point correlation functions) are expansions of the same function S ( a , a , a , a )( z , z , z , z )in di ff erent regions after taking the change of variables. For example, the parenthe-sized correlation function S ((42)1)(3 ⋆ ) ( a , a , a , a ; x , x , x , x ) in (0.4) is the expansionof S ( a , a , a , a )( z , z , z , z ) around {| x | >> | x | >> | x | , | x |} , where ( x , x , x , x ) = ( z , z − z , z , z − z ). We remark that for a vertex algebra, the composition of vertex op-erators Y ( Y ( a , x ) a , x ) is an analytic continuation of Y ( a , x ) Y ( a , x ) after taking thechange of variables x = x − x . Physically, the change of variables comes from the factthat any state in F is located at 0, i.e., lim z → Y ( a , z ) = a (the state-field correspondence).In Appendix, for each A ∈ Q , we give a combinatorial rule for the change of variablesand the region by using trees and in Section 1.3 we define the power series expansion ofa function in Cor , which gives e A : Cor → T A . Then, our formulation of the consistency of a four point function is stated as follows:For any A ∈ Q , the composition of the four point function S : F ⊗ → Cor and theexpansion map e A : Cor → T A is equal to the parenthesized correlation function S A : F ⊗ → T A , that is, S A = e A ◦ S . (0.5) In physics, the bootstrap hypothesis says that all consistencies follows from the con-sistency for special parenthesized correlation functions, S (21)(34) (s-channel), S (41)(23) (t-channel) and S (31)(24) (u-channel). The purpose of this paper is to define a full vertexalgebra by slightly modifying the bootstrap hypothesis as a generalization of a Z -gradedvertex algebra and to prove the consistency as a consequence of the axiom of a full vertexalgebra. In the theory of vertex algebras, it is commonly used the limit of a four pointcorrelation function as ( z , z ) → ( ∞ , generalized two point func-tion . Before explaining the definition of a full vertex algebra, we briefly describe the limit( z , z ) → ( ∞ ,
0) in relation to the consistency. Let a , a , a , a ∈ F with a ∈ F h , ¯ h .Then, the limit ( z , z ) → ( ∞ ,
0) of z h ¯ z h S ( a , a , a , a ) exists and is a linear combina-tion of ( z − z ) α z α z α (¯ z − ¯ z ) β ¯ z β ¯ z β f ( z / z ) , , (0.6)which is a real analytic function on Y = { ( z , z ) ∈ C | z , z , z , , z , } . Denoteby GCor the space of real analytic function spanned by (0.6).Since ( ∞ , z , z , ∈ X with | z | > | z | is in the convergent region of S ⋆ ))) ( a , a , a , a ),which is {| z | > | z | > | z | > | z |} , the limit of ( − h − ¯ h z h ¯ z h S ( a , a , a , a ) as ( z , z ) → ( ∞ ,
0) with | z | > | z | is equal to the formal limit,lim ( z , z ) → ( ∞ , ( − h − ¯ h z h ¯ z h S ⋆ ))) ( a , a , a , a ) = ( a , Y ( a , z ) Y ( a , z ) a ) . The convergent region of S ⋆ ))) ( a , a , a , a ) and S ⋆ )) ( a , a , a , a ) are {| z | > | z | > | z | > | z |} and {| z | > | z | > | z − z | , | z |} . Thus, the limits of ( − h − ¯ h z h ¯ z h S ( a , a , a , a )as ( z , z ) → ( ∞ ,
0) with | z | > | z | and | z | > | z − z | are given by ( a , Y ( a , z ) Y ( a , z ) a )and ( a , Y ( Y ( a , z ) a , z ) a ), respectively, where z = z − z . Furthermore, in the re-gions, {| z | > | z |} , {| z | > | z |} and {| z | > | z − z |} , z z is close to 0, ∞ and 1, respec-tively. Thus, the expansions of (0.6) in these regions are determined by the expansion of f ∈ F , , ∞ around { , , ∞} .0.3. Definition of full vertex algebra and main result.
Let us describe the precisedefinition of a full vertex algebra. A full vertex algebra is an R -graded vector space F = L h , ¯ h ∈ R F h , ¯ h with a distinguished vector ∈ F , and a vertex operator Y ( − , z ) : F → End F [[ z , ¯ z , | z | R ]] , a Y ( a , z ) = X r , s ∈ R a ( r , s ) z − r − ¯ z − s − satisfying the following conditions: For any a , b ∈ F , there exists N ∈ R such that a ( r , s ) b = unless r ≤ N and s ≤ N ; F h , ¯ h = h − ¯ h ∈ Z ; The vacuum vector satisfies Y ( , z ) = id F and Y ( a , z ) ∈ F [[ z , ¯ z ]] and lim z → Y ( a , z ) = a for any a ∈ F ; F h , ¯ h ( r , s ) F h ′ , ¯ h ′ ⊂ F h + h ′ − r − , ¯ h + ¯ h ′ − s − for any h , h ′ , ¯ h , ¯ h ′ , r , s ∈ R ; For any a , b , c ∈ F and u ∈ F ∨ = L h , ¯ h ∈ R F ∗ h , ¯ h , there exists µ ( z , z ) ∈ GCor such that u ( Y ( a , z ) Y ( b , z ) c ) = µ ( z , z ) | | z | > | z | , u ( Y ( Y ( a , z ) b , z ) c ) = µ ( z + z , z ) | | z | > | z | , (0.7) u ( Y ( b , z ) Y ( a , z ) c ) = µ ( z , z ) | | z | > | z | , where F ∗ h , ¯ h is the dual vector space and µ ( z , z ) | | z | > | z | is the expansion of µ in {| z | > | z |} .For a full vertex algebra F , we define linear maps D , ¯ D ∈ End F by Y ( a , z ) = a + Daz + ¯ Da ¯ z + . . . . Then, one can show that [ D , Y ( a , z )] = Y ( Da , z ) = d / dzY ( a , z ) and [ ¯ D , Y ( a , z )] = Y ( ¯ Da , z ) = d / d ¯ zY ( a , z ) (Proposition 2.1). Thus, if a , b ∈ ker ¯ D , then the generalized correlation func-tion (0.7) satisfies d / d ¯ z µ = d / d ¯ z µ =
0, that is, µ is a holomorphic function on Y . Weprove that if µ ∈ GCor is holomorphic, then µ ∈ C [ z ± , z ± , ( z − z ) ± ], which implies thatker ¯ D is a vertex algebra (Proposition 2.4). This reflects the fact that a function in F , , ∞ is holomorphic if and only if f ∈ C [ z ± , (1 − z ) − ]. Thus, in the definition of a full vertexalgebra, we replace the function space of a vertex algebra C [ z ± , (1 − z ) − ] with F , , ∞ ,which implies that the notion of a full vertex algebra is a generalization of the notion of a Z -graded vertex algebra.We remark that, for a vertex algebra, the pole of u ( Y ( a , z ) Y ( b , z ) c ) is ( z − z ) − k forsome k ∈ Z , which implies that ( z − z ) N [ Y ( a , z ) , Y ( b , z )] = ffi ciently large N ∈ Z ≥ . However, for a full vertex algebra, the pole of u ( Y ( a , z ) Y ( b , z ) c ) is a linear sumof ( z − z ) n | z − z | r for r ∈ R with n ∈ Z , e.g., 1 + | z − z | / + . . . . Hence, we couldnot simultaneously cancel all poles by multiplication by a polynomial. The number of theindex r ∈ R / Z appeared in ( z − z ) n | z − z | r reflects the number of intermediate states,which is mathematically the number of the irreducible components of the tensor productof modules. In the case of a full vertex algebra, the function space F , , ∞ is much morecomplicated, however, we can still describe all the consistencies of a four point functionin term of the expansions of the same function f ∈ F , , ∞ around { , , ∞} .In order to prove the consistency, we need the global conformal symmetry. A full vertexalgebra with an energy-momentum tensor, a pair of holomorphic and anti-holomorphicconformal vectors, which we call a full vertex operator algebra (see Section 3.1). Afull vertex operator algebra admits an action of the Virasoro algebras, Vir ⊕ Vir and, inparticular, a subalgebra, sl C ⊕ sl C = L i = − , , L ( i ) ⊕ L j = − , , L ( j ). A vector v ∈ F h , ¯ h is called a quasi-primary if L (1) v = L (1) v = F is generated by quasi-primary vectors as an sl C ⊕ sl C -module.A notion of a dual module, introduced in [FHL] for a vertex operator algebra, can begeneralized to a full vertex operator algebra and its modules. We give a criterion when afull vertex operator algebra is self-dual and QP-generated, which generalize a result of Li[L1]. The main result of this paper is that for a QP-generated self-dual full vertex operatoralgebra, all the consistencies of four point correlation functions hold (Theorem 3.2). Locality and the proof.
One can relax the assumption of a full vertex algebra, (0.7),to the existence of D , ¯ D ∈ End F and the following condition: For u ∈ F ∨ and a , a , a ∈ F , there exists µ ∈ GCor such that u ( Y ( a , z ) Y ( b , z ) c ) = µ ( z , z ) | | z | > | z | , u ( Y ( b , z ) Y ( a , z ) c ) = µ ( z , z ) | | z | > | z | . It is a generalization of the Goddard’s axiom of a vertex algebra (Proposition 4.1). Byusing this description, we construct a QP-generated self-dual full vertex operator algebraassociated with an even lattice.Hereafter, we recall the S -symmetry of the correlation function and then explain thekey idea of the proof of the consistency. The symmetric group S acts on X ⊂ ( C P ) bythe permutation, thus, on Cor . As a consequence of the consistency, four point correlationfunctions S : F ⊗ → Cor posses an S -symmetry, σ · S ( a , a , a , a ) = S ( a σ − , a σ − , a σ − , a σ − ) , ( S -symmetry)for any a , a , a , a ∈ F and σ ∈ S . Interestingly, in the Goddard’s axiom, we onlyassume that the correlation function is invariant under the permutation (23) ∈ S . Incontrast, four point correlation functions should be invariant under the S -symmetry. Weobserve that since the S -action commutes with the diagonal action of PSL C on X ⊂ ( C P ) , the homeomorphism ξ : PSL C \ X → C P \ { , , ∞} induces a S -action on C P \ { , , ∞} , which permutes { , , ∞} . Thus, the S -action is degenerate to the S -action, that is, the action of the Klein subgroup C × C ⊂ S is a part of the globalconformal symmetry. Since (34) · ξ = ξξ − and both ξ and ξξ − go to zero as ξ →
0, there isa simple relation between the expansion of f ◦ ξ in the domain | z | > | z | > | z | > | z | and | z | > | z | > | z | > | z | . This relation gives the skew-symmetry of a full vertex algebra F ,i.e., Y ( a , z ) b = exp( Dz + ¯ D ¯ z ) Y ( b , − z ) a for any a , b ∈ F , , ∞ . Since S is generated by the Klein subgroup together with (23) , (34) ∈ S , the axiom of a full vertex algebra and the global conformal symmetry implies that the S -symmetry of the correlation function (Theorem 3.1).Another important point is that in the definition of full vertex algebra, we considerthe generalized correlation function, whereas in the consistency, we need the four pointcorrelation functions. This problem is solved by the di ff erential equations derived fromthe global conformal symmetry.The bootstrap hypothesis is also proved by using this di ff erential equation. More pre-cisely, consider the following parenthesized correlation functions:( , (cid:16)(cid:16) a ( x ) a (cid:17) ( x ) a ( x ) a (cid:17) ( x ) )(s-channel) ( , (cid:16)(cid:16) a ( x ) a (cid:17) ( x ) a ( x ) a (cid:17) ( x ) )(t-channel) ( , (cid:16)(cid:16) a ( x ) a (cid:17) ( x ) a ( x ) a (cid:17) ( x ) ) . (u-channel) The equality of s-channel and t-channel (or s-channel and u-channel) are called a boot-strap equation . By using the di ff erential equation, we can relate them with the generalizedtwo point functions in (0.7) and one can show that under some assumptions if a vertex op-erator Y on F satisfies the bootstrap equation, then F is a full vertex algebra (Proposition4.7).0.5. Outline.
In Section 1, the definition of the space of correlation functions Cor andthe expansions e A : Cor → T A associated with parenthesized products are given. Prop-erties of the functions and expansions, e.g., relations among the expansions in di ff erentregions and di ff erential equations, holomorphic correlation functions, are studied. In Sec-tion 2, we define a full vertex algebra and study the consequence of the axiom by usingthe results in the previous section. In particular, we study a property of the translationmap D , ¯ D and show that ker ¯ D and ker D is a vertex algebra. A tensor product of full ver-tex algebras is also constructed here. Section 3 is devoted to studying the parenthesizedcorrelation functions and its consistency. The existence of bilinear form on a full vertexoperator algebra are also studied. In Section 4, a full vertex algebra is constructed fromthe Goddard’s axiom and the bootstrap equation and In Section 5, we construct an exam-ple of a QP-generated self-dual full vertex operator algebra from an even lattice, whichgeneralize lattice vertex algebras. In Appendix, the combinatorial rule for the expansionsis given. A cknowledgements The author would like to express his gratitude to his supervisor Masahito Yamazakiand Professor Yuji Tachikawa for useful discussions and to Shigenori Nakatsuka for read-ing the manuscript and valuable comments. This work was supported by World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan. The author wasalso supported by the Program for Leading Graduate Schools, MEXT, Japan.C ontents
Introduction 10.1. Space of four point correlation function 30.2. Consistency and parenthesized correlation functions 40.3. Definition of full vertex algebra and main result 60.4. Locality and the proof 80.5. Outline 9Acknowledgements 9Preliminaries and Notations 101. Correlation functions and formal calculus 121.1. Definition of four point functions 121.2. S -symmetry 141.3. Expansions 16 ff erential Equation and Convergence 241.6. Holomorphic correlation functions 261.7. Vacuum state and D -symmetry 281.8. Generalized two point Correlation function 302. Full vertex algebra 332.1. Definition of full vertex algebra 332.2. Holomorphic vertex operators 372.3. Tensor product of full vertex algebras 383. Correlation functions and full vertex algebras 393.1. Self-duality 393.2. Quasi-primary vectors 423.3. Parenthesized correlation functions and formal power series 443.4. Consistency of four point functions 464. Construction 524.1. Locality and Associativity I 524.2. Locality and Associativity II 544.3. Bootstrap 575. Example 585.1. Full vertex algebra and AH pair 595.2. Troidal compactification of string theory 635.3. Commutative algebra object in H-Vect 646. Appendix 656.1. Binary tree and expansions 65References 67P reliminaries and N otations We assume that the base field is C unless otherwise stated. Throughout of this paper, z and ¯ z are independent formal variables. We will use the notation z for the pair ( z , ¯ z ) and | z | for z ¯ z . For a vector space V , we denote by V [[ z , ¯ z , | z | R , . . . , z n , ¯ z n , | z n | R ]] the set of formalsums X s , ¯ s ,..., s n , ¯ s n ∈ R a s , ¯ s ,..., s n , ¯ s n z s ¯ z ¯ s . . . z s n ¯ z ¯ s n such that(1) a s , ¯ s ,..., s n , ¯ s n = s i − ¯ s i ∈ Z for all i = , . . . , n ;(2) { ( s , ¯ s , . . . , s n , ¯ s n ) ∈ R n | a s , ¯ s ,..., s n , ¯ s n , } is a countable set.We also denote by V (( z , ¯ z , | z | R , . . . , z n , ¯ z n , | z n | R )) the subspace of V [[ z , ¯ z , | z | R , . . . , z n , ¯ z n , | z n | R ]]spanned by the series satisfying(3) There exists N ∈ Z such that a s , ¯ s ,..., s n , ¯ s n = s i , ¯ s i ≥ N for all i = , . . . , n . For a series f ( z ) = P r , s ∈ R a r , s z r ¯ z s ∈ V (( z , ¯ z , | z | R )), The number of the elements { r ∈ R | a r , s , } in the image of the natural map R → R / Z is called an exponent of theseries . If the exponent of f ( z ) is l ∈ Z ≥ , then f ( z ) could be written as l X i = ∞ X n , m = a in , m z n ¯ z m | z | r i , where r i ∈ R and r i − r j < Z for any i , j .We will consider the following subspaces of V [[ z , ¯ z , | z | R ]]: V [[ z , ¯ z ]] = { X s , ¯ s ∈ Z ≥ a s , ¯ s z s ¯ z ¯ s | a s , ¯ s ∈ V } , V [ z ± , ¯ z ± ] = { X s , ¯ s ∈ Z a s , ¯ s z s ¯ z ¯ s | a s , ¯ s ∈ V , all but finitely many a s , ¯ s = } , V [ | z | R ] = { X r ∈ R a r z r ¯ z r | a r ∈ V , all but finitely many a r = } . We will also consider their combinations, e.g., V (( y / x , ¯ y / ¯ x , | y / x | R ))[ x ± , ¯ x ± , | x | R ] , which isa subspace of V [[ x , y , ¯ x , ¯ y , | x | R , | y | R ]] spanned by k X i = l X n , m = − l X r , s ∈ R a in , m , r , s x n + r i ¯ x m + r i ( y / x ) r (¯ y / ¯ x ) s for some k , l ∈ Z > and r i ∈ R and a in , m , r , s ∈ V such that a in , m , r , s = r − s ∈ Z andthere exists N such that a in , m , r , s = r ≥ N and s ≥ N .Let ddz and dd ¯ z be formal di ff erential operators acting on V [[ z , ¯ z , | z | R ]] by ddz X s , ¯ s ∈ R a s , ¯ s z s ¯ z ¯ s = X s , ¯ s ∈ R sa s , ¯ s z s − ¯ z ¯ s dd ¯ z X s , ¯ s ∈ R a s , ¯ s z s ¯ z ¯ s = X s , ¯ s ∈ R ¯ sa s , ¯ s z s ¯ z ¯ s − . Since ddz | z | s = s | z | s z − , the di ff erential operators ddz and dd ¯ z acts on all the above vectorspaces. Define a linear map lim z f ( w ) from a space of the formal variable ( z , ¯ z ) to a spaceof the formal variable ( w , ¯ w ) if the substitution of ( f ( w ) , f ( ¯ w )) into ( z , ¯ z ) is well-defined,that is, each coe ffi cient is a finite sum. For example, lim z z V (( z , ¯ z , | z | R )) → V (( z , ¯ z , | z | R ))is given by lim z z X r , s ∈ R a r , s z r ¯ z s = X r , s ∈ R ( − r − s a r , s z r ¯ z s . Since a r , s = r − s < Z , lim z z is well-defined. Another example islim x → x + y : V (( y / x , ¯ y / ¯ x , | y / x | R )) → V (( y / x , ¯ y / ¯ x , | y / x | R )) , which is defined bylim x → x + y (cid:16) X r , s ∈ R a r , s ( y / x ) r (¯ y / ¯ x ) s (cid:17) = X r , s ∈ R X i , j ∈ Z ≥ − ri ! − sj ! a r , s ( y / x ) r + i (¯ y / ¯ x ) s + j , where we used ( x + y ) s = P m ≥ (cid:16) sm (cid:17) x s − m y m . It is well-defined since a r , s = ffi cientlysmall r or s . We end this section by discussing a convergence of a formal power series in C (( z , ¯ z , | z | R , . . . , z n , ¯ z n , | z n | R )). Let f ∈ C (( z , ¯ z , | z | R , . . . , z n , ¯ z n , | z n | R )). Then, there exists N ∈ R such that | z | N | z | N · · · | z n | N f ( z , . . . , z n ) = X s , ¯ s ,..., s n , ¯ s n ∈ R s , ¯ s ,..., s n , ¯ s n ≥ a s , ¯ s ,..., s n , ¯ s n z s ¯ z ¯ s . . . z s n ¯ z ¯ s n . (0.8)We say the series f is absolutely convergent around 0 if | z | N | z | N · · · | z n | N f ( z , . . . , z n ) isabsolutely convergent in { ( z , z , . . . , z n ) ∈ C n | | z | , | z | , . . . , | z n | < R } for some R ∈ R > . Inthis case, f ( z , . . . , z n ) is compactly absolutely-convergent to a continuous function in theannulus { ( z , z , . . . , z n ) ∈ C n | < | z | , | z | , . . . , | z n | < R } .1. C orrelation functions and formal calculus In this section, we define the space of four point correlation functions, Cor and developformal calculus which we need to study a full vertex algebra. In Section 1.1, we defineCor and in Section 1.2, we study an action of the symmetric group S on Cor . Section1.3 and 1.7 is devoted to studying series expansions of a function in Cor . We study formaldi ff erential equations in Section 1.5 and holomorphic correlation function in Section 1.6.Certain limit of a four point correlation function is studied in Section 1.8. The reader whois only interested in the definition of a full vertex algebra can skip Section 1.3, 1.4, 1.5,1.6 and 1.7.1.1. Definition of four point functions.
For n ∈ Z ≥ , set X n = { ( z , . . . , z n ) ∈ ( C P ) n | z i , z j for any i , j } , called a space of ordered configurations of n points in the projectivespace C P . In this section, we define and study a space of four point correlation functionsin two dimensional conformal field theory, which are C -valued real analytic functions on X with possible singularity along { ( z , . . . , z ) ∈ ( C P ) | z i = z j } for 1 ≤ i < j ≤ C P is PSL C , whose action is called the linearfractional transformation and also called a global conformal symmetry. Define the realanalytic function ξ : X → P C \ { , , ∞} by ξ ( z , z , z , z ) = ( z − z )( z − z )( z − z )( z − z ) . It is easy to show that the map ξ : X → C P \ { , , ∞} is invariant under the diagonalaction of PSL C on X ⊂ ( C P ) , in fact, which gives a homeomorphism PSL C \ X → P C \ { , , ∞} (see, for example, [Y]). Remark 1.1.
The hypersurfaces in ( C P ) , { z = z } ∪ { z = z } and { z = z } ∪ { z = z } and { z = z } ∪ { z = z } are maps to and , ∞ respectively by ξ : ( C P ) → C P . Let α , . . . , α n ∈ C P and f be a C -valued real analytic function on C P \ { α , . . . , α n } .A chart ( χ, α ) of C P at a point α ∈ C P is a biholomorphism χ from an open subset U of C P to an open subset of C such that α ∈ U and χ ( α ) =
0. We say that f has a conformal singularity at α i if for any chart ( χ, α i ) of C P at α i , there exists a formal powerseries X r , s ∈ R a r , s p r ¯ p s ∈ C (( p , ¯ p , | p | R ))(1.1)such that it is compactly absolutely-convergent to f ◦ χ − ( p ) in the annulus { p ∈ C | < | p | < R } for some R ∈ R > (see Section 0.5). It is clear that the above condition isindependent of a choice of a chart and the coe ffi cients of the series is uniquely determinedby the chart.Denote by F , , ∞ the space of real analytic functions on C P \ { , , ∞} with possibleconformal singularities at { , , ∞} . Let p be a canonical coordinate of C ⊂ C P . Anon-trivial example of such functions is f Ising ( p ) = | − p − p | / + | + p − p | / , (1.2)which appears in a four point function of the 2 dimensional Ising model. We remark thatthe exponent of the series (1.1) is independent of the choice of the chart. An exponent ofa function f ∈ F , , ∞ is the maximal number of the exponents of the series expansion of f at { , , ∞} . For example, the expansion of f Ising for the charts p is2 + | p | / / − p / − ¯ p / + | p | / ( p + ¯ p ) / + p ¯ p / − p / − p / + . . . . (1.3)Since f Ising ( p ) satisfies f Ising ( p ) = f Ising (1 − p ) = ( z ¯ z ) / f Ising (1 / p ) , the exponent of f Ising is2. We remark that for a rational conformal field theory (or more generally quasi-rationalconformal field theory), the exponent is always finite.In this paper, we consider the following special charts of { , , ∞} :Chart(0 , , ∞ ) = { p , pp − , − p , − p , p , − p } . Recall that the ring of regular functions on the a ffi ne scheme C P \ { , , ∞} is C [ p ± , (1 − p ) ± ] and Chart(0 , , ∞ ) is a generator of the ring. It is easy to show that a function in C [ p ± , (1 − p ) ± ] has conformal singularities at { , , ∞} . Thus, C [ p ± , (1 − p ) ± ] ⊂ F , , ∞ .Conversely, from the existence of the expansion the following proposition follows: Proposition 1.1.
If f ∈ F , , ∞ is a holomorphic function on C P \ { , , ∞} , then f ∈ C [ p ± , (1 − p ) ± ] . We will consider the real analytic functions on X of the form: Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij f ◦ ξ ( z , z , z , z ) , (1.4)where f ∈ F , , ∞ and α i j , β i j ∈ R such that α i j − β i j ∈ Z for any 1 ≤ i < j ≤
4. In this paper,we always allow to have singularities around { z i = ∞} i = , , , ; The asymptotic behavior of(1.4) as z → ∞ is z α + α + α ¯ z β + β + β . By the condition α i j − β i j ∈ Z , we have( z i − z j ) α ij (¯ z i − ¯ z j ) β ij = | z i − z j | α ij (¯ z i − ¯ z j ) β ij − α ij , which implies that Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij is a single valued real analytic function on X .Let Cor be the space of C -valued real analytic functions spanned by (1.4). We alsoconsider the spaces of functions on X and X and X spanned by( z − z ) α (¯ z − ¯ z ) β , Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij , Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij where α i j , β i j ∈ R such that α i j − β i j ∈ Z for any 1 ≤ i < j ≤
4. We denote them byCor , Cor and Cor f , respectively. By the definition, we have: Lemma 1.1.
A function φ ∈ Cor is translation invariant, i.e., φ ( z , z , z , z ) = φ ( z + α, z + α, z + α, z + α ) for any α ∈ C . S -symmetry. Define the left action of the symmetric group S on ( C P ) by thepermutation, σ · ( z , z , z , z ) = ( z σ − , z σ − , z σ − , z σ − ) for ( z , z , z , z ) ∈ ( C P ) and σ ∈ S . Since the action commutes with the diagonal action of PSL C on X ⊂ ( C P ) ,we have an S -action on C P \ { , , ∞} . It is easy to show that each image of σ ∈ S inAut C P \{ , , ∞} is a linear fractional transformation and preserves { , , ∞} (see Remark1.1). Let Aut (0 , , ∞ ) be the subgroup of the linear fractional transformations PSL C which preserves { , , ∞} . Since the action of PSL C on C P is strictly 3-transitive, anelement of Aut (0 , , ∞ ) is uniquely determined by the action on the 3 points { , , ∞} .Thus, Aut (0 , , ∞ ) is isomorphic to the permutation group S . Hence, we have a grouphomomorphism t : S → Aut (0 , , ∞ ) and the kernel of this map is the Klein subgroup K = { , (12)(34) , (13)(24) , (14)(23) } .The action of S on X induces the dual action on the function space Cor by σ f ( − ) = f ( σ − − ) for f ∈ Cor and σ ∈ S , which also induces the action on the function space of C P \ { , , ∞} , F , , ∞ . Remark 1.2.
For σ ∈ S and Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij ∈ Cor f , σ · Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij = Π ≤ i < j ≤ ( z σ i − z σ j ) α ij (¯ z σ i − ¯ z σ j ) β ij . Regard the chart p ∈ Chart(0 , , ∞ ) as a function on C P \ { , , ∞} , for σ ∈ S ,we have σ · p ( − ) = p ( t − σ − ) = t − σ . p , where a bc d . p is defined by ap + bcp + d ∈ C ( p ) for a bc d ∈ PSL C . Lemma 1.2.
The explicit expressions of the map t : S → Aut (0 , , ∞ ) ⊂ PSL C and theorbit σ · p = t − σ . p isK (1) = , p p , (12) K (1 ∞ ) = − , p pp − , (23) K (0 ∞ ) = , p p , (13) K (01) = − , p − p , (123) K (01 ∞ ) = − , p p − p , (132) K (0 ∞ = −
11 0 , p − p . Furthermore, σ · ξ = t − σ .ξ for any σ ∈ S . Thus, the S -action induces an action on Chart(0 , , ∞ ), i.e., σ · χ ( p ) = χ ( p ( σ − − )) = χ ( t − σ . p ) for χ ( p ) ∈ Chart(0 , , ∞ ) and σ ∈ S . Hence, we have: Lemma 1.3.
The function spaces
Cor and Cor f and F , , ∞ are preserved by the action ofS . By the definition of a conformal singularity, for a chart χ ( p ) ∈ Chart(0 , , ∞ ) and f ∈ F , , ∞ , we have the expansion of f ◦ χ − ( p ) around p =
0, which defines a linear map j ( χ ( p ) , − ) : F , , ∞ → C (( p , ¯ p , | p | R )), that is, j ( χ ( p ) , f ) = X r , s ∈ R a r , s p r ¯ p s , where the right-hand-side is compactly absolutely convergent to f ( χ − ( p )) around p = t ∈ Aut (0 , , ∞ ), t · p = t − . p ∈ Chart(0 , , ∞ ) is a chart at t . t · p ) − : C P → C P is given by t − · p = t . p , e.g., − p is a chart at ∞ ∈ C P whoseinverse is p − p . Lemma 1.4.
For σ ∈ S , χ ( p ) ∈ Chart(0 , , ∞ ) and f ∈ F , , ∞ , j ( χ ( p ) , σ · f ) = j ( σ − · χ ( p ) , f ) . Proof.
We may assume that χ ( p ) = t . p for some t ∈ Aut (0 , , ∞ ). The series j ( t . p , σ f )is absolutely convergent to f ( t − σ t − . p ) around p =
0. Since f ( t − σ t − . p ) = f (( tt σ ) − . p ), wehave j ( t . p , σ f ) = j ( tt σ . p , f ) = j ( σ − · t . p , f ). (cid:3) The stabilizer subgroup of a point 0 ∈ C P in S (resp. Aut (0 , , ∞ )) is generated by theKlein subgroup K and (12) (resp. { , (0 ∞ ) } ), which is isomorphic to the dihedral group D . For σ ∈ K , the action of σ on F , , ∞ is trivial and for σ ∈ (12) K , the action of σ on F , , ∞ is (12) K · f ( p ) = f ( pp − ). Define the map lim p pp − : C (( p , ¯ p , | p | R )) → C (( p , ¯ p , | p | R ))by substituting − P n ≥ p n into p . Then, we have: Lemma 1.5.
For σ ∈ K , j ( σ · p , − ) = j ( p , − ) and j ((12) · p , − ) = lim p pp − j ( p , − ) . Expansions.
In this section, we consider expansions of a function in Cor . ByLemma 1.1, a function φ ∈ Cor can be expanded in three variables, e.g., z − z , z − z , z − z . Since φ is a multi-variable function, we need to determine the order of theexpansion. For example, we expand φ around z = z first and then around z = z , and z = z . By setting x = z − z , y = z − z , z = z − z , the resulting series is a formal powerseries of the variables ( x , ¯ x , y , ¯ y , z , ¯ z ), which is absolutely convergent in {| x | > | y | > | z |} .All such series expansions are described by parenthesized products of four symbols. Theabove case corresponds to 1((23)4). The innermost product (23) means that we expand φ around z − z first. Let P be the set of parenthesized products of four elements 1 , , , S naturally acts on P and P consists of the permutations of the following elements, called standard elements of P :(12)(34) , ((12)3)4 , (standard elements of P ) (1(23))4 , , . We introduce formal variables ( x A , y A , z A , ¯ x A , ¯ y A , ¯ z A ) for each A ∈ P . In this section, foreach A ∈ P , we define a series expansion e A : Cor → T A , where T A is a space of formal variables ( x A , y A , z A , ¯ x A , ¯ y A , ¯ z A ) defined below and show thatthe expansions are completely described by j ( χ ( p ) , − ) : F , , ∞ → C (( p , ¯ p , | p | R )) for some χ ( p ) ∈ Chart(0 , , ∞ ). Remark 1.3.
In introduction 0.2, we consider the expansions in four variables associatedwith an element in Q . Such expansions are described in Section 1.7 by using the resultsin this section. Set T ( x , y , z ) = C [[ y / x , y / x ]](( z / y , z / y , | z / y | R ))[ x ± , y ± , x ± , y ± , | x | R , | y | R ] , T f ( x , y , z ) = C [[ y / x , y / x , z / y , z / y ]][ x ± , y ± , z ± , x ± , y ± , ¯ z ± , | x | R , | y | R , | z | R ] , for the formal variables x , y , z , ¯ x , ¯ y , ¯ z .We first examine the case of A = φ ∈ Cor in the variable z = z − z (related to the innermost product (34)),then in y = z − z (related to 2(34)) and x = z − z (For the general rule forthe change of variables, see Appendix). The resulting formal power series is convergentin some open region in {| x | > | y | > | z |} , which define a linear map e : Cor → T ( x , y , z )such that for φ ( z , z , z , z ) ∈ Cor , the formal power series e ( φ ) is absolutely con-vergent to the function f ( z , z , z , z ) by taking the change of variables ( z − z , z − z , z − z ) = ( x , y , z ).For a function in Cor f ⊂ Cor , the map e is defined as a binomial expansion inthe domain | x | >> | y | >> | z | . For example, | z − z | r
7→ | x − z | r = | x | r X i , j ≥ ( − i ri ! rj ! x − i x − j z i z j , where the right hand side is absolutely convergent to | z − z | r in the open domain | z − z | > | z − z | and is an element of T f ( x , y , z ). We denote this map by e f : Cor f → T f ( x , y , z ) . Let f ∈ F , , ∞ . For f ◦ ξ ∈ Cor , expand ξ = ( z − z )( z − z )( z − z )( z − z ) in | x | >> | y | >> | z | , e.g., ξ = ( z − z )( z − z )( z − z )( z − z ) ( x − y ) z y ( x − z ) = z / y (1 − y / x ) × X i ≥ ( z / x ) i , which is equal to e f ( ( z − z )( z − z )( z − z )( z − z ) ), defined above. Define the map s : C (( p , ¯ p , | p | R )) → T ( x , y , z ) by substituting p = e f ( ( z − z )( z − z )( z − z )( z − z ) ) into P r , s ∈ R a r , s p r ¯ p s . Since p = e f ( ( z − z )( z − z )( z − z )( z − z ) ) is an element of z / y C [[ y / x , z / y ]] , by the definition of C (( p , ¯ p , | p | R )), s is well-defined. The following lemma is obvi-ous: Lemma 1.6. T ( x , y , z ) is a module over the C -algebra T f ( x , y , z ) . Then, e : Cor → T ( x , y , z ) is defined by e ( φ · f ◦ ξ )) = e f ( φ ) s ( j ( p , f ))for φ ∈ Cor f and f ∈ F , , ∞ , which is absolutely convergent to φ · f ◦ ξ in | x | >> | y | >> | z | . Here, | x | >> | y | >> | w | means some non-empty open domain in { ( z , z , z , z ) | | z − z | > | z − z | > | z − z |} . The map e isindependent of a choice of the decomposition φ · f ◦ ξ ∈ Cor , since the coe ffi cient of aconvergent series is unique.Second, we give the definition of e A in case of A = z − z , z − z , z − z ) ( x , y , z ) . The map e f : Cor f → T f ( x , y , z ) is defined by the binomial ex-pansion in the domain | x | >> | y | >> | z | as above. The expansion of ξ = ( z − z )( z − z )( z − z )( z − z ) is equal to ξ = ( x − y + z ) y ( x − y )( z + y ) = ( x − y + z ) y × X i , j ≥ ( − j x − i − y i − j − z j . Importantly, this is not in z / y C [[ y / x , z / y ]] . Thus, the substitution of this into j ( p , f ) is not well-defined for f ∈ F , , ∞ . The reasonis that when z = z − z goes to zero, ξ goes to 1. Thus, we have to consider theexpansion of f around 1 and the expansion of 1 − ξ = ( z − z )( z − z )( z − z )( z − z ) . Then, the expansion is1 − ξ = x z ( x − y )( y + z ) = z / y X i , j ≥ ( − j ( y / x ) i ( z / y ) j , which is an element of z / y C [[ y / x , z / y ]] . Define the map s : C (( p , ¯ p , | p | R )) → T ( x , y , z )by substituting p = e f (1 − ξ ) into P r , s ∈ R a r , s p r ¯ p s . Then, e : Cor → T ( x , y , z )is defined by e ( φ f ◦ ξ ) = e f ( φ ) s ( j (1 − p , f ))for φ ∈ Cor f and f ∈ F , , ∞ . Since j (1 − p , f ) is absolutely convergent to f (1 − p ), s ( j (1 − p , f ) is absolutely convergent to f ◦ ξ in | x | >> | y | >> | z | . Remark 1.4.
In the domain | x | >> | y | >> | z | , the di ff erence | z | = | z − z | is assumed to be small. Then, by Remark 1.1, ξ . Thus, we have to considerthe expansion of − ξ or − ξ − . If we choose − ξ − , we substitute e f (1 − ξ − ) intoj ( − p , f ) . Then, the resulting series is, roughly, convergent to f ( − (1 − p − ) ) = f ( p ) . Hence,both of the choices − ξ and − ξ − give the same result. The last example is the case of A = (12)(34), which has a special property. We considerthe following change of variables:( z − z , z − z , z − z ) ( x (12)(34) , y (12)(34) , z (12)(34) ) . Then, z − z x (12)(34) + y (12)(34) − z (12)(34) , z − z x (12)(34) + y (12)(34) , z − z x (12)(34) − z (12)(34) . Since there is no y (12)(34) − z (12)(34) or y (12)(34) + z (12)(34) in the above list, the image of e f (12)(34) : Cor f → T f ( x (12)(34) , y (12)(34) , z (12)(34) ) is in C [[ y / x , z / x , y / x , z / x ]][ x ± , y ± , z ± , x ± , y ± , z ± , | x | R , | y | R , | z | R ] , that is, both y (12)(34) and z (12)(34) are bounded below, which happens only for the S -conjugate elements of (12)(34). Set T fA ( x , y , z ) = C [[ y / x , z / x , y / x , z / x ]][ x ± , y ± , z ± , x ± , y ± , z ± , | x | R , | y | R , | z | R ] , for any A ∈ P which is S -conjugate of (12)(34). Then, the binomial expansion defines e (12)(34) : Cor → T f (12)(34) ( x (12)(34) , y (12)(34) , z (12)(34) ) and e (12)(34) ( ξ ) = y (12)(34) z (12)(34) x (12)(34) ( y (12)(34) + x (12)(34) − z (12)(34) ) = z (12)(34) x (12)(34) y (12)(34) x (12)(34) X n ≥ n X i = ( − n − i ni !(cid:16) y (12)(34) x (12)(34) (cid:17) n − i (cid:16) z (12)(34) x (12)(34) (cid:17) i . Let T ′ ( x , y , z ) be a subspace of C (( y / x , z / x , ¯ y / ¯ x , ¯ z / ¯ x , | y / x | R , | z / x | R )) spanned by X s , s , ¯ s , ¯ s ∈ R c s , ¯ s , s , ¯ s (cid:16) yx (cid:17) s (cid:16) ¯ y ¯ x (cid:17) ¯ s (cid:16) zx (cid:17) s (cid:16) ¯ z ¯ x (cid:17) ¯ s such that:(1) c s , ¯ s , s , ¯ s = s − ¯ s , s − ¯ s ∈ Z ;(2) There exists N ∈ R such that c s , ¯ s , s , ¯ s = s , ¯ s , s , ¯ s ≥ N ;(3) c s , ¯ s , s , ¯ s = s − s , ¯ s − ¯ s ∈ Z .Set T A ( x , y , z ) = T ′ ( x , y , z )[ x ± , y ± , z ± , ¯ x ± , ¯ y ± , ¯ z ± , | x | R , | y | R , | z | R ] , for any A ∈ P which is S -conjugate to (12)(34). Define the map s (12)(34) : C (( p , ¯ p , | p | R )) → T (12)(34) ( x (12)(34) , y (12)(34) , z (12)(34) )by substituting p = e f (12)(34) ( ξ ) into P r , s ∈ R a r , s p r ¯ p s . The following lemma is obvious: Lemma 1.7.
For any A ∈ P which is S -conjugate of (12)(34) , T fA ( x , y , z ) is a subspaceof T A ( x , y , z ) and T A ( x , y , z ) is a module over the C -algebra T fA ( x , y , z ) . Then, e (12)(34) : Cor → T (12)(34) ( x (12)(34) , y (12)(34) , z (12)(34) ) can be defined as above. Now, we will consider the general case. For standard elements of P , consider thefollowing change of variables:(12)(34) : ( x (12)(34) , y (12)(34) , z (12)(34) ) = ( z − z , z − z , z − z ) , ((12)3)4 : ( x ((12)3)4 , y ((12)3)4 , z ((12)3)4 ) = ( z − z , z − z , z − z ) , (1(23))4 : ( x (1(23))4 , y (1(23))4 , z (1(23))4 ) = ( z − z , z − z , z − z ) , x , y , z ) = ( z − z , z − z , z − z ) , x , y , z ) = ( z − z , z − z , z − z ) . For general A ∈ P , regard the above x A , y A , z A as polynomials of ( z , z , z , z ). Then, S naturally acts on them. For a standard element A and σ ∈ S , set ( x σ A , y σ A , z σ A ) = ( σ x A , σ y A , σ z A ) . For example,( x , y , z ) = (124) · ( z − z , z − z , z − z ) = ( z − z , z − z , z − z ) . For each A ∈ P which is not S -conjugate to (12)(34), set T A ( x A , y A , z A ) = T ( x A , y A , z A )and T fA ( x A , y A , z A ) = T f ( x A , y A , z A ) . Then, we have the binomial expansion convergentin | x A | >> | y A | >> | z A | , e fA : Cor f → T fA ( x A , y A , z A ) as above. Define the map T σ : T ( x A , y A , z A ) → T ( x σ A , y σ A , z σ A ) by ( x A , y A , z A , ¯ x A , ¯ y A , ¯ z A ) ( x σ A , y σ A , z σ A , ¯ x σ A , ¯ y σ A , ¯ z σ A ).Then, we have: Lemma 1.8.
For any σ ∈ S and A ∈ P and φ ∈ Cor f , T σ ◦ e A ( φ ) = e σ A ( σφ ) . By Remark 1.4, we need to choose an appropriate choice of ξ, ξξ − , − ξ, − ξ − , ξ − , − ξ for each A ∈ P , when we expand a function f ∈ F , , ∞ . The map τ : P → S is chosenfor standard elements as (12)(34) , ((12)3)4 , (1(23))4 (13) , (13) , , and for σ A ∈ P as τ ( σ A ) = σ · τ ( A ), where σ ∈ S and A is a standard element of P .Then, for standard element A , the expansion of τ ( A ) · ξ is given by: ξ = y (12)(34) z (12)(34) x (12)(34) ( y (12)(34) + x (12)(34) − z (12)(34) ) = z (12)(34) y (12)(34) X n ≥ n X i = ( − n − i ni ! x − n − y n − i (12)(34) z i (12)(34) ,ξ = x ((12)3)4 z ((12)3)4 ( z ((12)3)4 + y ((12)3)4 )( y ((12)3)4 + x ((12)3)4 ) = z ((12)3)4 / y (12)(34) X i , j ≥ ( − i + j x − i (12)(34) y i − j (12)(34) z j (12)(34) , − ξ = ( x (1(23))4 + y (1(23))4 ) z (1(23))4 y (1(23))4 ( x (1(23))4 + z (1(23))4 ) = z (1(23))4 / y (1(23))4 (1 − y (1(23))4 / x (1(23))4 ) X i ≥ ( − i x − i (1(23))4 z i (1(23))4 , − ξ = x z ( x − y )( y + z ) = z / y X i , j ≥ ( − j x − i y i − j z j ,ξ = ( x − y ) z y ( x − z ) = z / y (1 − y / x ) X i ≥ x − i z i . Thus, for σ ∈ S , by Lemma 1.8, e f σ A ( τ ( σ A ) · ξ ) = e f σ A ( στ ( A ) · ξ ) = T σ e fA ( τ ( A ) · ξ ) , whichimplies that e f σ A ( τ ( σ A ) · ξ ) ∈ z σ A / x σ A y σ A / x σ A C [[ y σ A / x σ A , z σ A / x σ A ]] if A is S -conjugateto (12)(34) and e f σ A ( τ ( σ A ) · ξ ) ∈ z σ A / y σ A C [[ y σ A / x σ A , z σ A / y σ A ]] otherwise.Define the map s A : C (( p , ¯ p , | p | R )) → T A ( x A , y A , w A )by substituting p = e fA ( τ ( A ) · ξ ) into P r , s ∈ R a r , s p r ¯ p s . Then, the composition of j ( τ ( A ) · p , − ) : F , , ∞ → C (( p , ¯ p , | p | R )) and s A define the map e A : Cor → T A ( x A , y A , z A ). Then, we have: Proposition 1.2.
For A ∈ P and φ ∈ Cor , e A ( φ ) ∈ T A ( x A , y A , z A ) is absolutely convergentto φ in some non-empty domain in | x A | > | y A | > | z A | and T σ e A ( φ ) = e σ A ( σ · φ ) .Proof. Let f ∈ F , , ∞ . Then, e A ( f ◦ ξ ) = s A ( j ( τ ( A ) · p , f )) = s A ( j ( t − τ ( A ) . p , f )) is, by Lemma1.2, absolutely convergent to f ( t τ ( A ) . ( τ ( A ) · ξ )) = f ( t τ ( A ) . t − τ ( A ) .ξ ) = f ( ξ ) . Furthermore, for σ ∈ S , since T σ e fA ( τ ( A ) ξ ) = e σ A f ( τ ( σ A ) ξ ), by Lemma 1.4, we have e σ A ( σ · f ◦ ξ ) = s σ A ( j ( σ · τ ( A ) · p , σ · f )) = T σ s A j ( τ ( A ) · p , f ) = T σ e A ( f ◦ ξ ) (cid:3) The symbols e A and x A , y A , z A is convenient to express the various expansions systemat-ically. However, in practice it is cumbersome to work in this notation. For A = (1(23))4, x A , y A , z A represents z − z , z − z , z − z . Thus, we introduce new formal variables z , z , z and set z = x (1(23))4 , z = y (1(23))4 , z = z (1(23))4 . Then, the map e A is a mapfrom Cor to T ( z , z , z ). For, φ ∈ Cor , we also denote e (1(23))4 ( φ ) by φ | | z | > | z | > | z | . Thenotations x A , y A , z A and z i j will be used interchangeably in this paper.We remark that we can define the expansion maps Cor , Cor in similar way, e.g.,Cor → T ( z , z ) , φ φ | | z | > | z | . Relations among expansions.
In this section, we establish relations among expan-sions e A for A ∈ P . For A , B ∈ P , define the map T ′ BA : T ( x A , y A , z A ) → T ( x B , y B , z B )by sending ( x A , y A , z A , ¯ x A , ¯ y A , ¯ z A ) into ( x B , y B , z B , ¯ x B , ¯ y B , ¯ z B ).We first construct a map T : T ( x , y , z ) → T ( x , y , z )such that T ◦ e = e . By definition, there should be a relation x = z − z = x − y , y = z − z = − y , z = z − z = z . Thus, define T : T ( x , y , z ) → T ( x , y , z ) by T = T ′ exp( − y ddx ) exp( − ¯ y dd ¯ x ) lim y y . We remark that the operator exp( y ddx ) : C (( y / x )) → C (( y / x )) changes the variable x into x + y , that is, exp( y ddx ) X n a n x − n y n = X n X i ≥ − ni ! a n x − n − i y n + i . For σ ∈ S , define T σ σ : T ( x σ , y σ , z σ ) → T ( x σ , y σ , z σ )by T σ σ = T σ ◦ T ◦ T σ − . Then, we have:
Lemma 1.9.
For any σ ∈ S , T σ σ ◦ e σ = e σ . Second, we construct T : T ( x (1(23))4 , y (1(23))4 , z (1(23))4 ) → T ( x , y , z )and T : T ( x ((12)3)4 , y ((12)3)4 , z ((12)3)4 ) → T ( x , y , z ). Similarly to theabove, by the relation x = z − z = − x (1(23))4 , y = z − z = y (1(23))4 , z = z − z = z (1(23))4 , and x = z − z = − x ((12)3)4 − y ((12)3)4 , y = z − z = − y ((12)3)4 , z = z − z = z ((12)3)4 . Thus, for σ ∈ S , set T = T ′ lim x (1(23))4 →− x (1(23))4 T = T ′ lim ( x ((12)3)4 , y ((12)3)4 ) → ( − x ((12)3)4 , − y ((12)3)4 ) exp y ((12)3)4 ddx ((12)3)4 exp ¯ y ((12)3)4 dd ¯ x ((12)3)4 and T σ σ (1(23))4 = T σ ◦ T ◦ T σ − , T σ σ ((12)3)4 = T σ ◦ T ◦ T σ − . Then, we have
Lemma 1.10.
For any σ ∈ S , T σ σ (1(23))4 ◦ e σ (1(23))4 = e σ and T σ σ ((12)3)4) ◦ e σ ((12)3)4) = e σ . By the above lemmas, all expansions except for e (12)(34) can be transformed into e .For e (12)(34) , the relation is x = z − z = x (12)(34) , y = z − z = y (12)(34) + x (12)(34) , z = z − z = z (12)(34) . It seems that we only need to consider the operator exp( − x (12)(34) ddy (12)(34) ), However, it isnot well-defined, since, informally, exp( x ddy )( x − y ) − = − y and exp( x ddy ) P n ≥ x − − n y n = P n ≥ P nk = x − − n ( y + x ) n whose coe ffi cient of x − is 1 + + . . . . We will discuss this issuein Section 1.5.Finally, we transform e into e . In this case, the relation is x = z − z = x − z , y = z − z = y − z , z = z − z = − z . Set T = T ′ lim z →− z exp( z ( ddy + ddx )) exp(¯ z ( dd ¯ y + dd ¯ x )) . Then,
Lemma 1.11. T e = e . Remark 1.5.
We remark that the existence of this transformation reflects the fact thatboth e and e correspond to the expansion of a function in F , , ∞ at the samepoint, ∈ C P . The Borcherds identity, an axiom of a vertex algebra, can be derivedfrom a property of holomorphic functions, the Cauchy integral formula, which relate theexpansions at the di ff erent points. However, for a real analytic function in F , , ∞ , it seemsdi ffi cult to compare the expansions at the di ff erent points. Formal Di ff erential Equation and Convergence. Since ξ : X → C P \ { , , ∞} is PSL C invariant, the image of an expansion e A satisfies di ff erential equations. We firstconsider the formal di ff erential equations for e . We abbreviate ( x , y , z )to ( x , y , z ). Set D = x d / dx + y d / dy + z d / dz , ¯ D = ¯ x d / d ¯ x + ¯ y d / d ¯ y + ¯ z d / d ¯ z , D = xd / dx + yd / dy + zd / dz , ¯ D = ¯ xd / d ¯ x + ¯ yd / d ¯ y + ¯ zd / d ¯ z , which acts on T ( x , y , z ) and Set S ( x , y , z ) = ∩ i ∈{ , } (ker D i ∩ ker ¯ D i ), which is the space ofthe formal solutions of the di ff erential equations. The following lemma follows from thePSL C -invariance of ξ : Lemma 1.12.
The image of s : C (( p , ¯ p , | p | R )) → T ( x , y , z ) is in S ( x , y , z ) . We will show that s : C (( p , ¯ p , | p | R )) → S ( x , y , z ) is an isomorphism of vectorspaces. Let f ( x , y , z ) ∈ S ( x , y , z ). Since D f = D f , we can assume that f ( x , y , z ) = X n , m , r , s ∈ R a n , m , r , s ( y / x ) n ( z / y ) m (¯ y / ¯ x ) r (¯ z / ¯ y ) s ∈ C [[ y / x , ¯ y / ¯ x ]](( z / x , ¯ z / ¯ x , | z / x | R ))[( y / x ) ± , (¯ y / ¯ x ) ± , | y / x | R ] . By D f = D f =
0, we have( n − m ) a n , m , r , s + ( m − a n , m − , r , s − ( n + a n + , m , r , s = , ( r − s ) a n , m , r , s + ( s − a n , m , r , s − − ( r + a n , m , r + , s = . By the definition of T ( x , y , z ), a n , m , r , s = n , m , r , s is su ffi ciently small. Fix r , s ∈ R and let n = min { n | a n , m , r , s , m ∈ R } . Then, since n a n , m , r , s = ( n − − m ) a n − , m , r , s + ( m − a n − , m − , r , s =
0, we have n =
0. If there exists n , m , r , s ∈ R such that n ∈ R \ Z and a n , m , r , s ,
0, then either a n − , m , r , s or a n − , m − , r , s is non-zero,which contradicts the fact that a n , m , r , s = ffi ciently small n . Thus, f ( x , y , z ) = P n , r ∈ Z ≥ P m , s ∈ R a n , m , r , s ( y / x ) n ( z / y ) m (¯ y / ¯ x ) r (¯ z / ¯ y ) s . Define the map v : S ( x , y , z ) → C (( p , ¯ p , | p | R ))by f ( x , y , z ) X m , s ∈ R a , m , , s p m ¯ p s , which is a formal limit of ( x , y , z ) ( ∞ , , p ). Lemma 1.13.
The map v is inverse to s .Proof.
Recall that e ξ = (1 − y / x ) z / y P m ≥ ( z / x ) m , which implies that v ( s ( p )) = p . Thus, v is a left inverse of s . Hence, it su ffi ces to show that v is in-jective. Let f ( x , y , z ) = P n , r ∈ Z ≥ P m , s ∈ R a n , m , r , s ( y / x ) n ( z / y ) m (¯ y / ¯ x ) r (¯ z / ¯ y ) s ∈ S ( x , y , z ) satisfy v ( f ( x , y , z )) =
0, that is, a , m , , r = m , r ∈ R . Then, it is easy to show that f = (cid:3) Corollary 1.1.
Let g ∈ S ( x , y , z ) . If there exists f ∈ F , , ∞ such that v ( g ) = j ( p , f ) ,then g = e ( f ◦ ξ ) . In particular, g is absolutely convergent to a function in Cor in | x | >> | y | >> | z | . We second consider the formal di ff erential equations for e (12)(34) . We abbreviate( x (12)(34) , y (12)(34) , z (12)(34) ) to ( x , y , z ) again. Set D ′ = x d / dx + y d / dy + z d / dz , ¯ D ′ = ¯ x d / d ¯ x + ¯ y d / d ¯ y + ¯ z d / d ¯ z , D ′ = xd / dx + yd / dy + zd / dz , ¯ D ′ = ¯ xd / d ¯ x + ¯ yd / d ¯ y + ¯ zd / d ¯ z , and let S ′ ( x , y , z ) be the space of the formal solutions of D ′ , D ′ , ¯ D ′ , ¯ D ′ in T (12)(34) ( x , y , z ).Similarly to the above, we have s (12)(34) : C (( p , ¯ p , | p | R )) → S ′ ( x , y , z ) . We will show that the map is an isomorphism of vector spaces by constructing the inversemap. By D ′ and ¯ D ′ , any element of S ′ ( x , y , z ) is of the form P n , m , r , s ∈ R a n , m , r , s ( y / x ) n ( z / x ) m (¯ y / ¯ x ) r (¯ z / ¯ x ) s . The map v (12)(34) : S ′ ( x , y , z ) → C (( p / , ¯ p / , | p | R )) defined by X n , m , r , s a n , m , r , s ( y / x ) n ( z / x ) m (¯ y / ¯ x ) r (¯ z / ¯ x ) s X n , m , r , s a n , m , r , s p n + m ¯ p r + s is well-defined by the definition of T (12)(34) ( x , y , z ). Furthermore, since e (12)(34) ( ξ ) = ( y / x )( z / x )(1 + ( z / x − y / z ) + ( z / x − y / z ) + . . . ), we have v (12)(34) ◦ s (12)(34) ( ξ ) = p , which implies that e (12)(34) is injective. Thus, it su ffi ces to show that e (12)(34) is surjective. Let g ( x , y , z ) = P n , m , r , s a n , m , r , s ( y / x ) n ( z / x ) m (¯ y / ¯ x ) r (¯ z / ¯ x ) s ∈ S ′ ( x , y , z ). Since it is in the kernel of D ′ and ¯ D ′ ,( n − m ) a n , m , r , s + na n − , m , r , s + ma n , m − , r , s = , ( r − s ) a n , m , r , s + ra n , m , r − , s + sa n , m , r , s − = . Lemma 1.14.
For non-zero g ( x , y , z ) ∈ S ( x , y , z ) . Set N = min { n + m | a n , m , r , s , for some r , s ∈ R } . and R = min { r + s | a n , N − n , r , s , for some n , r , s ∈ R } . Then, a N , N , R , R , anda n , N − n , r , R − r = for any n , N or r , R . Furthermore, N − R ∈ Z .Proof. Let n , r ∈ R satisfy a n , N − n , r , R − r ,
0. Then, by the above recurrence formula,2( N − n ) a n , N − n , r , R − r = n a n − , N − n , r , R − r + (2 N − n ) a n , N − n − , r , R − r =
0. Thus, n = N and r = R . Since g ( x , y , z ) ∈ T (12)(34) ( x , y , z ) , a n , m , r , s = n − r < Z or m − s < Z ,which implies that N − R ∈ Z . (cid:3) By the above lemma, p N ¯ p R ∈ C [ p , ¯ p , | p | R ]. Thus, we can replace g ( x , y , z ) by g ( x , y , z ) − e (12)(34) ( a N , N , R , R p N ¯ p R ), that is, g ( x , y , z ) − e (12)(34) ( a N , N , R , R p N ¯ p R ) ∈ S ′ ( x , y , z ). We re-peat this process for R and N . Thus, we have g ( x , y , z ) ∈ Im e (12)(34) , which implies that e (12)(34) is surjective and the image of v (12)(34) is in C (( p , ¯ p , | p | R )) ⊂ C (( p / , ¯ p / , | p | R )).Hence, we have: Lemma 1.15.
The map v (12)(34) : S ′ ( x , y , z ) → C (( p , ¯ p , | p | R )) is the inverse of s (12)(34) : C (( p , ¯ p , | p | R )) → S ′ ( x , y , z ) . Corollary 1.2.
Let g ′ ∈ S ′ ( x , y , z ) . If there exists f ∈ F , , ∞ such that v (12)(34) ( g ′ ) = j ( p , f ) ,then g ′ = e (12)(34) ( f ◦ ξ ) . In particular, g ′ is absolutely convergent to a function in Cor in | x | >> | y | >> | z | . Finally, we state a similar result on Cor . Set D ′′ = x d / dx + y d / dy , ¯ D ′′ = ¯ x d / d ¯ x + ¯ y d / d ¯ yD ′′ = xd / dx + yd / dy , ¯ D ′′ = ¯ xd / d ¯ x + ¯ yd / d ¯ y , and let S ′′ ( x , y ) be a space of the formal solutions of D ′ , D ′ , ¯ D ′ , ¯ D ′ in T ( x , y ) = C [[ y / x , ¯ y / ¯ x } [ x ± , ¯ x ± , | x | , y ± , ¯ y ± , | y | ] . Then, we have:
Lemma 1.16.
The space S ′′ ( x , y ) consists of the constant functions.Proof. Let f ∈ S ′′ ( x , y ) By D ′′ f = ¯ D ′′ f =
0, we can assume that f ( x , y ) = P r , s ∈ R a r , s ( y / x ) r (¯ y / ¯ x ) s .By D ′′ f = ¯ D ′′ f = a n , ¯ n satisfies ra r , s = ( r + a r + , s and sa r , s = ( s + a r , s + . Thus, a r , s = r , s ) = (0 , (cid:3) Holomorphic correlation functions.
In this section, we consider holomorphic cor-relation functions.The following proposition is important:
Proposition 1.3.
If d / dz a φ = for φ ∈ Cor and a ∈ { , , , } , then φ is in Cor f .Furthermore, φ is a finite sum of functions Π ≤ i < j ≤ ( z i − z j ) α ij ( z i − z j ) β ij such that α i j − β i j ∈ Z and β ai = for i , j = , , , .Proof. By the S -symmetry, we may assume that a =
2. By the definition of Cor , we canassume that φ ( z , z , z , z ) = P Nt = φ t ( z , z , z , z ) and φ t ( z , z , z , z ) = Π ≤ i < j ≤ ( z i − z j ) α tij ( z i − z j ) β tij f t ( ( z − z )( z − z )( z − z )( z − z ) )(1.5)and f t ∈ F , , ∞ and for χ ∈ { p , − p , p − } j ( χ, f t ) = ∞ X k = | p | r t ,χ k X n , m ≥ a t ,χ n , m p n ¯ p m . (1.6)We first prove that there exists integers n , n , n , N ∞ such that d / dz N ∞ (( z − z ) − n + ( z − z ) − n + ( z − z ) − n + φ ) =
0. Fix z , z , z . Since d / dz φ = φ is a holomorphic func-tion for the variable z on C \ { z , z , z } . Let n (resp. n , n ) be the largest integer smaller than any element of the set { α t + β t + r t , pk } t = ,..., N , k = , , ,... (resp. { α t + β t + r t , − pk } t = ,..., N , k = , , ,... and { α t + β t + r t , p − k } t = ,..., N , k = , , ,... ), which exists since r t ,χ k ∈ R isbounded below. Then, by (1.5) and (1.6), lim z → z | z − z | − n + f = z → z | z − z | − n + f = z → z | z − z | − n + f =
0. Hence, ( z − z ) − n + ( z − z ) − n + ( z − z ) − n + φ is a holomorphic function on C . Since d / dz ( z − z ) − n + ( z − z ) − n + ( z − z ) − n + φ =
0, we may assume without loss of generality that φ is a holomorphic functionfor the variable z on C . Let n ∞ be the largest integer smaller than any element of the set { α t + α t + α t + } t = ,..., N . Then, by (1.5), lim z →∞ | z | − n ∞ φ =
0. Hence, φ is a rationalfunction on the projective plane C P and holomorphic on C . Thus, φ is a polynomialwhose degree is less that n ∞ , which implies that d / dz n ∞ φ = e : Cor → T ( x , y , z ) andset ( x , y , z , ¯ x , ¯ y , ¯ z ) = ( x , y , z , ¯ x , ¯ y , ¯ z ). Since T ( x , y , z ) is acompletion of M r , s ∈ R z r ¯ z s C [[ y / x , ¯ y / ¯ x ]][ x ± , y ± , ¯ x ± , ¯ y ± , | x | R , | y | R ] , by d / dz n ∞ φ = d / dz φ = e ( φ ) ∈ n ∞ − M k = z k C [[ y / x , ¯ y / ¯ x ]][ x ± , y ± , ¯ x ± , ¯ y ± , | x | R , | y | R ] . Denote by pr n ∞ : T ( x , y , z ) → L n ∞ − k = z k C [[ y / x , ¯ y / ¯ x ]][ x ± , y ± , ¯ x ± , ¯ y ± , | x | R , | y | R ] the projec-tion of T ( x , y , z ) onto the coe ffi cients of 1 , z , . . . , z n ∞ − . Set z i j = z i − z j for i , j = , . . . , e ( φ ) = pr n ∞ e ( φ ) = pr n ∞ N X t = e f ( Π ≤ i < j ≤ ( z i j ) α tij (¯ z i j ) β tij ) s ( j (1 − p , f t ) , = pr n ∞ N X t = ∞ X k = e f ( Π ≤ i < j ≤ ( z i j ) α tij (¯ z i j ) β tij ) s ( X n , m ≥ a t , − pn , m p n ¯ p m | p | r t , − pk ) . (1.7)Since e f ( Π ≤ i < j ≤ z α tij i j ¯ z β tij i j ) ∈ z α t ¯ z β t C [[ y / x , z / y , ¯ y / ¯ x , ¯ z / ¯ y ]][ x ± , y ± | x | R , | y | R ] and s ( p a ¯ p a ′ ) ∈ z a ¯ z a ′ C [[ y / x , z / y , ¯ y / ¯ x , ¯ z / ¯ y ]][ x ± , y ± | x | R , | y | R ] for any a , a ′ ∈ R , pr n ∞ e f ( Π ≤ i < j ≤ z α tij i j ¯ z β tij i j ) s ( p a ¯ p a ′ ) is equal to zero unless α t + a ∈ Z ≤ n ∞ and β t + a ′ ∈ Z ≤ . Thus, the right hand side of (1.7)is finite sum, that is, = pr n ∞ N X t = X k = , , ,... r t , − pk + β t ∈ Z ≤ r t , − pk + α t ∈ Z ≤ n ∞ e f ( Π ≤ i < j ≤ ( z i j ) α tij (¯ z i j ) β tij ) s ( X n ∞ − α t − r t , − pk ≥ n ≥ − β t − r t , − pk ≥ m ≥ a t , − pn , m p n ¯ p m | p | r t , − pk ) . = pr n ∞ N X t = X k = , , ,... r t , − pk + β t ∈ Z ≤ r t , − pk + α t ∈ Z ≤ n ∞ X n ∞ − α t − r t , − pk ≥ n ≥ − β t − r t , − pk ≥ m ≥ e f ( Π ≤ i < j ≤ z α tij i j ¯ z β tij i j (1 − ξ ) r t , − pk + n (1 − ¯ ξ ) r t , − pk + m ) . Thus, the coe ffi cients of 1 , z , . . . , z n ∞ − in e ( φ ) is an element in Cor f . This finishesthe proof, the detailed verification of the assertion being left to the reader. (cid:3) Corollary 1.3.
If a function f ∈ Cor is independent of the variable z , i.e., d / dz f = d / d ¯ z f , then f ∈ Cor . Vacuum state and D -symmetry. In section 1.3, we consider the expansions of afunction in Cor in three variables associated with an element in A ∈ P . In this section,we consider expansions in four variables. Let Q be the set of parenthesized products offive elements 1 , , , , ⋆ with ⋆ at right most, e.g., 3((4((12) ⋆ )) (see Introduction 0.2).The permutation group S acts on Q , which fixes ⋆ , and Q consists of the permutationsof the following elements, called standard elements of Q :((12)(34)) ⋆, (12)((34) ⋆ ) , (12)(3(4 ⋆ )) , (((12)3)4) ⋆, ((12)3)(4 ⋆ ) , (standard elements of Q ) ((1(23))4) ⋆, (1(23))(4 ⋆ ) , (1((23)4)) ⋆, ⋆ ) , ⋆ )) , (1(2(34))) ⋆, ⋆ ) , ⋆ )) , ⋆ ))) . The rule for the change of variables is given in Appendix. In this section, we brieflyexplain that all the expansions associated with Q are given by the expansions associatedwith P .We start with the example 1(2(3(4 ⋆ ))) ∈ Q . In this case, we consider the expansion ofa function Cor in | z | > | z | > | z | > | z | . Set T ⋆ ( x , y , z , w ) = C [[ y / x , w / z , y / x , ¯ w / ¯ z ]](( z / y , ¯ z / ¯ y , | z / y | R ))[ x ± , y ± , z ± , x ± , y ± , z ± , | x | R , | y | R , | z | R ] , for the formal variables x , y , z , w , ¯ x , ¯ y , ¯ z , ¯ w . We recall that the expansion e is givenby the change of variables ( z − z , z − z , z − z ) ( z , z , z ) with the expansion in | z | > | z | > | z | . Then, let T ⋆ )))1(2(34)) : T ( z , z , z ) → T ⋆ ( z , z , z , z ) be the linearmap defined by T ⋆ )))1(2(34)) = lim ( z , z , z ) → ( z , z , z ) exp( − z ( d / dz + d / dz + d / dz )) exp( − ¯ z ( d / d ¯ z + d / d ¯ z + d / d ¯ z )) and set e ⋆ ))) = T ⋆ )))1(2(34)) ◦ e : Cor → T ⋆ ( z , z , z , z ) . Then, similarly to Lemma 1.9, we have:
Lemma 1.17.
For φ ∈ Cor , the formal power series e ⋆ ))) ( φ ) is absolutely convergentto φ ( z , z , z , z ) in | z | >> | z | >> | z | >> | z | . For the case of 1((2(34)) ⋆ ) ∈ Q , we consider the following expansion with the changeof variables( z , z , z , z ) = ( z , z − z , z − z , z ) , {| z | >> | z | >> | z | >> | z |} . Similarly to the above, such expansion e ⋆ ) : Cor → T ∗ ( z , z , z , z ) is given by e ⋆ ) = T ⋆ )1(2(34)) ◦ e , T ⋆ )1(2(34)) = lim z → z exp( − z d / dz ) exp( − ¯ z d / d ¯ z ) . In this way, we can define expansions and the space of formal power series for all elementsin Q by using the expansions defined in the previous section.Hereafter, we study e ⋆ ))) in more detail. Since the expansions and the derivationsare commute with each other, by Lemma 1.1, the image of e ⋆ ))) is in the kernel of ddz + ddz + ddz + ddz and dd ¯ z + dd ¯ z + dd ¯ z + dd ¯ z . Set T ⋆ ( z , z , z , z ) = { f ∈ T ⋆ ( z , z , z , z ) | ddz + ddz + ddz + ddz f = ddz + ddz + ddz + ddz f = } . Then, the image of T ⋆ )))1(2(34)) : T ( z , z , z ) → T ⋆ ( z , z , z , z ) is in T ⋆ ( z , z , z , z ),since [ ddz , exp( − z ( ddz + ddz + ddz ))] = − ( ddz + ddz + ddz ). Lemma 1.18.
The above map T ⋆ )))1(2(34)) : T ( z , z , z ) → T ⋆ ( z , z , z , z ) is an isomor-phism.Proof. By substituting z = ¯ z =
0, we obtain the left inverse of the map, which impliesthat the map is injective. Let f ( z , z , z , z ) = P n , m ≥ f n , m z n ¯ z m ∈ T ⋆ ( z , z , z , z ). It su ffi cesto show that f is in the image of the map. We may assume that f , =
0. Then, n f n , m = ( ddz + ddz + ddz ) f n − , m and m f n , m = ( dd ¯ z + dd ¯ z + dd ¯ z ) f n , m − . Thus, f n , m = n , m ≥ (cid:3) We end this section by studying the relation between e ⋆ ))) and e ⋆ ))) . The map e ⋆ ))) expands a function in | z | > | z | > | z | > | z | , whereas e ⋆ ))) expands in | z | > | z | > | z | > | z | . We consider the following involution: X → X , ( z , z , z , z ) ( z − , z − , z − , z − ) and set T ⋆ d ( x , y , z , w ) = C [[ y / x , w / z , y / x , ¯ w / ¯ z ]](( z / y , ¯ z / ¯ y , | z / y | R ))[ y ± , z ± , y ± , z ± , | y | R , | z | R ] , which is a subspace of T ⋆ ( x , y , z , w ). Define the map I d : T ⋆ d ( z , z , z , z ) → T ⋆ d ( z , z , z , z )by ( z , z , z , z ) ( z − , z − , z − , z − ). We observe that I d ( z − r e ⋆ ))) ( z − z ) r ) = I d ( X i ≥ ( − i ri ! z − i z i ) = X i ≥ ( − i ri ! z i z − i = z − r ( − r e ⋆ ))) (( z − z ) r ) . Let φ ( z , z , z , z ) = Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij f ◦ ξ ( z , z , z , z ) ∈ Cor , where f ∈ F , , ∞ and α i j , β i j ∈ R satisfy α i j − β i j ∈ Z for any 1 ≤ i < j ≤
4. Set P ( α, β, z ) = Π ≤ i < j ≤ ( − α ij − β ij ( z i z j ) α ij (¯ z i ¯ z j ) β ij . Then, we have: Lemma 1.19. I d ( P ( α, β, z ) − e ⋆ ))) ( φ )) = e ⋆ ))) ( φ ) . The above transformation corresponds to (14)(23) in S . It is well-known that theaxiom of a vertex algebra possesses S -symmetry (see [FHL]). We will later prove thatthe correlation functions is S -invariant by using the S -symmetry together with (14)(23)-symmetry above (see Proposition 3.7).1.8. Generalized two point Correlation function.
In the theory of a vertex algebra,four point correlation functions in the limit of ( z , z ) ( ∞ ,
0) are used, which we callgeneralized two point functions (see Introduction 0.2). This subsection is devoted tostudying the property of this generalized two point function. Set U ( y , z ) = C (( z / y , ¯ z / ¯ y , | z / y | R ))[ y ± , z ± , ¯ y ± , ¯ z ± , | y | R , | z | R ]and Y = { ( z , z ) ∈ C | z , z , z , , z , } . Let η ( z , z ) : Y → C P \ { , , ∞} be the real analytic function defined by η ( z , z ) = z z .For f ∈ F , , ∞ , f ◦ η is a real analytic function on Y . Denote by GCor the space of realanalytic functions on Y spanned by z α z α ( z − z ) α ¯ z β ¯ z β (¯ z − ¯ z ) β f ◦ η ( z , z ) , (1.8)where f ∈ F , , ∞ and α , α , α , β , β , β ∈ R satisfy α − β , α − β , α − β ∈ Z .Define the action of Aut (0 , , ∞ ) on Y by(01) · ( z , z ) = ( z , z − z ) , (0 ∞ ) · ( z , z ) = ( z , z ) , (1 ∞ ) · ( z , z ) = ( z − z , z ) , (01 ∞ ) · ( z , z ) = ( z − z , z ) , (10 ∞ ) · ( z , z ) = ( z , z − z ) , for ( z , z ) ∈ Y . Then, we have: Lemma 1.20.
The map η : Y → C P \{ , , ∞} commutes with the action of Aut (0 , , ∞ ) ,i.e., η ( t · ( z , z )) = t · z z for any ( z , z ) ∈ Y and t ∈ Aut (0 , , ∞ ) . In particular, for µ ∈ GCor , (cid:16) t · µ (cid:17) ( − ) = µ ( t − − ) ∈ GCor for any t ∈ Aut (0 , , ∞ ) . Let µ ( z , z ) = z α z α ( z − z ) α ¯ z β ¯ z β (¯ z − ¯ z ) β f ◦ η ( z , z ) in (1.8). The expansions of µ in {| z | > | z |} and {| z | > | z |} are respectively given by z α + α ¯ z β + β z α ¯ z β X i , j ≥ α i ! β j ! ( − z / z ) i ( − ¯ z / ¯ z ) j lim p → z / z j ( p , f )( − α − β z α ¯ z β z α + α ¯ z β + β X i , j ≥ α i ! β j ! ( − z / z ) i ( − ¯ z / ¯ z ) j lim p → z / z j ( p − , f ) , which define maps | | z | > | z | : GCor → U ( z , z ) , µ ( z , z ) µ ( z , z ) | | z | > | z | and | | z | > | z | : GCor → U ( z , z ) , µ ( z , z ) µ ( z , z ) | | z | > | z | . Since f ( z z ) = f ( z z + ( z − z ) ), the expansions of µ in {| z | > | z − z |} is given by z α + α ¯ z β + β z α ¯ z β X i , j ≥ α i ! β j ! ( z / z ) i (¯ z / ¯ z ) j lim p →− z / z j (1 − p − , f ) , where z = z − z . We denote it by | | z | > | z − z | : GCor → U ( z , z ) , µ ( z , z ) µ ( z , z ) | | z | > | z − z | . Then, we have:
Lemma 1.21.
For f ∈ F , , ∞ ,f ◦ η | | z | > | z | = lim p → z / z j (1 , f ) , f ◦ η | | z | > | z | = lim p → z / z j ( p − , f ) , f ◦ η | | z | > | z − z | = lim p →− z / z j (1 − p − , f ) . For a , a ′ ∈ R , define a C -linear map C a , a ′ ( x ) : T ( x , y , z ) → U ( y , z ) by taking the coe ffi -cient of x a ¯ x a ′ . Then, we have the C -linear map C a , a ′ ( x ) : T ( x , y , z ) → U ( y , z ). Proposition 1.4.
For φ ∈ Cor and a , a ′ ∈ R , there exists η ∈ GCor satisfying thefollowing conditions:(1) C a , a ′ ( x ) e φ = η | | z | > | z | ,(2) C a , a ′ ( x ) e φ = η | | z | > | z | ,(3) C a , a ′ ( x ) e φ = η | | z | > | z − z | ,where we identify ( y , z ) and ( y , z ) , ( y , z ) as ( z , z ) and ( z , z ) , ( z , z ) , respectively. Proof.
Since the expansion is linear, we may assume that φ = Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij f ◦ ξ , where f ∈ F , , ∞ and α i j − β i j ∈ Z . Set T g ( x , y , z ) = C [[ y / x , ¯ y / ¯ x ]](( z / y , z / y , | z / y | R ))[ y ± , z ± , y ± , ¯ z ± , | y | R , | z | R ] , which involves only x − n ¯ x − n ′ with n , n ′ ∈ Z ≥ , and c = α + α + α and c ′ = β + β + β . Let A = , , e A ( φ ) ⊂ z c ¯ z c ′ T g ( x A , y A , z A ) and, thus, C a , a ′ ( x A ) e A ( φ ) = c − a , c ′ − a ′ ∈ Z ≥ . Hence, we may assume that a = c − n and a ′ = c ′ − n ′ for some n , n ′ ∈ Z ≥ . Since lim z →∞ z − c ¯ z − c ′ φ exists and is equal to Π ≤ i < j ≤ ( z i − z j ) α ij (¯ z i − ¯ z j ) β ij f ( z − z z − z ) , C c − n , c ′ − n ′ ( x A ) e A ( φ ) is convergent to the real analytic function η ( z , z ) = n ! n ′ ! lim ( z , z , z , z ) → ( ∞ , z , z , ( − z d / dz ) n ( − z d / dz ) n ′ ( z − c ¯ z − c ′ φ ( z , z , z , z )) ∈ GCor , where we used − z ddz z − k = kz − ( k − . Since e is the expansion around | z − z | > | z − z | > | z − z | , in the limit of lim ( z , z , z , z ) → ( ∞ , z , z , , it gives the expansion around | z | > | z − z | . (cid:3) Similarly to the proof of Proposition 1.3, we have:
Lemma 1.22. If µ ∈ GCor satisfies dd ¯ z µ = , then µ ( z , z ) ∈ C [ z ± , ( z − z ) ± , z ± , ¯ z ± , | z | R ] .Furthermore, if dd ¯ z µ = dd ¯ z µ = , then µ ( z , z ) ∈ C [ z ± , z ± , ( z − z ) ± ] . Similarly to Section 1.5, let S g ( z , z ) be the subspace of U ( z , z ) consisting of f ∈ U ( z , z ) with ( z d / dz + z d / dz ) f = (¯ z d / d ¯ z + ¯ z d / d ¯ z ) f = . Clearly, the map s g : C (( p , ¯ p , | p | R )) → S g ( z , z ) defined by the substitution of z / z into p and the map v g : S g ( z , z ) → C (( p , ¯ p , | p | R )) defined by the limit ( z , z ) → (1 , p ) are mutually inverse.Then, we have: Lemma 1.23.
Let µ ∈ GCor satisfy ( z d / dz + z d / dz ) µ = (¯ z d / d ¯ z + ¯ z d / d ¯ z ) µ = .Then, there exists f ∈ F , , ∞ such that µ | | z | > | z | = lim p → z / z j ( p , f ) µ | | z | > | z − z | = lim p →− z / z j (1 − p − , g ) µ | | z | > | z | = lim p → z / z j ( p − , g ) . Proof.
Set f ( p ) = µ (1 , p ), which is a real analytic function on C P \ { , , ∞} . By theexistence of the expansion around { , , ∞} , f has conformal singularities at { , , ∞} .Thus, the assertion holds. (cid:3) We end this section by studying the behavior of the expansion | | z | > | z − z | of generalizedtwo point functions under the involution I Y : Y → Y , ( z , z ) ( z − , z − ), which is important to define a dual module for a full vertex algebra. The involution I Y acts onGCor by µ ( z , z ) µ ( z − , z − ) for µ ∈ GCor . Define the maplim ( z , z ) ( − z z z + z , z ) : U ( z , z ) → U ( z , z )by substituting ( − z / z P i ≥ ( − i ( z / z ) i , z − ) into z and z . The map is well-defined,since U ( z , z ) = C (( z / z , ¯ z / ¯ z , | z / z | ))[ z ± , z ± , ¯ z ± , ¯ z ± , | z | R , | z | R ] and lim ( z , z ) ( − z z z + z , z ) z / z = − z / z (1 − z / z + ( z / z ) − · · · ). Lemma 1.24.
For any µ ∈ GCor , lim ( z , z ) ( − z z z + z , z ) µ | | z | > | z − z | = I Y ( µ ) | | z | > | z − z | . Proof.
Set µ ′ ( z , z ) = µ ( z + z , z ). Since the series µ | | z | > | z − z | ∈ U ( z , z ) is absolutelyconvergent in | z | >> | z | , lim ( z , z ) ( − z z z + z , z ) µ | | z | > | z − z | is also absolutely convergent to µ ′ ( − z z ( z + z ) , z ) in | z | >> | z | , which is equal to µ ( − z z ( z + z ) + z , z ) = µ (( z + z ) − , z − ). Theseries I Y ( µ ) | | z | > | z − z | is also absolutely convergent to µ (( z + z ) − , z − ) in the same domain.Hence, they coincide with each other. (cid:3)
2. F ull vertex algebra
In this section, we introduce the notion of a full vertex algebra, which is a generalizationof a Z -graded vertex algebra.2.1. Definition of full vertex algebra.
For an R -graded vector space F = L h , ¯ h ∈ R F h , ¯ h ,set F ∨ = L h , ¯ h ∈ R F ∗ h , ¯ h , where F ∗ h , ¯ h is the dual vector space of F h , ¯ h . A full vertex algebra isan R -graded C -vector space F = L h , ¯ h ∈ R F h , ¯ h equipped with a linear map Y ( − , z ) : F → End( F )[[ z ± , ¯ z ± , | z | R ]] , a Y ( a , z ) = X r , s ∈ R a ( r , s ) z − r − ¯ z − s − and an element ∈ F , satisfying the following conditions:FV1) For any a , b ∈ F , there exists N ∈ R such that a ( r , s ) b = r ≥ N or s ≥ N ,that is, Y ( a , z ) b ∈ F (( z , ¯ z , | z | R ));FV2) F h , ¯ h = h − ¯ h ∈ Z ;FV3) For any a ∈ F , Y ( a , z ) ∈ F [[ z , ¯ z ]] and lim z → Y ( a , z ) = a ( − , − = a .FV4) Y ( , z ) = id ∈ End F ;FV5) For any a , b , c ∈ F and u ∈ F ∨ , there exists µ ( z , z ) ∈ GCor such that u ( Y ( a , z ) Y ( b , z ) c ) = µ | | z | > | z | , u ( Y ( Y ( a , z ) b , z ) c ) = µ | | z | > | z − z | , u ( Y ( b , z ) Y ( a , z ) c ) = µ | | z | > | z | , where z = z − z .FV6) F h , ¯ h ( r , s ) F h ′ , ¯ h ′ ⊂ F h + h ′ − r − , ¯ h + ¯ h ′ − s − for any r , s , h , h ′ , ¯ h , ¯ h ′ ∈ R . Remark 2.1.
Physically, the energy and the spin of a state in F h , ¯ h are h + ¯ h and h − ¯ h.Thus, the condition (FV2) implies that we only consider the particles whose spin is aninteger, that is, we consider only bosons and not fermions. The notion of a full supervertex algebra can be defined by modifying (FV5) and (FV2). Remark 2.2.
Define the linear map L (0) , L (0) ∈ End
F by L (0) v = hv and L (0) a = ¯ ha forany h , ¯ h ∈ R and a ∈ F h , ¯ h . Then, the condition (FV6) is equivalent to the the followingcondition: For any h , ¯ h ∈ R and a ∈ F h , ¯ h , [ L (0) , Y ( a , z )] = ( zd / dz + h ) Y ( a , z ) , [ L (0) , Y ( a , z )] = (¯ zd / d ¯ z + ¯ h ) Y ( a , z ) . Let ( F , Y , ) and ( F , Y , ) be full vertex algebras. A full vertex algebra homomor-phism from F to F is a linear map f : F → F such that(1) f ( ) = (2) f ( Y ( a , z ) − ) = Y ( f ( a ) , z ) f ( − ) for any a ∈ F .The notions of a subalgebra and an ideal are defined in the usual way.A module of a full vertex algebra F is an R -graded C -vector space M = L h , ¯ h ∈ R M h , ¯ h equipped with a linear map Y M ( − , z ) : F → End( M )[[ z ± , ¯ z ± , | z | R ]] , a Y M ( a , z ) = X r , s ∈ R a ( r , s ) z − r − ¯ z − s − satisfying the following conditions:FM1) For any a ∈ F and m ∈ M , there exists N ∈ R such that a ( r , s ) m = r ≥ N or s ≥ N ;FM2) M n , m = n − m ∈ Z ;FM3) Y M ( , z ) = id ∈ End M ;FM4) For any a , b ∈ F , m ∈ M and u ∈ M ∨ , there exists µ ∈ GCor such that u ( Y M ( a , z ) Y M ( b , z ) m ) = µ | | z | > | z | , u ( Y M ( Y M ( a , z ) b , z ) m ) = µ | | z > | z − z | , u ( Y M ( b , z ) Y M ( a , z ) m ) = µ | | z | > | z | ;FM5) F h , ¯ h ( r , s ) M h ′ , ¯ h ′ ⊂ M h + h ′ − r − , ¯ h + ¯ h ′ − s − for any r , s , h , h ′ , ¯ h , ¯ h ′ ∈ R .As a consequence of (FM1) and (FM5), we have: Lemma 2.1.
Let h i , ¯ h i ∈ R , a i ∈ F h i , ¯ h i (i = , ), m ∈ M h , ¯ h and u ∈ M ∗ h , ¯ h . Then,u ( Y ( a , z ) Y ( a , z ) m ) ∈ z h − h − h − h ¯ z ¯ h − ¯ h − ¯ h − ¯ h C (( z / z , ¯ z / ¯ z , | z / z | R )) .Proof. Set X s , ¯ s , ¯ s , ¯ s ∈ R c s , ¯ s , ¯ s , ¯ s z s ¯ z ¯ s z s ¯ z s = u ( Y ( a , z ) Y ( a , z ) m ) . Then, c s , ¯ s , s , ¯ s = u ( a ( − s − , − ¯ s − a ( − s − , − ¯ s − m ) . By (FM5), a ( − s − , − ¯ s − a ( − s − , − ¯ s − m ∈ M h + h + h + s + s , ¯ h + ¯ h + ¯ h + ¯ s + ¯ s . Hence, c s , ¯ s , s , ¯ s = h = h + h + h + s + s and ¯ h = ¯ h + ¯ h + ¯ h + ¯ s + ¯ s . Thus, wehave u ( Y ( a , z ) Y ( a , z ) m ) = z h − h − h − h ¯ z ¯ h − ¯ h − ¯ h − ¯ h X s , ¯ s ∈ R c s , ¯ s , ¯ s , ¯ s ( z / z ) s (¯ z / ¯ z ) s , where s = h − ( h + h + h + s ) and ¯ s = ¯ h − (¯ h + ¯ h + ¯ h + ¯ s ). By (FM1), the assertionholds. (cid:3) By Lemma 2.1 and Lemma 1.23, we have:
Lemma 2.2.
Let h i , ¯ h i ∈ R , a i ∈ F h i , ¯ h i (i = , ), m ∈ M h , ¯ h and u ∈ M ∗ h , ¯ h , there existsf ∈ F , , ∞ such thatz − h + h + h + h ¯ z − ¯ h + ¯ h + ¯ h + ¯ h u ( Y ( a , z ) Y ( b , z ) m ) = lim p → z / z j ( p , f ) , z − h + h + h + h ¯ z − ¯ h + ¯ h + ¯ h + ¯ h u ( Y ( Y ( a , z ) b , z ) m ) = lim p →− z / z j (1 − p − , f ) , z − h + h + h + h ¯ z − ¯ h + ¯ h + ¯ h + ¯ h u ( Y ( b , z ) Y ( a , z ) m ) = lim p → z / z j (1 / p , f ) . Let M , N be a F -module. A F -module homomorphism from M to N is a linear map f : M → N such that f ( Y M ( a , z ) − ) = Y N ( a , z ) f ( − ) for any a ∈ F .Let M be a F -module. According to [L1], a vector v ∈ M is said to be a vacuum-likevector if Y ( a , z ) v ∈ M [[ z , ¯ z ]] for any a ∈ F . Lemma 2.3.
Let v ∈ M be a vacuum-like vector and a , b ∈ F and u ∈ M ∨ and µ ∈ GCor satisfy u ( Y ( a , z ) Y ( a , z ) v ) = µ | | z | > | z | . Then, µ is a linear combination of the functions ofthe form ( z − z ) α z α (¯ z − ¯ z ) β ¯ z β , where α , β ∈ Z ≥ and α , β ∈ R satisfy α − β ∈ Z . Furthermore, the linear function F v : F → M defined by a a ( − , − v is a F-module homomorphism.Proof. By (FM4), u ( Y ( Y ( a , z ) a , z ) v ) = µ | | z | > | z − z | . Since v is a vacuum like vector, byLemma 2.1 p ( z , z ) = µ | | z | > | z − z | ∈ C [ z ± , ¯ z ± , | z | R , z , ¯ z ] ⊂ U ( z , z ), which proves the firstpart of the lemma. It su ffi ces to show that F v ( Y ( a , z ) a ) = Y ( a , z ) F v ( a ). Since u ( Y ( a , z ) Y ( a , z ) v ) = µ | | z | > | z | = lim z → ( z − z ) | | z | > | z | p ( z , z ) , we have u ( Y ( a , z ) Y ( a , z ) v ) = exp( − z d / dz − ¯ z d / d ¯ z ) u ( Y ( Y ( a , z ) a , z ) v ) . (2.1)Thus, Y ( a , z ) F v ( a ) = lim z u ( Y ( a , z ) Y ( a , z ) v ) = lim z exp( − z d / dz − ¯ z d / d ¯ z ) u ( Y ( Y ( a , z ) a , z ) v ) = F v ( Y ( a , z ) a ) . (cid:3) Let F be a full vertex algebra and D and ¯ D denote the endomorphism of F defined by Da = a ( − , − and ¯ Da = a ( − , −
2) for a ∈ F , i.e., Y ( a , z ) = a + Daz + ¯ Da ¯ z + . . . . Define Y ( a , − z ) by Y ( a , − z ) = P r , s ( − r − s a ( r , s ) z r ¯ z s , where we used a ( r , s ) = r − s < Z , which follows from (FV2) and (FV6). Proposition 2.1.
For a ∈ F, the following properties hold:(1) Y ( Da , z ) = d / dzY ( a , z ) and Y ( ¯ Da , z ) = d / d ¯ zY ( a , z ) ;(2) D = ¯ D = ;(3) [ D , ¯ D ] = ;(4) Y ( a , z ) b = exp( zD + ¯ z ¯ D ) Y ( b , − z ) a;(5) Y ( ¯ Da , z ) = [ ¯ D , Y ( a , z )] and Y ( Da , z ) = [ D , Y ( a , z )] .Proof. Let u ∈ F ∨ and a , b ∈ F and µ , µ ∈ GCor satisfy u ( Y ( a , z ) Y ( , z ) b ) = µ | | z | > | z | , u ( Y ( a , z ) Y ( b , z ) ) = µ | | z | > | z | . By (FV4) and (FV5), p ( z ) = µ | | z | > | z | ∈ C [ z ± , ¯ z ± , | z | R ]. Then, u ( Y ( Y ( a , z ) , z ) b ) = µ | | z | > | z − z | = lim z → z exp( z ddz ) exp(¯ z dd ¯ z ) p ( z ) . Thus, u ( Y ( Da , z ) b ) = lim z → z ddz p ( z ) = ddz u ( Y ( a , z ) b ), which implies that Y ( Da , z ) = ddz Y ( a , z ) and similarly Y ( ¯ Da , z ) = dd ¯ z Y ( a , z ).By (FV4), Y ( D , z ) = ddz Y ( , z ) =
0. Thus, by (FV3), D = ¯ D =
0. Since Y ( D ¯ Da , z ) = ddz dd ¯ z Y ( a , z ) = dd ¯ z ddz Y ( a , z ) = Y ( ¯ DDa , z ), we have [ D , ¯ D ] = µ | | z | > | z − z | ∈ C [ z , ¯ z ][ z ± , ¯ z ± , | z | R ]. Set p ( z , z ) = µ | | z | > | z − z | = u ( Y ( Y ( a , z ) b , z ) ). Since u ( Y ( Y ( b , − z ) a , z ) ) = p ( z , z − z ) | | z | > | z | , we have u ( Y ( a , z ) b ) = p ( z , = lim z → exp( z ddz + ¯ z dd ¯ z ) p ( z , z − z ) = lim z → exp( z ddz + ¯ z dd ¯ z ) u ( Y ( Y ( b , − z ) a , z ) ) = lim z → u ( Y (exp( z D + ¯ z ¯ D ) Y ( b , − z ) a , z ) ) = u (exp( z D + ¯ z ¯ D ) Y ( b , − z ) a ) . Finally, ddz Y ( a , z ) b = ddz exp( Dz + ¯ D ¯ z ) Y ( b , − z ) a = D exp( Dz + ¯ D ¯ z ) Y ( b , − z ) a − exp( Dz + ¯ D ¯ z ) Y ( Db , − z ) a = DY ( a , z ) b − Y ( a , z ) Db . (cid:3) Let ( V , Y , ) be a Z -graded vertex algebra. Then, by a standard result of the theory ofa vertex algebra (see for example [FLM, FB]), u ( Y ( a , z ) Y ( b , z ) c ) is an expansion of arational polynomial in C [ z ± , z ± , ( z − z ) − ] ⊂ GCor in | z | > | z | for any u ∈ V ∨ and a , b , c ∈ V . Thus, we have: Proposition 2.2. A Z -graded vertex algebra is a full vertex algebra. Let ( F , Y , ) be a full vertex algebra. Set ¯ F = F and ¯ F h , ¯ h = F ¯ h , h for h , ¯ h ∈ R . Define¯ Y ( − , z ) : ¯ F → End( ¯ F )[[ z , ¯ z , | z | R ]] by ¯ Y ( a , z ) = P s , ¯ s ∈ R a ( s , ¯ s )¯ z − s − z − ¯ s − . Let C : Y → Y bethe conjugate map ( z , z ) (¯ z , ¯ z ) for ( z , z ) ∈ Y . For u ∈ ¯ F ∨ and a , b , c ∈ ¯ F , let µ ∈ GCor satisfy u ( Y ( a , z ) Y ( b , z ) c ) = µ ( z , z ) | | z | > | z | . Then, u ( ¯ Y ( a , z ) ¯ Y ( b , z ) c ) = µ ◦ C ( z , z ).Since µ ◦ C ∈ GCor , we have: Proposition 2.3. ( ¯ F , ¯ Y , ) is a full vertex algebra. We call it a conjugate full vertex algebra of ( F , , Y ).2.2. Holomorphic vertex operators.
Let F be a full vertex algebra. A vector a ∈ F is said to be a holomorphic vector (resp. an anti-holomorphic vector) if Da = Da = a ∈ ker ¯ D . Then, since 0 = Y ( ¯ Da , z ) = d / d ¯ zY ( a , z ), we have a ( r , s ) = s = −
1. Hence, Y ( a , z ) = P n ∈ Z a ( n , − z − n − . Lemma 2.4.
Let a , b ∈ F. If ¯ Da = , then for any n ∈ Z , [ a ( n , − , Y ( b , z )] = X i ≥ ni ! Y ( a ( i , − b , z ) z n − i , Y ( a ( n , − b , z ) = X i ≥ ni ! ( − i a ( n − i , − z i Y ( b , z ) − Y ( b , z ) X i ≥ ni ! ( − i + n a ( i , − z n − i . Proof.
For any u ∈ F ∨ and c ∈ F , there exists µ ∈ GCor such that (FV5) holds. Since¯ Da =
0, by Proposition 2.1, d / d ¯ z µ ( z , z ) =
0. Then, by Lemma 1.22, µ ∈ C [ z ± , ( z − z ) ± , z ± , ¯ z ± , | z | R ]. Thus, by the Cauchy integral formula, the assertion holds. (cid:3) By Proposition 2.1, ¯ DY ( a , z ) b = Y ( ¯ Da , z ) b + Y ( a , z ) ¯ Db =
0. Thus, the restriction of Y on ker ¯ D define a linear map Y ( − , z ) : ker ¯ D → End ker ¯ D [[ z ± ]]. By the above Lemmaand Lemma 1.22, we have: Proposition 2.4. ker ¯
D is a vertex algebra and F is a ker ¯
D-module.
Lemma 2.5.
For a holomorphic vector a ∈ F and an anti-holomorphic vector b ∈ F, [ Y ( a , z ) , Y ( b , ¯ w )] = , that is, [ a ( n , − , b ( − , m )] = and a ( k , − b = for any n , m ∈ Z and k ∈ Z ≥ .Proof. By Lemma 2.4, it su ffi ces to show that a ( k , − b = k ≥
0. Since DY ( a , z ) b = [ D , Y ( a , z )] b + Y ( a , z ) Db = d / dzY ( a , z ) b , we have Da ( n , − b = − na ( n − , − b for any n ∈ Z . Thus, the assertion follows from (FV1). (cid:3) Tensor product of full vertex algebras.
In this section, we define a tensor productof full vertex algebras and study the subalgebra of a full vertex algebra generated byholomorphic and anti-holomorphic vectors.Let F be a full vertex algebra. For h , ¯ h ∈ R , the energy and spin of a vector in F h , ¯ h are h + ¯ h and h − ¯ h and the set { ( h , ¯ h ) ∈ R | F h , ¯ h , } is called a spectrum. The spectrum of F is said to be bounded below if there exists N ∈ R such that F h , ¯ h = h ≤ N or¯ h ≤ N and discrete if for any H ∈ R , P h + ¯ h < H dim F h , ¯ h is finite. Lemma 2.6.
If the spectrum of F is discrete, then for any N ∈ Z > and a , b ∈ F, thenumber of the set { ( s , ¯ s ) ∈ R | a ( s , ¯ s ) b , , − N ≤ s , ¯ s ≤ N } is finite.Proof. We may assume that a ∈ F h , ¯ h and b ∈ F h ′ , ¯ h ′ . Since a ( s , ¯ s ) b ∈ F h + h ′ − s − , ¯ h + ¯ h ′ − ¯ s − , theenergies of vectors { a ( s , ¯ s ) b | − N ≤ s , ¯ s ≤ N } are bounded by h + h ′ + ¯ h + ¯ h ′ + N − (cid:3) Let ( F , Y , ) and ( F , Y , ) be full vertex algebras and assume that the spectrum of F is discrete and the spectrum of F is bounded below. Define the linear map Y ( − , z ) : F ⊗ F → End F ⊗ F [[ z , ¯ z , | z | R ]] by Y ( a ⊗ b , z ) = Y ( a , z ) ⊗ Y ( b , z ) for a ∈ F and b ∈ F . Then, for a , c ∈ F and b , d ∈ F , Y ( a ⊗ b , z ) c ⊗ d = X s , ¯ s , r , ¯ r ∈ R a ( s , ¯ s ) c ⊗ b ( r , ¯ r ) d z − s − r − ¯ z − ¯ s − ¯ r − . By (FV1) and the above lemma, the coe ffi cient of z k ¯ z ¯ k is a finite sum for any k , ¯ k ∈ R .Thus, Y ( − , z ) is well-defined. For any h , ¯ h ∈ R , set ( F ⊗ F ) h , ¯ h = L a , ¯ a ∈ R F a , ¯ a ⊗ F h − a , ¯ h − ¯ a . Since the spectrum of F is bounded below, there exists N ∈ R such that( F ⊗ F ) h , ¯ h = L a , ¯ a ≤ N F a , ¯ a ⊗ F h − a , ¯ h − ¯ a . Since the spectrum of F is discrete, the sumis finite. Thus, ( F ⊗ F ) ∗ h , ¯ h = L a , ¯ a ∈ R ( F a , ¯ a ) ∗ ⊗ ( F h − a , ¯ h − ¯ a ) ∗ , which implies that F ∨ = ( F ) ∨ ⊗ ( F ) ∨ . Let u i ∈ ( F i ) ∨ and a i , b i , c i ∈ F i for i = ,
2. Since u ⊗ u ( Y ( a ⊗ a , z ) Y ( b ⊗ b , z ) c ⊗ c ) = u ( Y ( a , z ) Y ( b , z ) c ) u ( Y ( a , z ) Y ( b , z ) c ) , we have: Proposition 2.5.
Let ( F , Y , ) and ( F , Y , ) be full vertex algebras. If the spectrumof F is discrete and the spectrum of F is bounded below, then ( F ⊗ F , Y ⊗ Y , ⊗ ) is a full vertex algebra. Furthermore, if the spectrum of F and F are bounded below(resp. discrete), then the spectrum of F ⊗ F is also bounded below (resp. discrete). By Proposition 2.2 and Proposition 2.3, we have:
Corollary 2.1.
Let V , W be a Z ≥ -graded vertex algebras such that dim V n and dim W n are finite for any n ∈ Z ≥ . Then, V ⊗ ¯ W is a full vertex algebra with a discrete spectrum,where ¯ W is the conjugate full vertex algebra. Let F be a full vertex algebra. By Proposition 2.4, ker ¯ D and ker D are subalgebrasof F . Let ker ¯ D ⊗ ker D be the tensor product full vertex algebra. Define the linear map t : ker ¯ D ⊗ ker D → F by ( a ⊗ b ) a ( − , − b for a ∈ ker ¯ D and b ∈ ker D . Then, wehave: Proposition 2.6.
Let F be a full vertex algebra. Then, t : ker ¯ D ⊗ ker D → F is a fullvertex algebra homomorphism.Proof.
Let a , c ∈ ker ¯ D , b , d ∈ ker D . By Lemma 2.5 and Lemma 2.4, Y ( a ( − , − b , z ) = Y ( a , z ) Y ( b , ¯ z ) = Y ( b , ¯ z ) Y ( a , z ) . Thus, it su ffi ces to show that t ( a ⊗ b ( n , m ) c ⊗ d ) = t ( a ⊗ b )( n , m ) t ( c ⊗ d ) for any n , m ∈ Z .By Lemma 2.4 t ( a ⊗ b ( n , m ) c ⊗ d ) = t ( a ( n , − c ⊗ b ( − , m ) d ) = ( a ( n , − c )( − , − b ( − , m ) d = X i = ni ! ( − i ( a ( n − i , − c ( − + i , − + c ( − + n − i , − a ( i , − b ( − , m ) d . Since b ( − , m ) d ∈ ker D , by Lemma 2.5, t ( a ⊗ b ( n , m ) c ⊗ d ) = a ( n , − c ( − , − b ( − , m ) d = a ( n , − b ( − , m ) c ( − , − d = t ( a ⊗ b )( n , m ) t ( c ⊗ d ) . Thus, the assertion holds. (cid:3)
We remark that if ker ¯ D ⊗ ker D is simple, then the above map is injective.3. C orrelation functions and full vertex algebras Self-duality.
A full vertex operator algebra (full VOA) is a full vertex algebra F with a holomorphic vector ω ∈ F and an anti-holomorphic vector ¯ ω ∈ F satisfying thefollowing conditions:FVO1) There exist a pair of scalars ( c , ¯ c ) ∈ C such that[ L ( n ) , L ( m )] = ( n − m ) L ( n + m ) + n − n δ n + m , c , [ L ( n ) , L ( m )] = ( n − m ) L ( n + m ) + n − n δ n + m , c holds for any n , m ∈ Z , where L ( n ) = ω ( n + , −
1) and L ( n ) = ¯ ω ( − , n + D = L ( −
1) and ¯ D = L ( − L (0) | F h , ¯ h = h and L (0) | F h , ¯ h = ¯ h for any h , ¯ h ∈ R ;FVO4) F h , ¯ h is a finite dimensional vector space for any h , ¯ h ∈ R ;FVO5) The spectrum of F is bounded below, that is, there exists N ∈ R such that F h , ¯ h = h ≤ N or ¯ h ≤ N .The pair of scalars ( c , ¯ c ) is called a central charge and the pair ( ω, ¯ ω ) is called an energy-momentum tensor of the full vertex operator algebra F . A module M of a full vertexalgebra F is said to be a module of a full vertex operator algebra if it satisfiesFVOM1) L (0) | M h , ¯ h = h and L (0) | M h , ¯ h = ¯ h for any h , ¯ h ∈ R ; FVOM2) M h , ¯ h is a finite dimensional vector space for any h , ¯ h ∈ R ;FVOM3) The spectrum of M is bounded below.Let F be a full VOA and M be a F -module. By Lemma 2.4, we have: Lemma 3.1.
For h , ¯ h ∈ R and a ∈ F h , ¯ h , [ L (0) , Y ( a , z )] = Y (( L (0) + zL ( − a , z ) = ( zd / dz + h ) Y ( a , z ) , [ L (0) , Y ( a , z )] = Y (( L (0) + ¯ zL ( − a , z ) = (¯ zd / d ¯ z + ¯ h ) Y ( a , z ) , [ L (1) , Y ( a , z )] = Y (( L (1) + zL (0) + z L ( − a , z ) = ( z d / dz + hz ) Y ( a , z ) + Y ( L (1) a , z ) , [ L (1) , Y ( a , z )] = Y (( L (1) + zL (0) + ¯ z L ( − a , z ) = (¯ z d / d ¯ z + h ¯ z ) Y ( a , z ) + Y ( L (1) a , z ) . Set M ∨ = L h , ¯ h ∈ R M ∗ h , ¯ h and let <> : M ∨ × M → C be a canonical pairing. Define S z : M → M [ z ± , ¯ z ± , | z | R ], by S z a = exp( zL (1) + ¯ zL (1))( − h − ¯ h z − h ¯ z − h , for h , ¯ h ∈ R and a ∈ M h , ¯ h , where we used the fact that L (1) and L (1) are locally nilpotentby (FVOM3). Define the vertex operator Y M ∨ ( − , z ) : F → End M ∨ [[ z ± , ¯ z ± , | z | R ]] by < Y M ∨ ( a , z ) u , v > ≡ < u , Y M ( S z a , z − ) v >, for a ∈ F and v ∈ M and u ∈ M ∨ .We will prove the following Proposition: Proposition 3.1. ( M ∨ , Y M ∨ ( − , z )) is a module of the full VOA F. Properties of the operators { L ( i ) } i = − , , acting on a vertex operator algebra which satisfythe equations in Lemma 3.1 are studied in [FHL], which can be easily generalized to afull vertex operator algebra. Lemma 3.2 ([FHL]) . For a full vertex operator algebra F and its module M, the followingequations hold:(1) [ L (0) , exp( L (1) z )] = − L (1) z exp( L (1) z ) .(2) For any a ∈ F, S z Y M ( a , z ) = Y M ( S z + z a , − z z ( z + z ) ) S z ,where z + z is expanded in | z | > | z | .proof of Proposition 3.1. Let u ∈ F ∨ and v ∈ M and a , b ∈ F . By Lemma 2.5 and (FVO1), L (1) ω = L (1) ¯ ω = L (1) ω = L (1) ¯ ω =
0. Thus, < Y ( ω, z ) u , v > = < u , z Y ( ω, z − ) v > , whichimplies that < L ( n ) u , v > = < u , L ( − n ) v > . By Lemma 3.1 and Lemma 3.2, Proposition2.1, < [ L (0) , Y M ∨ ( a , z )] u , v > − < Y M ∨ ( L (0) a , z ) u , v > = < u , Y ( S z a , z − ) L (0) v > − < L (0) u , Y ( S z a , z − ) v > − < u , Y ( S z L (0) a , z − ) v > = − < u , Y (( L (0) + z − L ( − S z a , z − ) v > − < u , Y ( S z L (0) a , z − ) v > = < u , Y ( L (1) zS z a , z − ) v > − < u , Y ( S z L (0) a , z − ) v > − z − < u , [ L ( − , Y ( S z a , z − )] v > . Since zd / dz < u , Y ( S z a , z − ) v > = < u , Y ( L (1) zS z a , z − ) v > − < u , Y ( L (1) zS z L (0) a , z − ) v > − z − < u , [ L ( − , Y ( S z a , z − )] v >, we have [ L (0) , Y M ∨ ( a , z )] = Y M ∨ ( L (0) a , z ) + d / dzY M ∨ ( a , z ). Thus, by Remark 2.2, F h , ¯ h ( r , s ) M ∨ h ′ , ¯ h ′ ⊂ M ∨ h + h ′ − r − , ¯ h + ¯ h ′ − s − . (FM2), (FM3), (FVOM1), (FVOM2) and (FVOM3) clearly follows.(FM1) follows from (FM5) together with (FVOM3). Thus, it su ffi ces to show (FM4). By(FM4), there exists µ ∈ GCor such that < u , Y ( S z − b , z ) Y ( S z − a , z ) v > = µ | | z | > | z | ,< u , Y ( S z − a , z ) Y ( S z − b , z ) v > = µ | | z | > | z | , lim z → ( z + z ) | | z | > | z | < u , Y ( Y ( S z − a , z ) S z − b , z ) v > = µ | | z | > | z − z | . Then, < Y ( a , z ) Y ( b , z ) u , m > = I Y ( µ ) | | z | > | z | and < Y ( b , z ) Y ( a , z ) u , m > = I Y ( µ ) | | z | > | z | . Toshow (FM4), it su ffi ces to show that I Y ( µ ) | | z | > | z − z | = < Y ( Y ( a , z ) b , z ) u , v > . By Lemma1.24, I Y ( µ ) | | z | > | z − z | = lim ( z , z ) ( − z z z + z , z ) µ | | z | > | z − z | = lim ( z , z ) ( − z z z + z , z ) lim z → z + z < u , Y ( Y ( S z − a , z ) S z − b , z ) v > = < u , Y ( Y ( S z + z a , − z z ( z + z ) ) S z b , z − ) v > . By Lemma 3.2, < Y ∨ M ( Y ( a , z ) b , z ) u , v > = < u , Y ( S z Y ( a , z ) b , z − ) v > = < u , Y ( Y ( S z + z a , − z z ( z + z ) ) S z b , z − ) v > . Thus, the assertion holds. (cid:3)
An invariant bilinear form on a full vertex operator algebra F is a bilinear form ( − , − ) : F × F → C such that ( Y ( a , z ) b , c ) = ( b , Y ( S z a , z − ) c )holds for any a , b , c ∈ F . By the proof of Proposition 3.1, ( L ( n ) a , b ) = ( a , L ( − n ) b ), whichimplies that F h , ¯ h and F h ′ , ¯ h ′ are orthogonal to each other unless ( h , ¯ h ) = ( h ′ , ¯ h ′ ).The following proposition is a straightforward generalization of Li’s result on the in-variant bilinear form on a vertex operator algebra (see [L1]): Proposition 3.2.
There is a one-to-one correspondence between Hom C ( F , / ( L (1) F , + L (1) F , ) , C ) and the space of invariant bilinear forms on F. According to [L1], we will use the following lemma:
Lemma 3.3.
Let M be an F-module. A vector v ∈ M is a vacuum-like vector if and onlyif L ( − v = L ( − v = . Proof. If v is a vacuum-like vector, then Y ( ω, z ) v ∈ F [[ z ]]. Thus, L ( − v = L ( − v =
0. Assume that L ( − v = L ( − v =
0. Then, L ( − Y M ( a , z ) v = [ L ( − , Y M ( a , z )] v = d / dzY M ( a , z ) v . Thus, similarly to the proof of Lemma 2.5, the asser-tion holds. (cid:3) proof of Proposition 3.2. Let I F be the space of invariant bilinear forms on F and ( − , − ) ∈ I F . Then, ( , − ) : F → C , a ( , a ) satisfies ( , F h , ¯ h ) = h , ¯ h ) , (0 , L ( − = L ( − =
0, we have ( , L (1) − ) = ( , L (1) − ) =
0. Thus, we have a linear map ρ : I F → Hom C ( F , / ( L (1) F , + L (1) F , ) , C ). Assume that ( , − ) =
0. Then, for any a , b ∈ F , ( a , b ) = lim z → ( Y ( a , z ) , b ) = lim z → ( , Y ( S z a , z − ) b ) =
0. Hence, ρ is injective.Let u ∈ Hom C ( F , / ( L (1) F , + L (1) F , ) , C ). Then, u is an element of F ∗ , ⊂ F ∨ .Since < L ( − u , a > = < u , L (1) a > for any a ∈ F , by Lemma 3.3, u is a vacuum-likevector. Hence, by Lemma 2.3, we have a F -module homomorphism F u : F → F ∨ , a a ( − , − u . Define the bilinear form ( − , − ) u : F × F → C by ( a , b ) u = < F u ( a ) , b > for a , b ∈ F . Then, ( Y ( a , z ) b , c ) u = < F u ( Y ( a , z ) b ) , c > = < Y F ∨ ( a , z ) F u ( b ) , c > = < F u ( b ) , Y ( S z a , z − ) c > = ( b , Y ( S z a , z − ) c ) u , which implies that ( − , − ) u is an invariant bilinear form. Since ( , − ) u = < F u ( ) , b > = < u , b > , ρ (( − , − ) u ) = u . Thus, ρ is an isomorphism. (cid:3) Similarly to [FHL], we have:
Proposition 3.3.
An invariant bilinear form on F is symmetric.
Proposition 3.4.
A full vertex operator algebra F is simple if and only if F ∨ is simple. A full vertex operator algebra F is said to be self-dual if F is isomorphic to F ∨ as an F -module, or equivalently, there exists a non-degenerate invariant bilinear form on F . Corollary 3.1.
If F is simple and F , = C and L (1) F , = L (1) F , = , then F isself-dual. For a self-dual full vertex operator algebra F and a ∈ F , ( , Y ( a , z ) ) = ( , exp( Dz + ¯ D ¯ z ) a ) = (exp( L (1) z + L (1) z ) , a ) = ( , a ). Thus, we have: Lemma 3.4.
For a ∈ F, ( , Y ( a , z ) ) is equal to ( , a ) ∈ C . Quasi-primary vectors.
For h , ¯ h ∈ R , set QF h , ¯ h = { v ∈ F h , ¯ h | L (1) v = L (1) v = } . A homogeneous vector in QF = L h , ¯ h ∈ R QF h , ¯ h is called a quasi-primary vector. Recallthat L ( i ) , L ( i ) ( i = − , ,
1) generate the Lie algebra sl ⊕ sl . Following the terminologyof [L1], we call a full vertex operator algebra F QP-generated if F is generated by QF asa sl ⊕ sl -module, or equivalently, F = C [ L ( − , L ( − QF . Proposition 3.5.
A self-dual full vertex operator algebra F is QP-generated if and onlyif the following conditions hold:(1) F h , ¯ h = if h ∈ Z < or ¯ h ∈ Z < .(2) L (1) F , n = and L (1) F n , = for any n ∈ Z .(3) L ( − F , n = and L ( − F n , = for any n ∈ Z . Lemma 3.5.
If F is self-dual and QP-generated, then the restriction of the invariantbilinear form on QF is non-degenerate. Furthermore, Im L ( − ∩ ker L (1) = and Im L ( − ∩ ker L (1) = hold.Proof. Assume that a ∈ QF h , ¯ h satisfy ( a , b ) = b ∈ QF h , ¯ h . Since ( a , L ( − b ) = ( L (1) a , b ) = a , L ( − b ) = ( L (1) a , b ) = b ∈ F , by F = C [ L ( − , L ( − QP , a =
0, which proves the first part of the lemma. Set QF ′ = { v ∈ F | L (1) v = } . Since F = C [ L ( − L ( − QF and L (1) C [ L ( − QF =
0, we have F = C [ L ( − QF ′ . Then,by the same proof, the restriction of the invariant bilinear form on QF ′ is non-degenerate.Since Im L ( − ∩ ker L (1) is in the radical of this invariant bilinear form, the assertionholds. (cid:3) proof of Proposition 3.5. Suppose that F is QP-generated. Let 0 , v ∈ QF − n , m for n ∈ Z > . Then, L (1) L ( − n + v =
0. By the representation theory of sl and Lemma3.5, we have 0 , L ( − n v ∈ F n , m and L ( − n + v =
0. Since L ( − n + v = L (0) L ( − n v =
0, a contradiction. Hence, QF − n , m = n ∈ Z > .Let v ∈ F , n . Since L (1) L ( − v =
0, by Lemma 3.5, L ( − v =
0, which implies (3).Since F = C [ L ( − , L ( − QF , by (1) and (3), F , n = L k ≥ L ( − k QF , n − k . Since L (1)commutes with L ( − L (1) F , n =
0, thus, (2).Suppose that F satisfies (1), (2) and (3). Let a , b ∈ QF n , m with n , m ∈ R \ Z < . Since( L ( − k L ( − l a , L ( − k L ( − l b ) = k ! l ! Π k − ≥ i ≥ l − ≥ j ≥ (2 n + i )(2 m + j )( a , b ) , the restriction of the bilinear form on L ( − k L ( − l Q n , m is non-degenerate if ( − , − ) | Q n , m isnon-degenerate. Let h , ¯ h ∈ R satisfy F h , ¯ h , F h − k , ¯ h − l = k , l ∈ Z > . Then, QF h , ¯ h = F h , ¯ h , which implies that ( − , − ) | QF h , ¯ h is non-degenerate. Since QF h + , ¯ h = { v ∈ F h + , ¯ h | ( v , L ( − F h , ¯ h ) = } , QF h + , ¯ h is non-degenerate and F h + , ¯ h = QF h + , ¯ h ⊕ L ( − Q h , ¯ h .Similarly, we have F a + k , b + l = L ≤ i ≤ k ≤ j ≤ l L ( − k − i L ( − l − j QF a + i , b + j by the induction on k , l ∈ Z ≥ . (cid:3) By Proposition 3.5 and Lemma 3.1, we have:
Corollary 3.2.
Suppose that simple full vertex operator algebra F satisfies the followingconditions:PN1) F , = C ;PN2) F n , m = if n ∈ Z < or m ∈ Z < ;PN3) L (1) F , n = and L (1) F n , = for any n ∈ Z ;PN4) L ( − F , n = and L ( − F n , = for any n ∈ Z . Then, F is self-dual and QP-generated.
Parenthesized correlation functions and formal power series.
Let F be a self-dual vertex operator algebra and denote Y ( a , ¯ x ) by a ( x ) in this section for a ∈ F . For n ∈ Z > , let P n be the set of parenthesized products of n elements 1 , , . . . , n , e.g.,((32)1)((47)(65)) ∈ P and Q n be the set of parenthesized products of n + , , . . . , n , ⋆ with ⋆ at right most, e.g., ((32)1)((4(65)) ⋆ ) ∈ Q . We can naturally asso-ciate a parenthesized n point correlation function to each element in P n and Q n (see alsoIntroduction 0.2 and Section 1.3). To give a simple example, an element (((31)6)(24))(57) ∈ P and a , . . . , a ∈ F define the following parenthesized 7 point correlation function: S (((31)6)(24))(57) ( a , a , a , a , a , a , a ) : = (cid:16) , "(cid:18)(cid:16) a ( x ) a (cid:17) ( x ) a (cid:19) ( x ) a ( x ) a ( x ) a ( x ) a (cid:17) . For an element in Q n and a , . . . , a n ∈ F , we consider parenthesized n + a , . . . , a n , where corresponds to ⋆ , e.g., for (((31)6)(24))(5 ⋆ ) ∈ Q , S (((31)6)(24))(5 ⋆ ) ( a , a , a , a , a , a ) : = (cid:16) , "(cid:18)(cid:16) a ( x ) a (cid:17) ( x ) a (cid:19) ( x ) a ( x ) a ( x ) a ( x ) (cid:17) . In this subsection, we study the space of formal variables for each A ∈ P n and A ∈ Q n .We start from A = . . . ( n − n )) . . . )) ∈ P n . In this case, for a , . . . , a n ∈ F , theparenthesized correlation function is S ... ( n − n )) ... )) ( a , . . . , a n ) = ( , a ( x ) a ( x ) . . . a n − ( x n − ) a n ) . Set T ... ( n − n )) ... )) ( x , . . . , x n − ) = C [[ x / x , ¯ x / ¯ x ]](( x / x , ¯ x / ¯ x , | x / x | R , . . . , x n − / x n − , ¯ x n − / ¯ x n − , | x n − / x n − | R ))[ x ± , ¯ x ± , | x | R , x ± , ¯ x ± , | x | R ] . For h , ¯ h , h ′ , ¯ h ′ ∈ R and a ∈ F h , ¯ h and b ∈ F h ′ , ¯ h ′ , since( , Y ( a , z ) b ) = ( Y ( S z a , z − ) , b ) = (exp( Dz − + ¯ D ¯ z − ) S z a , b )and F h , ¯ h and F h ′ , ¯ h ′ is orthogonal if ( h , ¯ h ) , ( h ′ , ¯ h ′ ), we have: Lemma 3.6.
For h , ¯ h , h ′ , ¯ h ′ ∈ R and a ∈ F h , ¯ h and b ∈ F h ′ , ¯ h ′ , ( , Y ( a , z ) b ) = unlessh − h ′ ∈ Z and ¯ h − ¯ h ′ ∈ Z . Then, we have:
Lemma 3.7.
For a , . . . , a n ∈ F,S ... ( n − n )) ... )) ( a , . . . , a N ; x , . . . , x n − ) ∈ T ... ( n − n )) ... )) ( x , . . . , x n − ) . Proof.
We may assume that a i ∈ F h i , ¯ h i for h i , ¯ h i ∈ R and i = , . . . , n . Set h = h + · · · + h n and ¯ h = ¯ h + · · · + ¯ h n and X s , ¯ s ,..., ¯ s n , ¯ s n ∈ R c s , ¯ s ,..., s n − , ¯ s n − z s ¯ z ¯ s . . . z s n − n − ¯ z s n − n − = ( , Y ( a , z ) Y ( a , z ) . . . Y ( a n − , z n − ) a n ) . Then, c s , ¯ s ,..., s n − , ¯ s n − = ( , a ( − s − , − ¯ s − a ( − s − , − ¯ s − . . . a n − ( − s n − − , − ¯ s n − − a n ) . By (FVO5), there exists N ∈ R such that F h , ¯ h = h ≤ N or ¯ h ≤ N . Since for any k = , , . . . , n − a k ( − s k − , − ¯ s k − a k − ( − s k − − , − ¯ s k − − . . . a n − ( − s n − − , − ¯ s n − − a n is in F h k + ··· + h n + s k + ··· + s n − , ¯ h k + ··· + ¯ h n + ¯ s k + ··· + ¯ s n − , by Lemma 3.6 and (FVO5), c s , ¯ s ,..., s n − , ¯ s n − = = h + s + · · · + s n − , 0 = ¯ h + ¯ s + · · · + ¯ s n − and ( h − h ) + s + · · · + s n − − h ∈ Z ,(¯ h − ¯ h ) + ¯ s + · · · + ¯ s n − − ¯ h ∈ Z and h k + · · · + h n + s k + · · · + s n − ≥ N , ¯ h k + · · · + ¯ h n + ¯ s k + · · · + ¯ s n − ≥ N for any k = , . . . , n −
1. Thus, we have( , Y ( a , x ) Y ( a , x ) . . . Y ( a n − , x n − ) a n ) = x − h ¯ x − ¯ h X n , m ∈ Z s , ¯ s ,..., s n , ¯ s n ∈ R c s , ¯ s ,..., s n − , ¯ s n − x x − h + h + n ¯ x ¯ x − ¯ h + h + m . . . x n − x n − s n − ¯ x n − ¯ x n − s n − , where s = − h − s − · · · − s n − , ¯ s = − ¯ h − ¯ s − · · · − ¯ s n − , s = − h + h − s − · · · − s n − + n and ¯ s = − ¯ h + h − ¯ s − · · · − ¯ s n − + m . Thus, the assertion follows. (cid:3) We have the embedding P n → Q n defined by A ( A ) ⋆ for A ∈ P n and surjection d ⋆ : Q n → P n defined by deleting ⋆ from A ∈ Q n . By lemma 3.4, we have: Lemma 3.8.
For A ∈ P n and a , . . . , a n ,S A ( a , . . . , a n ) = S ( A ) ⋆ ( a , . . . , a n ) . By Proposition 2.1, Y ( a , z ) = exp( Dz + ¯ D ¯ z ) a and [ D , Y ( a , z )] = d / dzY ( a , z ), anyparenthesized n point correlation function for A ∈ Q n is an expansion of a parenthesizedcorrelation function for d ⋆ ( A ) ∈ P n .For example, let us consider the case of A = ⋆ ))) ∈ Q . In this case, S ⋆ ))) ( a , a , a , a ; z , z , z , z ) = ( , a ( z ) a ( z ) a ( z ) a ( z ) ) . Since( , a ( z ) a ( z ) a ( z ) a ( z ) ) = ( , a ( z ) a ( z ) a ( z ) exp( Dz + ¯ D ¯ z ) a ) = exp( − z ( d / dz + d / dz + d / dz )) exp( − ¯ z ( d / d ¯ z + d / d ¯ z + d / d ¯ z )( , a ( z ) a ( z ) a ( z ) a ) = exp( − z ( d / dz + d / dz + d / dz )) exp( − ¯ z ( d / d ¯ z + d / d ¯ z + d / d ¯ z ) S ( a , a , a , a ; z , z , z ) , by Lemma 1.18, we have: Lemma 3.9.
For a , a , a , a ,S ⋆ ))) ( a , a , a , a ; z , z , z , z ) = T ⋆ )))1(2(34)) S ( a , a , a , a ; z , z , z ) . Hereafter, we concentrate on the case of n = A ∈ P . Similarly to the proof ofLemma 3.7, we can show that S A ( a , a , a , a ) ∈ T A , where T A is the space of formalpower series defined in Section 1.3.For example, for (12)(34), we have: Lemma 3.10.
For any a , a , a , a ∈ F, ( , Y ( Y ( a , x ) a , x ) Y ( a , x ) a ) ∈ T (12)(34) ( x , x , x ) .Proof. We may assume that a i ∈ F h i , ¯ h i for h i , ¯ h i ∈ R and i = , . . . ,
4. Set h = h + · · · + h and ¯ h = ¯ h + · · · + ¯ h and X s , ¯ s ,..., ¯ s , ¯ s ∈ R c s , ¯ s , s , ¯ s , s , ¯ s x s ¯ x ¯ s x s ¯ x s x s ¯ x s = ( , Y ( Y ( a , x ) a , x ) Y ( a , x ) a ) . Then, c s , ¯ s ,..., s , ¯ s = ( , (cid:16) a ( − s − , − ¯ s − a (cid:17) ( − s − , − ¯ s − a ( − s − , − ¯ s − a ) . By (FVO5), there exists N ∈ R such that F h , ¯ h = h ≤ N or ¯ h ≤ N . Thus, similarlyto the proof of Lemma 3.7, c s , ¯ s ,..., s , ¯ s = = h + ( s + s + s ), h + h + s − h − h − s ∈ Z and h + h + s , h + h + s ≥ N .Thus, there exists N ′ ∈ R such that( , Y ( Y ( a , x ) a , x ) Y ( a , x ) a ) = x − h ¯ x − ¯ h (cid:16) x x (cid:17) − h − h + h + h (cid:16) ¯ x ¯ x (cid:17) − ¯ h − ¯ h + ¯ h + ¯ h X s ′ , ¯ s ′ , s , ¯ s ∈ R c s , ¯ s ,..., s , ¯ s (cid:16) x x (cid:17) s ′ (cid:16) ¯ x ¯ x (cid:17) ¯ s ′ (cid:16) x x (cid:17) s (cid:16) ¯ x ¯ x (cid:17) ¯ s where s = − h − s − s , ¯ s = − ¯ h − ¯ s − ¯ s , s ′ = h + h − h − h + s and ¯ s ′ = ¯ h + ¯ h − ¯ h − ¯ h + ¯ s and the sum runs through s ′ , s , ¯ s ′ , ¯ s ≥ N ′ and s ′ − s ∈ Z , ¯ s ′ − ¯ s ∈ Z . Thus, the assertionholds. (cid:3) Consistency of four point functions.
Let F be a self-dual full vertex operator al-gebra. In this section, we will prove the main result of this paper, the consistency of fourpoint correlation functions. The following lemma follows from Lemma 3.1: Lemma 3.11.
For h i , ¯ h i ∈ R and a i ∈ F h i , ¯ h i and b i ∈ QF h i , ¯ h i ,(1) ( n X i = z i d / dz i + h i )( , Y ( a , z ) . . . Y ( a n − , z n − ) a n ) = , ( n X i = ¯ z i d / d ¯ z i + ¯ h i )( , Y ( a , z ) . . . Y ( a n − , z n − ) a n ) = (2) ( n X i = z i d / dz i + h i z i )( , Y ( b , z ) . . . Y ( b n − , z n − ) b n ) = , ( n X i = ¯ z i d / d ¯ z i + h i ¯ z i )( , Y ( b , z ) . . . Y ( b n − , z n − ) b n ) = . For h i , ¯ h i ∈ R with h i − ¯ h i ∈ Z ( i = , , , Q ( h , z ) = ( z − z ) − h ( z − z ) h − h − h − h ( z − z ) h − h + h − h ( z − z ) h − h − h + h (¯ z − ¯ z ) − h (¯ z − ¯ z ) ¯ h − ¯ h − ¯ h − ¯ h (¯ z − ¯ z ) ¯ h − ¯ h + ¯ h − ¯ h (¯ z − ¯ z ) ¯ h − ¯ h − ¯ h + ¯ h ∈ Cor f . Then, by an easy computation, we have:
Lemma 3.12.
For h i , ¯ h i ∈ R with h i − ¯ h i ∈ Z (i = , , , ), as a function on X , Q ( h , z ) satisfies ( X i = z i d / dz i + h i ) Q ( h , z ) = . ( X i = ¯ z i d / d ¯ z i + ¯ h i ) Q ( h , z ) = . ( X i = z i d / dz i + z i h i ) Q ( h , z ) = . ( X i = ¯ z i d / d ¯ z i + z i ¯ h i ) Q ( h , z ) = . Remark 3.1.
We can choose another Q ( h , z ) ∈ Cor which satisfies the above di ff erentialequations. In fact, we can multiply Q ( h , z ) by C [ ξ ± , ¯ ξ ± , | ξ | R ] . We also remark that Π ≤ i < j ≤ ( z i − z j ) − h i − h j + h / (¯ z i − ¯ z j ) − ¯ h i − ¯ h j + ¯ h / , is commonly used in physics, where h = h + h + h + h and ¯ h = ¯ h + ¯ h + ¯ h + ¯ h .However, since ( h − ¯ h ) / is not always integer, this function is not in Cor f . Proposition 3.6.
For h i , ¯ h i ∈ R and a i ∈ QF h i , ¯ h i (i = , , , ), the following conditionshold:(1) ( , Y ( a , z ) a ) = ( a , a )( − h − ¯ h z − h ¯ z − h . In particular, it is zero unless h = h and ¯ h = ¯ h .(2) There exists C a , a , a ∈ C such that ( , Y ( a , z ) Y ( a , z ) a ) = C a , a , a Π ≤ i < j ≤ ( z i − z j ) h + h + h − h i − h j (¯ z i − ¯ z j ) ¯ h + ¯ h + ¯ h − h i − h j | | z | > | z | . (3) There exists f ∈ F , , ∞ such that ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = (cid:16) Q ( h , z ) f ◦ ξ (cid:17) | | z | > | z | > | z | . In particular, this series is absolutely convergent to the function in
Cor .Proof. Since ( , Y ( a , z ) a ) = ( Y ( S z a , z − ) , a ) = ( − h − ¯ h z − h ¯ z − h (exp( L ( − z − + L ( − z − ) a , a ) = ( − h − ¯ h z − h ¯ z − h ( a , exp( L (1) z − + L (1)¯ z − ) a ) = ( − h − ¯ h z − h ¯ z − h ( a , a ),(1) holds. By Lemma 3.7, the product of the formal power series Π ≤ i < j ≤ ( z i − z j ) − h − h − h + h i + h j (¯ z i − ¯ z j ) − ¯ h − ¯ h − ¯ h + h i + h j | | z | > | z | and ( , Y ( a , z ) Y ( a , z ) a ) is in C [[ z / z , ¯ z / ¯ z ]][ z ± , z ± , ¯ z ± , ¯ z ± , | z | R , | z | R ]and is in the kernel of the formal di ff erentials D ′′ , ¯ D ′′ , D ′′ , ¯ D ′′ by Lemma 3.11 and thesimilar result to Lemma 3.12. Hence, it is constant C a , a , a ∈ C by Lemma 1.16, whichimplies that (2) holds.Similarly, set φ = Q ( h , z ) − | | z | > | z | > | z | ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) , which is an element of T ( z , z , z ) and in the kernel of the formal di ff erential D , ¯ D , D , ¯ D .Thus, by Lemma 1.13, v φ ∈ C (( p , ¯ p , | p | R )) determines φ . Recall that v : S ( z , z , z ) → C (( p , ¯ p , | p | R )) is defined by the evaluation ( z , z , z ) ( ∞ , , p ).Since ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = ( − h − ¯ h z − h ¯ z − h (exp( L ( − z + L ( − z ) a , Y ( a , z ) Y ( a , z ) a )and Q ( h , z ) = z − h z h − h − h − h (1 − z / z ) − h (1 − z / z ) h − h − h + h ¯ z − h ¯ z ¯ h − ¯ h − ¯ h − ¯ h (1 − ¯ z / ¯ z ) − h (1 − ¯ z / ¯ z ) ¯ h − ¯ h − ¯ h + ¯ h . the formal limit of φ as z → ∞ is( − h − ¯ h z − h + h + h + h ¯ z − ¯ h + ¯ h + ¯ h + ¯ h ( a , Y ( a , z ) Y ( a , z ) a )and v ( φ ) = ( − h − ¯ h p − h + h + h + h ¯ p − ¯ h + ¯ h + ¯ h + ¯ h ( a , Y ( a , Y ( a , p ) a ) . By Lemma 2.2, there exists f ∈ F , , ∞ such that v ( φ ) = j ( p , f ). By Corollary 1.1,we have φ = e ( f ). (cid:3) For h i , ¯ h i ∈ R and a i ∈ QF h i , ¯ h i ( i = , , , z →∞ Q ( h , z ) − ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = ( − h − ¯ h z − h + h + h + h ¯ z − ¯ h + ¯ h + ¯ h + ¯ h ( a , Y ( a , z ) Y ( a , z ) a )lim z →∞ Q ( h , z ) − ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = ( − h − ¯ h z − h + h + h + h ¯ z − ¯ h + ¯ h + ¯ h + ¯ h ( a , Y ( a , z ) Y ( a , z ) a ) . Thus, by Lemma 2.2, we have: Lemma 3.13.
For h i , ¯ h i ∈ R and a i ∈ QF h i , ¯ h i (i = , , , ), there exists f ∈ F , , ∞ suchthat ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = e ( Q ( h , z ) f ◦ ξ )( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = e ( Q ( h , z ) f ◦ ξ ) . Proposition 3.7.
Let F be a self-dual QP-generated full vertex operator algebra. Then,for any a i ∈ F i , there exists φ ∈ Cor such that ( , Y ( a σ , z σ ) . . . Y ( a σ , z σ ) ) = e σ ⋆ ))) ( φ ) , for any σ ∈ S . In particular, this series is absolutely convergent to a function in Cor .Proof. First, we assume that a i ∈ QF h i , ¯ h i for i = , , ,
4. Set Q = Q ( h , z ) and h = h + h + h + h and ¯ h = ¯ h + ¯ h + ¯ h + ¯ h . Then, by Proposition 3.6, there exists f ∈ F , , ∞ such that ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = e ( Q f ). Then, by Lemma 1.18 andLemma 3.9, we have( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) ) = T ⋆ )))1(2(34)) ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = T ⋆ )))1(2(34)) e ( Q f ) = e ⋆ ))) ( Q f ) . Let f , f ∈ F , , ∞ satisfy e ( Q f ) = ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) e ( Q f ) = ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) ) . We will show that f and f , f are the same function. Then, since (34) , (14)(23) , (23) ∈ S is a generator of S , by Lemma 3.13, the assertion holds for quasi-primary vectors. ByProposition 2.1 and Lemma 1.11, e ( Q f ) = ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = ( , Y ( a , z ) Y ( a , z ) exp( z D + ¯ z ¯ D ) Y ( a , − z ) a ) = exp( − z ( d / dz + d / dz ) − ¯ z ( d / d ¯ z + d / d ¯ z ))( , Y ( a , z ) Y ( a , z ) Y ( a , − z ) a ) = T e ( Q f ) , which implies that f = f .Since e ⋆ )) ( Q f ) = ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) ) = ( − h − ¯ h Π ≤ i ≤ z − h i i ¯ z − h i i ( , Y ( a , z − ) Y ( a , z − ) Y ( a , z − ) Y ( a , z − ) ) , by Lemma 1.19, e ⋆ )) ( Q f ) = ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) ) = I d (cid:16) ( − h − ¯ h Π ≤ i ≤ z h i i ¯ z h i i ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) ) (cid:17) = I d (cid:16) ( − h − ¯ h Π ≤ i ≤ z h i i ¯ z h i i e ( Q f ) (cid:17) = e ⋆ )) ( Q f ) , which implies f = f .We now turn to the case that a i ∈ F h i , ¯ h i . Since F is QP-generated, we may assume that a i = D n i ¯ D ¯ n i b i for some n i , ¯ n i ∈ Z ≥ and b i ∈ QF h i − n i , ¯ h i − ¯ n i . Since( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) ) = ( , Y ( D n ¯ D ¯ n b , z ) Y ( D n ¯ D ¯ n b , z ) D n ¯ D ¯ n Y ( b , z ) Y ( D n ¯ D ¯ n b , z ) ) = Π ≤ i ≤ d / dz n i i d / d ¯ z ¯ n i i ( , Y ( b , z ) Y ( b , z ) Y ( b , z ) Y ( b , z ) ) , by Proposition 3.6, there exists φ ∈ Cor such that ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) ) = e ⋆ ))) ( φ ). The S -invariance follows from the S -invariance for quasi-primary vec-tors. (cid:3) Hence, we can define the linear map S : F ⊗ → Cor by e ⋆ ))) ( S ( a , . . . , a )) = ( , Y ( a , z ) . . . Y ( a , z ) ). We recall that the symmetric group S acts on Cor (see Sec-tion 1.2). Then, the S -symmetry of the correlation function follows: Theorem 3.1.
For a self-dual QP-generated full vertex operator algebra F and a , a , a , a ∈ F and σ ∈ S , σ · S ( a , a , a , a ) = S ( a σ − , a σ − , a σ − , a σ − ) . Proof.
By Proposition 1.2 and Proposition 3.7, e ⋆ ))) ( σ · S ( a , a , a , a )) = T σ e σ − ⋆ ))) S ( a , a , a , a ) = T σ ( , Y ( a σ − , z σ − ) Y ( a σ − , z σ − ) Y ( a σ − , z σ − ) Y ( a σ − , z σ − ) ) = ( , Y ( a σ − , z ) Y ( a σ − , z ) Y ( a σ − , z ) Y ( a σ − , z ) ) = e ⋆ ))) S ( a σ − , a σ − , a σ − , a σ − ) . (cid:3) Now, the result for the consistency of four point correlation functions in conformal fieldtheory on C P can be stated as follows: Theorem 3.2.
Let F be a self-dual QP-generated full vertex operator algebra. Then,S A = e A ◦ S for any A ∈ Q . Lemma 3.14.
For a i ∈ QF h i , ¯ h i (i = , , , ),S (12)(34) ( a , a , a , a ) = ( − h − ¯ h z − h ¯ z − h z − h + h ¯ z − h + h (1 − z / z ) − h (1 − ¯ z / ¯ z ) − h (cid:16) a , Y (cid:16) a , − z (1 + z / z ) (cid:17) Y ( a , z − z / z ) a (cid:17) Proof.
By the invariance,( , Y ( Y ( a , z ) a , z ) Y ( a , z ) a ) = ( Y ( S z Y ( a , z ) a , z − ) , Y ( a , z ) a ) = (exp( z − D + ¯ z − ¯ D ) S z Y ( a , z ) a , Y ( a , z ) a ) = ( a , Y ( S z a , z − ) S † z (exp( z − L (1) + ¯ z − L (1)) Y ( a , z ) a ) , where S † z = ( − z ) L (0) ( − ¯ z ) L (0) exp( L ( − z + L ( − z ). By [FHL], z L (0) Y ( a , z ) z − L (0) = Y ( z L (0) a , zz )exp( zL (1)) Y ( a , z ) exp( − zL (1)) = Y (exp( z (1 − zz ) L (1))(1 − zz ) − L (0) a , z / (1 − zz )) . Thus, the assertion holds. (cid:3) proof of Theorem 3.2.
We only prove the Theorem for A ∈ P . The general result can beobtained by using Proposition 2.1 and the similar argument as Lemma 3.9. For the case of A = , ((12)3)4 and (1(23))4, the assertion follows from Lemma 1.9 and Lemma1.10. We will prove the theorem for A = (12)(34). Similarly to the proof of Theorem 3.7,we may assume that a i ∈ QF h i , ¯ h i for i = , , ,
4. Set Q = Q ( h , z ) and let f ∈ F , , ∞ satisfy S ( a , a , a , a ) = Q f ( ξ ). Then, by Lemma 3.1 and Lemma 3.10, e (12)(34) ( Q − )( , Y ( Y ( a , z ) a , z ) Y ( a , z ) a )is in S ′ ( z , z , z ). Thus, by Lemma 1.15, e (12)(34) ( Q − )( , Y ( Y ( a , z ) a , z ) Y ( a , z ) a )is uniquely determined by the limit of ( z , z , z ) → (1 , q , q ). By Lemma 3.14, the limitis equal to( − h − ¯ h q − h + h + h + h (1 + q ) h − h − h + h (1 − q ) − h ¯ q − h + h + ¯ h + ¯ h (1 + ¯ q ) ¯ h − ¯ h − ¯ h + ¯ h (1 − ¯ q ) − h · ( a , Y ( a , − (1 + q − )) Y ( a , q / (1 − q )) a ) . (q-series)Since the formal limit (we omit the anti-holomorphic part)lim ( z , z , z , z ) → ( ∞ , − (1 + q − ) , q / (1 − q ) , z − h ¯ z − h Q ( z ) − = q − q − h + h + h + h ( − q − (1 + q )) h − h − h + h ( − q (1 − q )) − h + h − h + h = ( − h − h q − h + h + h + h (1 + q ) h − h − h + h (1 − q ) − h and 2 h − h − h − h ∈ Z , we can verify that the series q-series is equal to the formallimit of Q ( z ) − | | z | > | z | > | z | > | z | ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) )as ( z , z , z , z ) → ( ∞ , − (1 + q − ) , q / (1 − q ) , ξ ( z , z , z , z ) as( z , z , z , z ) → ( ∞ , − (1 + q − ) , q / (1 − q ) ,
0) is q and Q ( z ) − | | z | > | z | > | z | > | z | ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) ) = e ⋆ ))) ( f ( ξ )) , v (12)(34) (cid:16) e (12)(34) ( Q − )( , Y ( Y ( a , z ) a , z ) Y ( a , z ) a ) (cid:17) = lim p → q j ( p , f ) . Hence, ( , Y ( Y ( a , z ) a , z ) Y ( a , z ) a ) = e (12)(34) ( Q f ( ξ )). (cid:3)
4. C onstruction
In this section, we generalize the Goddard’s axiom for a full vertex algebra (Section4.1 and Section 4.2). We also show that if a vertex operator Y ( − , z ) satisfies the bootstrapequation then it gives a full vertex algebra.A full prevertex algebra is an R -graded C -vector space F = L h , ¯ h ∈ R F h , ¯ h equippedwith a linear map Y ( − , z ) : F → End( F )[[ z ± , ¯ z ± , | z | R ]] , a Y ( a , z ) = X r , s ∈ R a ( r , s ) z − r − ¯ z − s − and an element ∈ F , such thatPV1) For any a , b ∈ F , there exists N ∈ R such that a ( r , s ) b = r ≤ N or s ≤ N ;PV2) F h , ¯ h = h − ¯ h ∈ Z ;PV3) For any a ∈ F , Y ( a , z ) ∈ F [[ z , ¯ z ]] and lim z → Y ( a , z ) = a ( − , − = a ;PV4) Y ( , z ) = id ∈ End F ;PV5) F h , ¯ h ( r , s ) F h ′ , ¯ h ′ ⊂ F h + h ′ − r − , ¯ h + ¯ h ′ − s − for any r , s , h , ¯ h , h ′ , ¯ h ′ ∈ R .A full prevertex algebra ( F , Y , ) is said to be translation covariant if there exist linearmaps D , ¯ D ∈ End F such thatT1) D = ¯ D = a ∈ F , [ D , Y ( a , z )] = ddz Y ( a , z ) and [ ¯ D , Y ( a , z )] = dd ¯ z Y ( a , z );4.1. Locality and Associativity I.
Let ( F , Y , , D , ¯ D ) be a translation covariant full pre-vertex algebra.In this section, we consider the following conditions:FL) For any a , a , a ∈ F and u ∈ F ∨ , there exists µ ∈ GCor such that u ( Y ( a , z ) Y ( a , z ) a ) = µ | | z | > | z | u ( Y ( a , z ) Y ( a , z ) a ) = µ | | z | > | z | andFA) For any a , a , a ∈ F and u ∈ F , there exists µ ∈ GCor such that u ( Y ( a , z ) Y ( a , z ) a ) = µ | | z | > | z | u ( Y ( Y ( a , z ) a , z ) a ) = µ | | z | > | z − z | andFSS) Y ( a , z ) a = e Dz + ¯ D ¯ z Y ( a , − z ) a for any a , a ∈ F .The condition (FL) (resp. (FA) and (FSS)) is a generalization of the locality (resp.associativity, skew-symmetry) of a vertex algebra. Lemma 4.1.
If a translation covariant full prevertex algebra ( F , Y , , D , ¯ D ) satisfies con-dition (FL), then the skew-symmetry, (FSS), holds. Proof.
Since DY ( a , z ) = d / dzY ( a , z ) , we have Y ( a , z ) = exp( Dz + ¯ D ¯ z ) a , which impliesthat DF h , ¯ h ⊂ F h + , ¯ h and ¯ DF h , ¯ h ⊂ F h , ¯ h + . Let a i ∈ F h i , ¯ h i ( i = ,
2) and u ∈ F ∗ h , ¯ h . Then, u ( Y ( a , z ) Y ( a , z ) ) = u ( Y ( a , z ) exp( Dz + ¯ D ¯ z ) a ) = lim z → ( z − z ) | | z | > | z | u (exp( Dz + ¯ D ¯ z ) a ) Y ( a , z ) a ) . Set h = h + h − h and ¯ h = ¯ h + ¯ h − ¯ h . Then, by (PV5), u (exp( Dz + ¯ D ¯ z ) a ) Y ( a , z ) a ) = X s , ¯ s ∈ R X n , ¯ n ∈ Z ≥ n !¯ n ! u ( D n ¯ D ¯ n a ( s , ¯ s ) a ) z − ¯ s − ¯ z − ¯ s − z n ¯ z ¯ n = X n , ¯ n ∈ Z ≥ n !¯ n ! u ( D n ¯ D ¯ n a ( h + n − , ¯ h + ¯ n − a ) z − h − n ¯ z − ¯ h − ¯ n z n ¯ z ¯ n . By (PV1), there exists an integer N such that a ( h + n − , ¯ h + ¯ n − a = n ≥ N or ¯ n ≥ N . Thus, z N + h ¯ z ¯ N + ¯ h u (exp( Dz + ¯ D ¯ z ) a ) Y ( a , z ) a ) ∈ C [ z , z , ¯ z , ¯ z ].Hence, by (FL), { ( z − z ) N + h (¯ z − ¯ z ) ¯ N + ¯ h }| | z | > | z | u ( Y ( a , z ) Y ( a , z ) ) = { ( z − z ) N + h (¯ z − ¯ z ) ¯ N + ¯ h }| | z | > | z | u ( Y ( a , z ) Y ( a , z ) ) , which implies( − z ) N + h ( − ¯ z ) ¯ N + ¯ h u ( Y ( a , z ) a ) = lim z → { ( z − z ) N + h (¯ z − ¯ z ) ¯ N + ¯ h }| | z | > | z | u ( Y ( a , z ) Y ( a , z ) ) = lim z → lim z → ( z − z ) | | z | > | z | z N + h ¯ z ¯ N + ¯ h u (exp( Dz + ¯ D ¯ z ) a ) Y ( a , z ) a ) = ( − z ) N + h ( − ¯ z ) ¯ N + ¯ h u (exp( Dz + ¯ D ¯ z ) a ) Y ( a , − z ) a ) . Hence, the assertion holds. (cid:3)
Then, we have:
Proposition 4.1.
Assume that a translation covariant full prevertex algebra ( F , Y , , D , ¯ D ) satisfies the condition (FL) and the spectrum of F is bounded below, that is, there existsN ∈ R such that F h , ¯ h = for any h ≤ N or ¯ h ≤ N. Then, ( F , Y , ) is a full vertex algebra.Proof. Let a , a , a ∈ F and u ∈ F ∨ . By (FL), there exists µ ∈ GCor such that u ( Y ( a , z ) Y ( a , z ) a ) = µ | | z | > | z | u ( Y ( a , z ) Y ( a , z ) a ) = µ | | z | > | z | . By Lemma 4.1, u ( Y ( a , z ) Y ( a , z ) a ) = lim z → ( z − z ) | | z | > | z | u (exp( Dz + ¯ D ¯ z ) Y ( a , z ) Y ( a , − z ) a ) . By the assumption, u ( D n ¯ D ¯ n − ) = ffi ciently large n or ¯ n in Z ≥ . Then, by (FL), u (exp( Dz + ¯ D ¯ z ) Y ( a , z ) Y ( a , − z ) a ) = µ | | z | > | z | u (exp( Dz + ¯ D ¯ z ) Y ( a , − z ) Y ( a , z ) a ) = µ | | z | > | z | . Since u (exp( Dz + ¯ D ¯ z ) Y ( a , − z ) Y ( a , z ) a ) = u ( Y ( Y ( a , z ) a , z ) a ), (FV5) holds. (cid:3) In the above proposition, we assume that the spectrum of F is bounded below. However,in the proof, we only use the property that for any a , a , a ∈ F and u ∈ F ∨ , there exits N such that u ( D n ¯ D m Y ( a , z ) Y ( b , z ) c ) = n ≥ N or m ≥ N . Thus, we have: Proposition 4.2.
Let ( F , Y , , D , ¯ D ) be a translation covariant full prevertex algebra sat-isfies the condition (FL) and M be an abelian group. Suppose that there exists an M × R -grading on F, F = L α ∈ M , h , ¯ h ∈ R F α h , ¯ h such that:(1) F h , ¯ h = L α ∈ M F α h , ¯ h for any h , ¯ h ∈ R and α ∈ M(2) F α h , ¯ h ( r , s ) F α ′ h ′ , ¯ h ′ ⊂ F α + α ′ h + h ′ − r − , ¯ h + ¯ h ′ − s − for any r , s , h , ¯ h , h ′ , ¯ h ′ ∈ R and α, α ′ ∈ M;(3) For any α , there exists N such that F α h , ¯ h = for any h ≥ N or ¯ h ≥ N.Then, F is a full vertex algebra.
Proposition 4.3.
Assume that a translation covariant full prevertex algebra ( F , Y , , D , ¯ D ) satisfies the condition (FA) and the skew-symmetry. Then, ( F , Y , ) is a full vertex algebra.Proof. We first prove that Y ( Da , z ) = d / dzY ( a , z ) for any a ∈ F . For any a , b ∈ F and u ∈ F ∨ , by (FFA), there exists µ ∈ GCor such that u ( Y ( a , z ) b ) = µ | | z | > | z | u ( Y ( Y ( a , z ) , z ) b ) = µ | | z | > | z − z | . By (PV5), u ( Y ( a , z ) b ) = p ( z ) ∈ C [ z ± , ¯ z ± , | z | R ]. Thus, µ | | z | > | z − z | = lim z → ( z + z ) | | z | > | z | p ( z ) = exp( z d / dz + ¯ z d / d ¯ z ) p ( z ) . Hence, u ( Y ( Y ( a , z ) , z ) b ) = exp( z d / dz + ¯ z d / d ¯ z ) u ( Y ( a , z ) b ) , which implies that Y ( Da , z ) = d / dz Y ( a , z ) and similarly Y ( ¯ Da , z ) = d / d ¯ z Y ( a , z ).Now, we will show the assertion. Let a , a , a ∈ F and u ∈ F ∨ . By (FA), there exists µ ∈ GCor such that u ( Y ( a , z ) Y ( a , z ) a ) = µ | | z | > | z | u ( Y ( Y ( a , z ) a , z ) a ) = µ | | z | > | z − z | . By the skew symmetry, u ( Y ( Y ( a , z ) a , z ) a ) = u ( Y (exp( Dz + ¯ D ¯ z ) Y ( a , − z ) a , z ) a ) = lim z → ( z + z ) | | z | > | z | u ( Y ( Y ( a , − z ) a , z ) a ) . Then, applying (FA) again, (FV5) holds. (cid:3)
Locality and Associativity II.
In this section, we give a su ffi cient condition in orderto construct a full vertex algebra by using four point functions. We remark that for a self-dual QP-generated full vertex operator algebra F , there is the linear map ( , − ) : F → C such that Proposition 3.7 holds.In this section, we assume that ( F , Y , , D , ¯ D ) is a translation covariant full prevertexalgebra equipped with a linear map <> : F → C such that E1) F h , ¯ h is a finite dimensional vector space for any h , ¯ h ∈ R ;E2) < > = < Da > = < ¯ Da > = a ∈ F ;E3) < F h , ¯ h > = h , ¯ h ) , (0 , N ∈ R such that F h , ¯ h = h ≤ N or ¯ h ≤ N .We remark that if F , = C , then <> is simply the projection F → F , .For ( F , Y , , D , ¯ D , <> ), set R F ,<> = { b ∈ F | < Y ( a , z ) b > = a ∈ F } . We say ( F , Y , , D , ¯ D , <> ) is non-degenerate if R F ,<> = a , a , a , a ∈ F , there exists φ ∈ Cor such that < Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) > = e ⋆ ))) φ< Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) > = e ⋆ ))) φ andFFA) For any a , a , a , a ∈ F , there exists φ ∈ Cor such that < Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) > = e ⋆ ))) φ< Y ( a , z ) Y ( a , z ) Y ( a , z ) Y ( a , z ) > = e ⋆ ))) φ. By Corollary 1.3 and (PV4), we have:
Lemma 4.2.
Let ( F , Y , , D , ¯ D , <> ) satisfy (FFL) or (FFA). Then, for a , a , a ∈ F, thereexists φ ∈ Cor such that for any σ ∈ S , < Y ( a σ , z σ ) Y ( a σ , z σ ) Y ( a σ , z σ ) > = e σ ⋆ )) φ. Lemma 4.3.
Let ( F , Y , , D , ¯ D , <> ) satisfy (FFL) or (FFA). Then, for a , a , a ∈ F, < Y ( a , z ) Y ( a , z ) a > = < Y ( a , z ) exp( Dz + ¯ D ¯ z ) Y ( a , − z ) a > . Proof.
Let φ ∈ Cor such that < Y ( a , z ) Y ( a , z ) a > = e ( φ ). Similarly to the proofof Lemma 1.11, e = lim z →− z z → ( z − z ) | | z | > | z | e . Thus, by Lemma 4.2 and (E2), < Y ( a , z ) Y ( a , z ) a > = e ( φ ) = exp( − z d / dz − ¯ z d / d ¯ z ) < Y ( a , z ) Y ( a , − z ) a > = < exp( − Dz − ¯ D ¯ z ) Y ( a , z ) exp( Dz + ¯ D ¯ z ) Y ( a , − z ) a > = < Y ( a , z ) exp( Dz + ¯ D ¯ z ) Y ( a , − z ) a > . (cid:3) We remark that we cannot obtain the skew-symmetry from the above lemma, sincethe skew-symmetry holds only in < − > ∈ F ∨ . However, if ( F , Y , , D , ¯ D , <> ) is non-degenerate, then, by (E1) and (PV5), F ∨ is spanned by { < a ( s , ¯ s ) − > } a ∈ F , s , ¯ s ∈ R . Thus, wehave: Lemma 4.4. If ( F , Y , , D , ¯ D , <> ) is non-degenerate and satisfy (FFL) or (FFA), then theskew-symmetry holds. Proposition 4.4. If ( F , Y , , D , ¯ D , <> ) is non-degenerate and satisfy (FFL) or (FFA), then ( F , Y , ) is a full vertex algebra.Proof. First, we assume that (FFL) holds. Then, by Lemma 1.4, the condition (FL) holds.Thus, by Proposition 4.1, F is a full vertex algebra. Next, we assume that (FFA) holds.Then, for any a , a , a , a ∈ F , there exists φ ∈ Cor such that < Y ( a , z ) Y ( a , z ) Y ( a , z ) a > = e φ< Y ( a , z ) Y ( a , z ) Y ( a , z ) a > = e φ. Then, by Lemma 4.4 and Lemma 1.11, < Y ( a , z ) Y ( a , z ) Y ( a , z ) a > = e φ. ByLemma 4.4 and (E2), < Y ( a , z ) Y ( a , z ) Y ( a , z ) a > = < Y ( a , z ) exp( Dz + ¯ D ¯ z ) Y ( Y ( a , z ) a , − z ) a > = exp( z d / dz + ¯ z d / d ¯ z ) < Y ( a , z ) Y ( Y ( a , z ) a , − z ) a > . Then, by Lemma 1.9, < Y ( a , z ) Y ( Y ( a , z ) a , z ) a > = e φ. By Lemma 1.4 again, the condition (FA) holds. Thus, by Proposition 4.3, F is a fullvertex algebra. (cid:3) In the rest of this section, we study the condition that ( F , Y , , D , ¯ D , <> ) is non-degenerate.A two-sided ideal of ( F , Y , , D , ¯ D , <> ) is a subspace I ⊂ F such that I = L h , ¯ h ∈ R I ∩ F h , ¯ h ,i.e., a graded subspace, and for any v ∈ I and a ∈ F , Y ( a , z ) v , Y ( v , z ) a ∈ I (( z , ¯ z , | z | R )).Then, we have: Lemma 4.5. If ( F , Y , , D , ¯ D , <> ) satisfy (FFL) or (FFA), the subspace R F ,<> is a two-sided ideal of F.Proof. By (PV5) and (E3), R F ,<> is a graded subspace of F . Let a , a ∈ F and v ∈ R F ,<> .It su ffi ces to show that < Y ( a , z ) Y ( a , z ) Y ( v , z ) > = < Y ( a , z ) Y ( v , z ) Y ( a , z ) > = < Y ( a , z ) b > = < Y ( b , − z ) a > for any a , b ∈ F . Since < Y ( v , z ) Y ( a , z ) Y ( a , z ) > = < Y ( Y ( a , z ) Y ( a , z ) , − z ) v > =
0, by Lemma 4.2, theassertion holds. (cid:3) By the above lemma, the vertex operator Y ( − , z ) : F / R F ,<> → End ( F / R F ,<> )[[ z , ¯ z , | z | R ]], D , ¯ D ∈ F / R F ,<> and <> : F / R F ,<> → C are induced from ( F , Y , , D , ¯ D , <> ) and are non-degenerate. We remark that by the construction, a parenthesized correlation function of F / R F ,<> and F are the same. Thus, we have: Proposition 4.5.
Let ( F , Y , , D , ¯ D , <> ) satisfy (FFL) or (FFA). Then, F / R F ,<> is a fullvertex algebra. ( F , Y , , D , ¯ D , <> ) is said to be simple if there is no proper left ideal of F . By the abovelemma, we have: Proposition 4.6.
Let ( F , Y , , D , ¯ D , <> ) satisfy (FFL) or (FFA). If ( F , Y , , D , ¯ D , <> ) issimple, then it is non-degenerate. Conversely, if ( F , Y , , D , ¯ D , <> ) is non-degenerate andF , = C , then ( F , Y , , D , ¯ D , <> ) is simple.Proof. By Lemma 4.5, R F ,<> is a left ideal. If F is simple, then R F ,<> = F , Y , , D , ¯ D , <> ) is non-degenerate and F , = C and let I ⊂ F be a left ideal which does not contain . Since I ∩ F , = < Y ( a , z ) v > = a ∈ F and v ∈ I . Hence, I ⊂ R F ,<> = (cid:3) Bootstrap.
In this section, we study the bootstrap equation. Set L ( − = D and L ( − = ¯ D and let L (0) , L (0) ∈ End F be linear maps defined by L (0) | F h , ¯ h = h and L (0) | F h , ¯ h = ¯ h for any h , ¯ h ∈ R . We assume that ( F , Y , , D , ¯ D ) is a translation covariant fullprevertex algebra equipped with linear maps L (1) , L (1) ∈ End F and a non-degeneratesymmetric bilinear form ( − , − ) : F × F → C such that:(1) There exists N ∈ R such that F h , ¯ h = h ≤ N or ¯ h ≤ N ;(2) dim F h , ¯ h is finite for any h , ¯ h ∈ R ;(3) For any i , j = − , ,
1, [ L ( i ) , L ( j )] = ( i − j ) L ( i + j ) , [ L ( i ) , L ( j )] = ( i − j ) L ( i + j ) , [ L ( i ) , L ( j )] = i = − , , a , b ∈ F ,( L ( i ) a , b ) = ( a , L ( − i ) b )( L ( i ) a , b ) = ( a , L ( − i ) b );(5) For k = − , , L ( k ) , Y ( a , z )] = k + X i = k + i + ! Y ( L ( i + a , z ) z − i [ L ( k ) , Y ( a , z )] = k + X i = k + i + ! Y ( L ( i + a , z )¯ z − i ;(6) L (1) = L (1) = (7) For any a , b , c ∈ F , ( a , Y ( b , z ) c ) = ( Y ( S z b , z − ) a , c ) . Let ( F , Y , , L ( i ) , L ( i ) , ( − , − )) satisfy the above condition. For any a , a , a , a ∈ F ,define s,t,u-channels by S (21)(34) ( a , a , a , a ) = (cid:16) , (cid:16)(cid:16) a ( z ) a (cid:17) ( z ) a ( z ) a (cid:17) (s-channel) S (41)(32) ( a , a , a , a ) = (cid:16) , (cid:16)(cid:16) a ( z ) a (cid:17) ( z ) a ( z ) a (cid:17) (t-channel) S (31)(24) ( a , a , a , a ) = (cid:16) , (cid:16)(cid:16) a ( z ) a (cid:17) ( z ) a ( z ) a (cid:17) . (u-channel)( F , Y , , L ( i ) , L ( i ) , ( − , − )) is said to satisfy a bootstrap equation for the s-channel and t-channel (resp. for the s-channel and u-channel) if for any a , a , a , a ∈ F , there exists φ ∈ Cor such that S (21)(34) ( a , a , a , a ) = e (21)(34) ( φ ) and S (41)(32) ( a , a , a , a ) = e (41)(32) ( φ )(resp. S (21)(34) ( a , a , a , a ) = e (21)(34) ( φ ) and S (31)(24) ( a , a , a , a ) = e (31)(24) ( φ )).The notion, QP-generated, is defined for ( F , Y , , L ( i ) , L ( i ) , ( − , − )) similarly to the fullvertex operator algebra (see Section 3.2).Let ( F , Y , , L ( i ) , L ( i ) , ( − , − )) be QP-generated and satisfy the bootstrap equation for thes-channel and t-channel. Set < − > = ( , − ). We will show that ( F , Y , , D , ¯ D , <> ) satisfythe assumption of Proposition 4.4. It su ffi ces to show that (FFA) holds. Similarly tothe proof of Proposition 3.7, we may assume that all a i ∈ F h i , ¯ h i are quasi-primary states.Then, by PSL C -symmetry, in order to verify (FFA), it su ffi ces to show that there exists f ∈ F , , ∞ such that Q ( h , z ) − ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = e ( f ◦ ξ ) , Q ( h , z ) − ( , Y ( a , z ) Y ( a , z ) Y ( a , z ) a ) = e ( f ◦ ξ ) . By Lemma 3.14, we can calculate v ( Q ( h , z ) − S ) and v ( Q ( h , z ) − S )similarly to the proof of Theorem 3.2, which are equal to j ( p , f ) and j (1 − p , f ). Thebootstrap equation for the s-channel and u-channel implies the same result. The detailverification is left to the reader. Thus, we have: Proposition 4.7. If ( F , Y , , L ( i ) , L ( i ) , ( − , − )) is QP-generated and satisfy the bootstrapequation for the s-channel and t-channel or the s-channel and u-channel, then F is a fullvertex algebra.
5. E xample
In this section, we construct a full vertex algebras from an AH pair, which is introducedin [Mo1]. One can construct an AH pair from an even lattice, called twisted group algebra[FLM, Mo1]. The full vertex algebra constructed from an even lattice is appeared in thetoroidal compactification of string theory and is shown to be a QP-generated self-dualfull vertex operator algebra. A conceptual origin of AH pairs is provided in Section 5.3,where we show that an AH pair is a commutative algebra object in some braided tensorcategory. Full vertex algebra and AH pair.
We first recall a concept of an AH pair intro-duced in [Mo1]. Let H be a finite-dimensional vector space over R equipped with anon-degenerate symmetric bilinear form ( − , − ) : H × H → R and A a unital associativealgebra over C with the unity 1. Assume that A is graded by H as A = L α ∈ H A α .We will say that such a pair ( A , H ) is an AH pair if the following conditions hold:AH1) 1 ∈ A and A α A β ⊂ A α + β for any α, β ∈ H ;AH2) A α A β , α, β ) ∈ Z ;AH3) For v ∈ A α , w ∈ A β , vw = ( − ( α,β ) wv .An AH pair ( A , H ) is called an even AH pair ifEAH ( α, α ) ∈ Z for any α ∈ H with A α , P ( H ) be the set of linear maps p ∈ End H such that:P1) p is a projection, that is, p = p ;P2) The subspaces ker p and ker(1 − p ) are orthogonal to each other.Let P > ( H ) be the subset of P ( H ) such thatP3) ker(1 − p ) is positive definite and ker p is negative-definite.Let ( A , H ) be an even AH pair and p ∈ P ( H ). In this section, we construct a full vertexalgebra F A , H , p for the triple ( A , H , p ).Set ¯ p = − p and H l = ker ¯ p and H r = ker p . Define the new symmetric bilinear forms( − , − ) p : H × H → R by ( h , h ′ ) p = ( ph , ph ′ ) − ( ¯ ph , ¯ ph ′ ) . By (P1) and (P2), ( − , − ) p is non-degenerate. Let ˆ H p = L n ∈ Z H ⊗ C t n ⊕ C c be the a ffi neHeisenberg Lie algebra associated with ( ˆ H p , ( − , − ) p ) and ˆ H p ≥ = L n ≥ H ⊗ C t n ⊕ C c asubalgebra of ˆ H p . Define the action of ˆ H p ≥ on A by ca = ah ⊗ t n a = , n ≥ , ( h , α ) p a , n = α ∈ H and a ∈ A α . Let F A , H , p be the ˆ H p -module induced from A and M H , p be thesubmodule of F A , H , p generated by the unit 1 ∈ A as a ˆ H p -module. Denote by h ( n ) the action of h ⊗ t n on F A , H , p for n ∈ Z . For h ∈ H , set h ( z , ¯ z ) = X n ∈ Z (( ph )( n ) z − n − + ( ¯ ph )( n )¯ z − n − ) ∈ End F A , H , p [[ z ± , ¯ z ± ]] h + ( z ) = X n ≥ (( ph )( n ) z − n − + ( ¯ ph )( n )¯ z − n − ) h − ( z ) = X n ≥ (( ph )( − n − z n + ( ¯ ph )( − n − z n ) . E + ( h , z ) = exp (cid:18) − X n ≥ ( ph ( n ) n z − n + ¯ ph ( n ) n ¯ z − n ) (cid:19) E − ( h , z ) = exp (cid:18) − X n ≤− ( ph ( n ) n z − n + ¯ ph ( n ) n ¯ z − n ) (cid:19) . For h r ∈ H r and h l ∈ H l , h r ( z ) and h l ( z ) are denoted by h r ( z ) and h l ( z ).Then, similarly to the case of a lattice vertex algebra, we have: Lemma 5.1.
For any h , h ∈ H,E + ( h , z ) E − ( h , z ) = (cid:18) X n , ¯ n ≥ ( ph , ph ) p n ! ( ¯ ph , ¯ ph ) p ¯ n ! z n z − n ¯ z ¯ n ¯ z − ¯ n (cid:19) E − ( h , z ) E + ( h , z ) . We remark that the formal power series P n , ¯ n ≥ (cid:16) ( ph , ph ) p n (cid:17)(cid:16) ( ¯ ph , ¯ ph ) p ¯ n (cid:17) z n z − n ¯ z ¯ n ¯ z − ¯ n is equal to(1 − z / z ) ( ph , ph ) p (1 − ¯ z / ¯ z ) ( ¯ ph , ¯ ph ) p | | z | > | z | . Let α ∈ H and a ∈ A α . Denote by l a ∈ End A the left multiplication by a anddefine the linear map l a z p α ¯ z ¯ p α : A → A [ z , ¯ z , | z | R ] by l a z p α ¯ z ¯ p α b = z ( p α, p β ) p ¯ z ( ¯ p α, ¯ p β ) p ab for β ∈ H and b ∈ A β . Assume that ab ,
0. Then, ( α, β ) ∈ Z and since z ( p α, p β ) p ¯ z ( ¯ p α, ¯ p β ) p = | z | ( p α, p β ) p ¯ z ( ¯ p α, ¯ p β ) p − ( p α, p β ) p and ( ¯ p α, ¯ p β ) p − ( p α, p β ) p = − ( α, β ) ∈ Z , we have l a z p α ¯ z ¯ p α b ∈ A [ z , ¯ z , | z | R ]. Then, set a ( z ) = E − ( α, z ) E + ( α, z ) l a z p α ¯ z ¯ p α , which is a linear map F A , H , p → F A , H , p [[ z , ¯ z , | z | R ]] . By Poincar´e-Birkho ff -Witt theorem, F A , H , p (cid:27) M H , p ⊗ A , that is, F A , H , p is spanned by { h l ( − n − . . . h ll ( − n l − h r ( − ¯ n − . . . h kr ( − ¯ n k − a } , where h il ∈ H l , n i ∈ Z ≥ and h jr ∈ H r , ¯ n j ∈ Z ≥ for any 1 ≤ i ≤ l and 1 ≤ j ≤ k and a ∈ A . Then, a map Y : F A , H , p → End F A , H , p [[ z , ¯ z , | z | R ]] is defined inductively asfollows: For a ∈ A , define Y ( a , z ) by Y ( a , z ) = a ( z ). Assume that Y ( v , z ) is already definedfor v ∈ F A , H , p . Then, for h r ∈ H r and h l ∈ H l and n , ¯ n ∈ Z ≥ , Y ( h l ( − n − v , z ) and Y ( h r ( − ¯ n − v , z ) is defined by Y ( h l ( − n − v , z ) = / n ! (cid:16) ddz n h − l ( z ) (cid:17) Y ( v , z ) + Y ( v , z )1 / n ! (cid:16) ddz n h + l ( z ) (cid:17) Y ( h r ( − ¯ n − v , z ) = / ¯ n ! (cid:16) dd ¯ z ¯ n h − r (¯ z ) (cid:17) Y ( v , z ) + Y ( v , z )1 / ¯ n ! (cid:16) dd ¯ z ¯ n h + r (¯ z ) (cid:17) . By the direct calculus, we can easily prove the following lemma: Lemma 5.2.
For any v ∈ F A , H , p and h l ∈ H l and h r ∈ H r , [ h l ( n ) , Y ( v , z )] = X i ≥ ni ! Y ( h l ( i ) v , z ) z n − i , [ h r ( n ) , Y ( v , z )] = X i ≥ ni ! Y ( h r ( i ) v , z )¯ z n − i . By the above lemma,[ h + l ( z ) , Y ( v , z )] = X i ≥ , n ≥ ni ! Y ( h l ( i ) v , z ) z − n − z n − i = X i ≥ Y ( h l ( i ) v , z ) 1 i ! ddz i X n ≥ z − n − z n = X i ≥ Y ( h l ( i ) v , z ) 1( z − z ) i + | | z | > | z | . Thus, we have:
Lemma 5.3.
For any v ∈ F A , H , p and h l ∈ H l , [ h + l ( z ) , Y ( v , z )] = X i ≥ Y ( h l ( i ) v , z ) 1( z − z ) i + | | z | > | z | , [ Y ( v , z ) , h − l ( z )] = X i ≥ Y ( h l ( i ) v , z ) ( − i + ( z − z ) i + | | z | > | z | . Set = ⊗ ,ω = / dim H l X i = h il ( − h il , ¯ ω = / dim H r X j = h jr ( − h jr , where h il and h jr is an orthonormal basis of H l ⊗ R C and H r ⊗ R C with respect to the bilinearform ( − , − ) p .We will prove the following proposition by using Proposition 4.2: Proposition 5.1.
For an even AH pair ( A , H ) and p ∈ P ( H ) , ( F A , H , p , Y , ) is a full vertexalgebra. Set F = F A , H , p and D = ω (0) and ¯ D = ¯ ω (0) and F α h , ¯ h = { v ∈ F | ω (1) v = hv , ¯ ω (1) v = ¯ hv , h (0) v = ( α, h ) p v for any h ∈ H } and F h , ¯ h = M α ∈ H F α h , ¯ h , for h , ¯ h ∈ R and α ∈ H . Then, F = L h , ¯ h ∈ R L α ∈ H F α h , ¯ h . In order to prove ( F , Y , ) is a fullvertex algebra, it su ffi ces to show that ( F , Y , , D , ¯ D ) satisfies the condition in Proposition4.2. An easy computation show that ( F , Y , , D , ¯ D ) is a translation covariant full prevertexalgebra and F = L h , ¯ h ∈ R L α ∈ H F α h , ¯ h satisfies the conditions (1),(2) and (3) in Proposition4.2. Thus, it su ffi ces to show that the condition (FL).For α ∈ H and n , m ∈ Z ≥ , it is easy to show that F α ( p α, p α ) p + n , ( ¯ p α, ¯ p α ) p + m is spannedby { h l ( − i ) . . . h kl ( − i k ) h r ( j ) . . . h lr ( j l ) a } , where k , l ∈ Z ≥ , h al ∈ H l , h br ∈ H r , a ∈ A α , i a , j b ∈ Z ≥ , i + · · · + i k = n and j + · · · + j l = m for any a = , . . . , k and b = , . . . , l .Then, F α = M n , m ∈ Z ≥ F α ( p α, p α ) p + n , ( ¯ p α, ¯ p α ) p + m and F α ( p α, p α ) p , ( ¯ p α, ¯ p α ) p = A α . Let a ∗ ∈ A ∨ = L α ∈ H ( A α ) ∗ and < a ∗ , − > be the linear map F → C defined by thecomposition of the projection F = A ⊕ L n , m ∈ Z ≥ ( n , m ) , (0 , F α ( p α, p α ) p + n , ( ¯ p α, ¯ p α ) p + m → A and a ∗ : A → C . Then, it is easy to verify < a ∗ , − > is a highest weight vector, that is, < a ∗ , h ( − n ) − > = n ≥ h ∈ H . Thus, for any α ∈ H , we have: E + ( α, z ) = ,< a ∗ , E − ( α, z ) − > = < a ∗ , − > . Thus, by using the above fact and Lemma 5.1, we have:
Lemma 5.4.
For α , α , α , α ∈ H and a ∗ ∈ ( A α ) ∗ and a i ∈ A α i (for i = , , ), < a ∗ , Y ( a , z ) Y ( a , z ) a > = z ( p α , p α ) p z ( p α , p α ) p ( z − z ) ( p α , p α ) p | | z | > | z | ¯ z ( ¯ p α , ¯ p α ) p ¯ z ( ¯ p α , ¯ p α ) p (¯ z − ¯ z ) ( ¯ p α , ¯ p α ) p | | z | > | z | < a ∗ , a a a > . Assume that none of a a , a a , a a is equal to zero. Then, ( α , α ) , ( α , α ) , ( α , α ) ∈ Z by (AH2). Set φ = z ( p α , p α ) p z ( p α , p α ) p ( z − z ) ( p α , p α ) p | | z | > | z | ¯ z ( ¯ p α , ¯ p α ) p ¯ z ( ¯ p α , ¯ p α ) p (¯ z − ¯ z ) ( ¯ p α , ¯ p α ) p | | z | > | z | , which is a function in GCor . The above lemma implies that < a ∗ , Y ( a , z ) Y ( a , z ) a > = ( − ( α ,α ) φ | | z | > | z | < a ∗ , a a a > . By the definition of an AH pair, a a = ( − ( α ,α ) a a .Thus, (FL) holds. If one of a a , a a , a a is equal to zero, then both < a ∗ , a a a > and < a ∗ , a a a > are zero, which implies (FL). For u ∈ F ∨ and v , v , v ∈ F , set S ( u ; v , v , v ) = u ( Y ( v , z ) Y ( v , z ) v ) . We can compute S ( u ; v , v , v ) by using Lemma 5.4 and Lemma 5.3. We observe that L ( F α h , ¯ h ) ∗ is spanned by { < a ∗ , h ( i ) . . . h k ( i k ) − > } , where a ∗ ∈ ( A α ) ∗ and i , . . . , i k Z ≥ and h , . . . , h k ∈ H . Let u ∈ F ∨ and v , v , v ∈ F and h l ∈ H l and n ∈ Z ≥ . Then, by Lemma5.3, S ( u ( h l ( n ) − ); v , v , v ) = u ( h l ( n ) Y ( v , z ) Y ( v , z ) v ) = S ( u ; v , v , h l ( n ) v ) + X i ≥ ni !(cid:16) S ( u ; h l ( i ) v , v , v ) z n − i + S ( u ; v , h l ( i ) v , v ) z n − i (cid:17) . Since the above formula is symmetric for v and v , in order to show (FL), we may assumethat u = a ∗ ∈ A ∨ . Similarly, for n ∈ Z ≥ , a ∗ ( Y ( h l ( − n − v , z ) Y ( v , z ) v ) = X i ≥ − i − n !(cid:16) ( − n ( z − z ) n + i + | | z | > | z | a ∗ ( Y ( v , z ) Y ( h l ( i ) v , z ) v ) + z n + i + a ∗ ( Y ( v , z ) Y ( v , z ) h l ( i ) v ) (cid:17) and a ∗ ( Y ( v , z ) Y ( h l ( − n − v , z ) v ) = X i ≥ − i − n !(cid:16) ( − n ( z − z ) n + i + | | z | > | z | a ∗ ( Y ( h l ( i ) v , z ) Y ( v , z ) v ) + z n + i + a ∗ ( Y ( v , z ) Y ( v , z ) h l ( i ) v ) , which implies that a ∗ ( Y ( h l ( − n − v , z ) Y ( v , z ) v ) and a ∗ ( Y ( v , z ) Y ( h l ( − n − v , z ) v )are the expansion of the same function in the di ff erent region | z | > | z | and | z | > | z | if(FL) holds for ( a ∗ , v , h l ( i ) v , v ) and ( a ∗ , v , v , h l ( i ) v ) for any i ≥
0. In this way, we caninductively show the assumption (FL), by Lemma 5.4 and Lemma 5.3.5.2.
Troidal compactification of string theory.
In this section, we construct a full ver-tex operator algebra associated with an even lattice. An even lattice is a free abelian group L of finite rank equipped with a bilinear map ( − , − ) : L × L → Z such that ( α, α ) ∈ Z forany α ∈ L . Set H L = L ⊗ Z R . Then, a bilinear form on H L is induced by the bilinear formon L , denoted by ( − , − ) again. An even lattice is called non-degenerate if ( H L , ( − , − )) isnon-degenerate. Let L be a non-degenerate even lattice. In [FLM], they construct an AHpair associated with an even lattice: Proposition 5.2. [FLM]
There exists a unital associative algebra C [ ˆ L ] = L α ∈ L C e α suchthat:(1) e is the unit;(2) e α e β , for any α, β ∈ L;(3) e α e β = ( − ( α,β ) e β e α for any α, β ∈ L. In fact, a unital associative algebra satisfies the conditions in the above proposition isunique up to isomorphism as an AH pair ([FLM, Mo1]). The algebra C [ ˆ L ] is called atwisted group algebra, which is clearly an even AH pair. For p ∈ P ( H L ), set F L , H L , p = F C [ ˆ L ] , H L , p . By Proposition 5.1, we have: Proposition 5.3.
For a non-degenerate even lattice L and p ∈ P ( H L ) , F L , H L , p is a fullvertex algebra. Furthermore, if p ∈ P > ( H L ) , then ( F L , H L , p , Y , , ω H L , ¯ ω H L ) is a self-dualQP-generated full vertex operator algebra.Proof. Let p ∈ P > ( H L ) and set F = F L , H L , p . Since the spectrum of F L , H L , p is { ( p α, p α ) / + n , − ( ¯ p α, ¯ p α ) / + m } α ∈ L , n , m ∈ Z ≥ and the bilinear form ( p − , p − ) on H l and − ( ¯ p − , ¯ p − ) on H r is positive definite, the spectrum is discrete, thus, bounded below. It is easy to show that( F , Y , , ω, ¯ ω ) is a simple full vertex operator algebra. To verify the assertion, it su ffi cesto show that F satisfies the assumption of Corollary 3.2. By the positivity of the bilinearforms, (PN1) and (PN2) is clear. Let 0 , v ∈ F α , n for α ∈ L . Then, ( p α, p α ) = p α =
0, which implies that Dv = v ∈ F α , n for α ∈ L . Suppose that L (1) v ,
0. Then, L (1) v ∈ F α , n implies that p α =
0, that is, α ∈ H r . Thus, F α , n is spannedby { h l ( − e α } h l ∈ H l and ( α, h l ) =
0. Then, for h l ∈ H l , L (1) h l ( − e α = h l (0) e α = ( h l , α ) e α =
0, a contradiction, which implies (PN3). Hence, the assertion holds. (cid:3)
Let ( n , m ) be the signature of H L . Then, the orthogonal group O ( H L ) (cid:27) O ( n , m ; R )acts on P > ( H L ) by g · p = g ◦ p ◦ g − for p ∈ P > ( H L ) and g ∈ O ( H L ). By the elemen-tary calculus in linear algebra, we can show that the action is transitive. The stabilizersubgroup of p is O ( n ; R ) × O ( m ; R ). Thus, P > ( H L ) is homeomorphic to the Grassma-nian O ( n , m ; R ) / O ( n ; R ) × O ( m ; R ). Hence, we construct a family of full vertex operatoralgebras which is parametrized by the Grassmanian O ( n , m ; R ) / O ( n ; R ) × O ( m ; R ).Let II , be the unique even unimodular lattice with the signature (1 , II , = Z z ⊕ Z w and ( z , z ) = ( w , w ) = z , w ) = − N ∈ Z > , set II N , N = II , N . Then, we have a family of full vertex operator algebras of the central charge ( N , N )parametrized by O ( N , N ; R ) / O ( N ; R ) × O ( N ; R ). Remark 5.1.
Let
Aut II N , N be the lattice automorphism group of II N , N , which is isomor-phic to O ( N , N ; Z ) , the Z -value points of the algebraic group O ( N , N ) . Then, we canprove that if σ ∈ Aut II N , N then F II N , N , H IIN , N , p is isomorphic to F II N , N , H IIN , N ,σ p σ − as a fullvertex operator algebra. Thus, the true parameter space isO ( N , N ; Z ) \ O ( N , N ; R ) / O ( N ; R ) × O ( N ; R ) . This statement is proved in [Mo2] in more general setting. We also remark that thisconformal field theory and parameter space appear in the toroidal compactification ofstring theory [Polc1] . Commutative algebra object in H-Vect.
Let H be a finite-dimensional real vectorspace with a non-degenerate symmetric bilinear form( − , − ) : H × H → R . Let H-Vect be the category whose object is an H -graded vector space over C and mor-phism is a grading preserving C -linear map. Let V = L α ∈ H V α and W = L β ∈ H W β be H -graded vector spaces. A tensor product of H -graded vector spaces are defined by thetensor product of vector spaces with the H -grading:( V ⊗ W ) α = M α ′ ∈ H V α ′ ⊗ W α − α ′ . Then, H-Vect is a strict monoidal category whose unit is C = C . Define a braiding B V , W : V ⊗ W → W ⊗ V by v α ⊗ w β exp(( α, β ) π i ) w β ⊗ v α for α, β ∈ H , v α ∈ V α and w β ∈ W β . Then, it is easy to show that B satisfies the hexagon identity, thus (H-Vect , B )is a braided tensor category. Let A be a unital commutative associative algebra object inH-Vect and a ∈ A α and b ∈ A β for α, β ∈ H . If ab ,
0, then, by the commutativity, ab = exp(( α, β ) π i ) ba and ba = exp(( α, β ) π i ) ab . Thus, ( α, β ) ∈ Z and ab = ( − ( α,β ) ba ,which implies that A is an AH pair. Thus, we have: Proposition 5.4.
A unital commutative associative algebra object in H-Vect is an AH pair,and vice versa.
6. A ppendix
Binary tree and expansions.
Let F be a self-dual full vertex operator algebra anddenote Y ( a , x ) by a ( x ) for a ∈ F .For n ∈ Z > , let P n be the set of parenthesized products of n elements 1 , , . . . , n , e.g.,(((31)6)(24))(57).We can naturally associate a tree and a parenthesized correlation function (see Section6.1) to each element in P n . For example, an element (((31)6)(24))(57) ∈ P defines thefollowing binary tree: 3 1 6 2 4 5 7and the parenthesized correlation function: S (((31)6)(24))(57) ( a , a , a , a , a , a , a ) : = (cid:16) , "(cid:18)(cid:16) a ( x ) a (cid:17) ( x ) a (cid:19) ( x ) a ( x ) a ( x ) a ( x ) a (cid:17) , where a , . . . , a ∈ F .We conjecture that S (((31)6)(24))(57) ( a , a , a , a , a , a , a ) is the expansion of a real ana-lytic function φ ( z , . . . , z ) on X in the domain { ( x , . . . , x ) ∈ C | {| x | > | x | > | x | > | x |} ∩ {| x | > | x |} ∩ {| x | > | x |}} after the change of variables:( x , x , x , x , x , x ) = ( z − z , z − z , z − z , z − z , z − z , z − z ) . This conjecture is proved in the case where n ≤ n = i jjFig.1 i j z i − z j jFig.2We remark that the tree associated with A ∈ P n is a full binarytree, i.e., every node has either 0 or 2 children. A node with nochildren is called a leaf. We will inductively assign a number toeach edge of the tree and a formal variable to a node of the tree.First, the number i is assigned to the unique edge connected toleaf labeled by i . On each node, the number of the above edgeis equal to the number of the right below edge, which determine the numbers of the alledges (Fig.1).Then, we assigned the formal variables z i − z j on the node whose left below edge islabeled by i and right below edge is labeled by j and change the variable x i into z i − z j (Fig.2).For example, for (((31)6)(24))(57) ∈ P , we have: z − z z − z z − z z − z z − z z − z z i − z j is achild of z k − z l , then | z k − z l | > | z i − z j | or in the other coordinate | x k | > | x i | .We also consider the special case that the right most state is the vacuum state. Let Q n be the set of parenthesized products of n + , , . . . , n , ⋆ with ⋆ at the rightmost, e.g., 2(3(1(4 ⋆ ))) ∈ Q . Then, the parenthesized correlation function associated with A ∈ Q n is defined by S A ( a , . . . , a n , ), that is, we insert the vacuum state at ⋆ . We denoteit by S A ( a , . . . , a n ) again. For example, for (((31)6)(24))(5 ⋆ ) ∈ Q and a , . . . , a ∈ F ,we have S (((31)6)(24))(5 ⋆ ) ( a , a , a , a , a , a ) : = , "(cid:18)(cid:16) a ( x ) a (cid:17) ( x ) a (cid:19) ( x ) a ( x ) a ( x ) a ( x ) ! , The rule for the change of variable is given by the substitution of z ⋆ = P n + . For (((31)6)(24))(5 ⋆ ) ∈ P , the rule for the change of variables is:( x , x , x , x , x , x ) = ( z − z , z − z , z − z , z − z ⋆ , z − z ⋆ , z − z ) . Then, the rule for the change of variables for (((31)6)(24))(5 ⋆ ) ∈ Q is given by( x , x , x , x , x , x ) = ( z − z , z − z , z − z , z , z , z − z ) . 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