Lie Conformal Algebra and Dual Pair Type Realizations of Some Moonshine Type VOAs, and Calculations of the Correlation Functions
aa r X i v : . [ m a t h . QA ] M a r Lie Conformal Algebra and Dual Pair TypeRealizations of Some Moonshine Type VOAs,and Calculations of the Correlation Functions
Hongbo Zhao
A vertex operator algebra (VOA) V is called a CFT type VOA if V is Z ≥ -graded with V = C
1. A moonshine type VOA is a CFT type VOA such that V = { } . It is well known that for a moonshine type VOA V , the space V ishas a commutative but not necessarily associative algebraic structure with theproduct given by a ◦ b := a (1) b, and we call V the Greiss algebra of V .Finite dimensional Jordan algebras are examples of algebras which are com-mutative but in general not assocaitive, so we are interested in moonshine typeVOAs whose Greiss algebras are finite dimensional simple Jordan algebras. Inparticular, we focus on Hermitian type Jordan algebras.The examples of moonshine type VOAs whose Greiss algebras are Hermitiantype Jordan algebras were first given by Lam in [Lam96] and [Lam99]. In[AM09], Ashihara and Miyamoto constructed VOAs V J ,r parametrized by r ∈ C whose Greiss algebras are type B Jordan algebras J and properties of V J ,r werestudied by Niibori and Sagaki in [NS10]. In [Zha18], we constructed the simpleVOAs ¯ V J ,r , r ∈ Z =0 for type B Jordan algebras, and we constructed V J ,r forthe remaining cases of Hermitian Jordan algebras in [Zha19].This paper is a continuation of the papers [Zha16], [Zha18], [Zha19]. Themain tools we use here are Lie conformal algebras (LCA) and certain free fields.We give LCA realizations of the VOAs V J ,r , and free field realizations of thesimple VOAs ¯ V J ,r using dual pair type constructions. Moreover, we give adifferenet approach to calculating correlation functions of the VOAs V J ,r , usingour realization of the simple VOAs ¯ V J ,r .Recall that a Lie conformal algebra (LCA) is a C [ ∂ ]-module V equippedwith a C -linear map called the λ -bracket [ · λ · ] : V ⊗ V → C [ λ ] ⊗ V such that forany a, b, c ∈ V the following identities hold:[( ∂a ) λ b ] = − λ [ a λ b ] , [ a λ b ] = − [ b − λ − ∂ a ] , [ a λ [ b µ c ]] = [ b µ [ a λ c ]] + [[ a λ b ] λ + µ c ] .
1t is noted that we can get an LCA from a vertex algebra by forgetting thenegative integer products and only consider the non-negative integer products.More precisely, let V be a vertex algebra, then V is also an LCA with ∂ = T and the λ -bracket b λ a = X k ≥ λ k k ! b ( k ) a. (1.1)For our purpose we need the concept of vertex operator super algebras(VOSAs) which generalizes VOAs slightly, and fields we need are also under-stood in the super sense. Following [Kac98], given a space V , a collection offields S ( z ) = { a ( z ) | a ( z ) ∈ End ( V )[[ z, z − ]] } over V is called a free field, if forany three fields a ( z ) , b ( z ) , c ( z ) ∈ S ( z ) we have[[ a ( z ) , b ( w )] , c ( x )] = 0 . Free fields generate many important VO(S)As. The most fundamental examplesare bc -system VOSAs and βγ -system VOAs. For our purpose, the free fields weneed are the generating fields of Heisenberg VOAs and Symplectic FermionsVOSAs. They are analogues of the free fields generating bc -system and βγ -system VOSAs.The main content of this paper is organized as follows. In Section 2, weintroduce some LCAs together with their central extensions, and in Section3 we construct VOAs V J ,r using the LCAs introduced in Section 2, and thefirst main result is Theorem 1. In all these constructions we use free fields,and the way we construct V J ,r is different from the ones given in [AM09] and[Zha18]. In Section 4 we constrcut the simple VOAs ¯ V J ,r where r ∈ Z =0 , andthey are all dual pair type constructions, which are summarized in Theorem2 and Theorem 3. As an application, we calculate the correlation functions ofgenerating fields of the VOAs V J ,r , giving Theorem 4, where free fields and thedual pair constructions play key roles. In this section we briefly review the construction of certain free fields whichin particular generate Heisenberg and Symplectic Fermion VO(S)As. To beconvenient we use the language of superspaces. Throughout this paper, theground field is C and all concepts involving vector spaces are understood in thesuper sense.Let h = h ¯0 ⊕ h ¯1 be a superspace with sdim ( h ) = ( p | q ) , such that h is equipped with a non-degenerate supersymmetric bilienar form( · , · ). We define the Lie superalgebraˆ h := h ⊗ C [ t, t − ] ⊕ C K a ( m ) , b ( n )] = mδ m + n, ( a, b ) K, [ a ( m ) , K ] = 0 , a ( m ) := a ⊗ t m . The even part of this Lie superalgebra is h ¯0 ⊗ C [ t, t − ] ⊕ C K and the odd part is h ¯1 ⊗ C [ t, t − ] . The sub Lie superalgebra ˆ h − := h ⊗ t − C [ t − ] ⊕ C K is supercommutative, and we check that the supersymmetric tensor space S (ˆ h − )is a VOSA. The generaing fields are Y ( a, z ) = X k ∈ Z a ( k ) z − k − , and the whole VOSA structure is obtained from the normal ordered products.It is clear that { a ( z ) | a ∈ h } forms a free field, and this will be used in Section 5.When h is purely even the corresponding VOA is exactly the HeisenbergVOA, while h is purely odd the corresponding VOSA is the symplectic FermionVOSA. As we mentioned in the introduction, there are another two importantexamples called bc -systems and βγ systems, but we do not need these two ex-amples in this paper and their relations will be discussed elsewhere.Recall that for any vertex algebra V , it is automatically an LCA by (1.1).We note that for the VOSAs S (ˆ h − ), the subspace \ S ( h ) := span { a ( − m ) b ( − n ) · , | a, b ∈ V } is even which is also closed under the k -th products for all k ≥
0, therefore \ S ( h )is an LCA. The one dimensional subspace C S ( h ) = \ S ( h ) / C ≃ span { a ( − m ) b ( − n ) · | a, b ∈ V } is also an LCA. Moreover, \ S ( h ) is a nontrivial central extension of S ( h ), andfor x, y ∈ S ( h ) the corresponding non-trivial 2-cocycle c ( x, y ) is given by c ( x, y ) := 1( | x | + | y | − x ( | x | + | y | − y (2.1)3here ˆ x and ˆ y denote arbitrary choices of preimages of x and y in \ S ( h ), andfor x = a ( − m ) b ( − n ) · | x | := m + n denotes the degree of x . In other words, let π : \ S ( h ) → C C
1, then c ( x, y ) λ | x | + | y |− = π ([ˆ x λ ˆ y ]) . For our purpose we also need to introduce another LCA describing the type A case. Let h be an even vector space and h ∗ be the corresponding dual space,then there is a unique way to assign a supersymmetric bilinear form ( · , · ) on h ⊕ h ∗ such that h , h ∗ are isotropic subspaces, h ⊕ h ∗ is even, and( a, b ∗ ) = h b ∗ , a i . We note that there is a C × -action on h ⊕ h ∗ : c · ( a, b ∗ ) = ( ca, c − b ∗ ) , (2.2)and we have two fixed point LCAs S ( h ⊕ h ∗ ) C × and \ S ( h ⊕ h ∗ ) C × . It followsthat S ( h ⊕ h ∗ ) C × = span { a ( − m ) b ∗ ( − n ) · | a ∈ h , b ∗ ∈ h ∗ } , \ S ( h ⊕ h ∗ ) C × := span { a ( − m ) b ∗ ( − n ) · , | a ∈ h , b ∗ ∈ h ∗ } . We remark that we can also impose h and h ⊕ h ∗ to be odd spaces, but there isonly a minus sign difference on the level and we omit this equivalent choice.We also remark that the LCAs S ( h ) and S ( h ⊕ h ∗ ) C × introduced here arenot new. It can be shown that if sdim ( h ) = ( N |
0) or (0 | N ), then S ( h ⊕ h ∗ ) C × ≃ gc N,Ix , (2.3)and S ( h ) ≃ ( oc N,Jx , h = (0 | N ) ,spc N,Ix , h = ( N | , (2.4)where gc N,Ix , oc N,Jx and spc
N,Ix are certain sub LCAs of the general linearconformal algebra gc N introduced in [BKL03]. Moreover, all these LCAs aresimple, and the 2-cocycle c ( x, y ) is the same as the 2-cocyle introduced in [Kac98]up to a constant scalar. 4or oc N,jx and spc
N,Ix we set L a,b ( m, n )1 = 12 a ( − m ) b ( − n ) · , and in particular L a,b := L a,b ( − , − . Similarly for gc N,Ix , L a,b ∗ ( m, n )1 = 12 a ( − m ) b ∗ ( − n ) · , L a,b ∗ := L a,b ∗ ( − , − . The following lemma will be used later
Lemma 1.
For N ≥ , the LCAs gc N,Ix , oc N,Jx and spc
N,Ix are generated bytheir degree two subspaces S ( h ) := span { L a,b | a, b ∈ h } S ( h ⊕ h ∗ ) C × := span { L a,b ∗ | a ∈ h , b ∗ ∈ h ∗ } , through the identifications (2.3), (2.4), as LCAs. Proof.
We only prove the case for spc
N,Ix with N = 2, h purely even, andgeneral cases are proved in a similar way. The computation is similar to theproof of Theorem 1.1, (3), of [Zha16]. We assume that h is spanned by the basis a, b with ( a, a ) = ( b, b ) = 1 , ( a, b ) = 0 . We note that L a,a , L b,b are mutually orthogonal Virasoro elements, and L a,b ( − m, − n )1 = L a,a (0) m − L b,b (0) n − ( m − n − L a,b , therefore L a,b ( − m, − n )1 can be generated. We also note that L rb,a (1) L ra,b ( − m, − n )1 = mL rb,b ( − m, − n )1 + nL ra,a ( − m, − n )1 , ( L rb,a ( − , − L ra,b ( − m, − n )1 = m ( m − L rb,b ( − m, − n )1¸ − n ( n + 1) L ra,a ( − m, − n )1hold by a direct computation, therefore L a,a ( − m, − n )1 and L b,b ( − m, − n )1 canalso be generated by solving the equation, and we conclude the proof.We have mentioned in [Zha18] and [Zha19] that the condition N ≥ V J ,r Using Lie Conformal Alge-bras
In this section we use Lie conformal algebras gc N,Ix , oc N,Jx , spc N,Ix , and the2-cocycle (2.1) to construct the VOAs V J ,r .5e first briefly review some facts about Hermitian Jordan algebras whichwill be used later. Let J X be a Hermitian Jordan algebra where X denotes thetype, X = A, B, C . Let d denote the rank of J X , then J X can be describedusing tensors. Let h be a superspace with a non-degenerate supersymmetricbilinear form ( · , · ) according to the type X : sdim ( h ) = ( d | , X = A, (0 | d ) , X = C, ( d | , X = B. (3.1)Then we identify the dual space h ∗ with h , and as vector spaces the HermitianJordan algebras are J X = ( h ⊗ h ∗ ≃ S ( h ⊕ h ∗ ) C × , X = A,S ( h ) , X = B, C.
We introduce elements L a,b ∗ := a ⊗ b ∗ ∈ J A (3.2)in type A case, and L a,b := ( a ⊗ b + b ⊗ a, X = B,a ⊗ b − b ⊗ a, X = C (3.3)in B, C cases. Then J X = ( span { L a,b ∗ | a ∈ h , b ∗ ∈ h ∗ } , X = A,span { L a,b | a, b ∈ h } , X = B, C. and in any cases the Jordan product is given by A ◦ B := 12 ( AB + BA )where A, B are viewed as elements in
End ( h ) ≃ h ⊗ h with the associativeproduct ( a ⊗ b )( u ⊗ v ) = ( b, u )( a ⊗ v ) . We briefly recall the construction of V J X ,r , where the case for X = B was given in [AM09] and the cases for X = B, C were given in [Zha19]. Thedetails can be found in [Zha19], Section 3 and Section 4. Let L X be the infinitedimensional Lie algebra which is chosen according to the type X : L X = gl ∞ , X = A, sp ∞ , X = B, so ∞ , X = C.
6e write L for short to agree with the notation in [Zha19] if there is no ambi-guity. By making a certain Lie algebra decomposition L = L − M L + and defining a one dimensional L + -module spanned by v r , we have an induced L -module M r : M r := U ( L ) ⊗ U ( L + ) C v r ∼ = U ( L − ) v r . It was shown in [AM09] and [Zha19] that M r is a vertex operator algebra,denoted by V J X ,r .Now we give the LCA realization of V J ,r , which is the main result of thissection. We first review a general construction of vertex algebras starting fromcertain LCAs with a non-trivial 2-cocyle. Recall that we can construct ‘formaldistribution Lie algebras’, and moreover, vertex algebras from LCAs ([Kac98],Section 2.7). Let C e an LCA. We consider the ‘affinization’ˆ C := C ⊗ C [ t, t − ] , and define the quotient space Lie ( C ) := ˆ C/span { ( ∂a ) t n + nat n − | a ∈ C } . Then
Lie ( C ) is a Lie algebra with the following Lie bracket which can be verifiedfrom the axioms of LCAs:[ at m , bt n ] := X k ≥ (cid:18) mk (cid:19) ( a ( k ) b ) t m + n − k . Moreover, if C is a free C [ ∂ ]-module generated by A , C = C [ ∂ ] ⊗ A, then Lie ( C ) ≃ A ⊗ C [ t, t − ]with a derivation ∂ = − ∂ t . For each a ∈ A we can associate a Lie ( C )-valuedformal power series a ( z ) := X k ( at k ) z − k − , and it is checked that any two power series a ( z ) and b ( w ) are mutually local.Let c ( · , · ) be a 2-cocyle of C and K be the central element. In our cases, itis always possitble to find a non-trivial one with ∂K = 0. Then Kt n = 0 for all n = − , and we have a centrally extended Lie algebra Lie ( C M C K ) = Lie ( C ) ⊕ C Kt − · , · ] ′ :[ x, y ] ′ = [ x, y ] + c ( x, y ) Kt − . Take a split
Lie ( C M C K ) := Lie ( C M C K ) + ⊕ Lie ( C M C K ) − where Lie ( C M C K ) + := span { at n | a ∈ A, n ≥ } ⊕ C Kt − ,Lie ( C M C K ) − := span { at n | a ∈ A, n < } . Define a one dimensional
Lie ( C L C K ) + -module spanned by 1: at n · , for all a ∈ AK, n ≥ , Kt − · r · . Then we construct a (level r ) vertex algebra as an induced Lie ( C L C K )-module V ert r ( C ) := U ( Lie ( C M C K )) ⊗ U ( Lie ( C L C K ) + ) C ≃ U ( Lie ( C M C K ) − ) · Lie ( C )-valued power series a ( z ), a ∈ C are generating mutu-ally local fields over V ert r ( C ). The whole vertex algebra structure on V ert r ( C )is determined by normal ordered products according the the reconstruction the-orem [Kac98]. It follows that C is a sub space of V ert r ( C ) through C → Ct − ⊆ V ert r ( C ) , which will be used in Section 5.For a CFT type VOA V , we can get an LCA V / C c ( · , · ) in the same way as (2.1). We note that in general, the vertexalgebra V ert r ( V / C V instead, and for particular examples of Heisenberg andsymplectic Fermion VO(S)As, we choose sub LCAs gc N,Ix , oc N,Jx and spc
N,Ix .We choose the LCA according to the type X . Let C X denote the followingLCAs according to the type X : C X = ( S ( h ) , X = B, C,S ( h ⊕ h ∗ ) C × , X = A. in which the type of h is determined by (3.1). Let c ( · , · ) be the 2-cocycle (2.1),then we have the vertex algebra V ert r ( C X ) . We have the following lemma 8 emma 2.
The Lie algebra
Lie ( C X ⊕ C K ) acts on M r with K = r This is verified by a direct comparison and calculation.Using Lemma 2 the following theorem is a direct consequence of our con-struction.
Theorem 1.
The
Lie ( C X ⊕ C K ) -action on M r extends to a VOA isomorphism V ert r ( C X ) → V J X ,r . which is induced by ( L a,b t − · L a,b , X = B, C,L a,b ∗ t − · L a,b ∗ , X = A. The following proposition follows easily from Lemma 1 and Theorem 1, andthe case for X = B was proved in [NS10] as Proposition 3.1. Proposition 1.
The VOAs V = V J X ,r are generated by their degree two sub-spaces V . ¯ V J ,r In [NS10] the authors proved that the VOA V J B ,r is simple if and only if r / ∈ Z ,and in [Zha18], we give the explicit construction of the simple quotients ¯ V J B ,r for all r ∈ Z =0 using dual pair constructions. In [Zha19] we show that V J A ,r and V J C ,r are also simple if and only if r / ∈ Z . The main result of this sectionis to give a uniform description of the simple quotients ¯ V J X ,r , X = A, B, C .The main idea is the following: Let h and h ′ be two superspaces of types( p | q ) and ( p ′ | q ′ ) , with non-degenerate supersymmetric bilinear forms ( · , · ) and( · , · ) ′ respectively. Then the tensor product space h ⊗ h ′ is a supersymmetric space of type ( pp ′ + qq ′ | pq ′ + p ′ q ) with the non-degeneratesupersymmetric bilinear form ( · , · ) ⊗ ( · , · ) ′ . On the level of VOSAs, we usethe VOSA associated with h ⊗ h ′ described in Section 2, and we consider theaction of the ‘Lie supergroup’ Osp ( h ′ ) on h ′ , where the Lie supergroup Osp ( h ′ ) isunderstood as a Harish-Chandra pair [Zha18]. To construct the simple quotientsof the VOA V J X ,r when r ∈ Z =0 , the type of h is related to the type X of theJordan algebra, and the type of h ′ is related to the level r .We first describe the cases X = B, C . Let d denote the rank of the Jordanalgebra J X , and r be the level, then we choose the type of h and h ′ as follows sdim ( h ) = ( ( d | , X = B, (0 | d ) , X = C, sdim ( h ′ ) = ( r | , r ∈ Z ≥ , (0 | − r ) , r < , r ∈ Z , (1 | − r + 1) , r < , r ∈ Z + 1 . It is clear that for our choices of h ′ , the following relation always holds r = p ′ − q ′ . Theorem 2.
For X = B, C and r ∈ Z =0 , ¯ V J X ,r ≃ S ( \ ( h ⊗ h ′ ) − ) Osp ( h ′ ) . We describe ¯ V J X ,r more explicitly using invariant theory for Lie supergroups Osp ( h ′ ) (See for example, [LZ16],[Ser01]). For any m, n ∈ Z we introduceelements L ra,b ( m, n ) := X i ( a ⊗ e i )( m )( b ⊗ e ∗ i )( n )which are elements in the enveloping algebra U (( \ h ⊗ h ′ ) − ). It follows that S ( \ ( h ⊗ h ′ ) − ) Osp ( h ′ ) =span { L ra ,b ( − m , − n ) · · · L ra k ,b k ( − m k , − n k ) · | a i , b i ∈ h , m i , n i ≥ } , and we also use L ra,b to denote the element L ra,b ( − , − · ∈ S ( \ ( h ⊗ h ′ ) − ) Osp ( h ′ ) if there is no confusion. Proposition 2.
For X = B, C , there is a VOA homomorphism V J X ,r → S ( \ ( h ⊗ h ′ ) − ) Osp ( h ′ ) . which is induced by φ : L a,b L ra,b . Proof.
It is easy to verify that for all k ≥ φ ( L a,b ( k ) L u,v ) = φ ( L a,b )( k ) φ ( L u,v )by setting K = r , therefore it extends to an LCA homomorphism φ : L a,b ( − m, − n )1 L ra,b ( − m, − n )1 , V J X ,r → S ( \ ( h ⊗ h ′ ) − ) Osp ( h ′ ) by Theorem 1 and Proposition 1.It is clear that V J ,r has a unique simple quotient, and to conclude the proofof Theorem 2 we need to prove the following.10 roposition 3. The fixed point VOA S ( \ ( h ⊗ h ′ ) − ) Osp ( h ′ ) . is simple. The proof for type B case has already been given in [Zha18], and the proofof the general case is similar. First, we note that the VO(S)A S ( \ ( h ⊗ h ′ ) − )is simple, and there is an invariant bilienar form over them [KR87]. The mainpoint is that in any cases, Osp ( h ′ ) acts semisimplely on S ( \ ( h ⊗ h ′ ) − ) which alsopreserves the invariant bilinear form. For simplicity we set M := S ( \ ( h ⊗ h ′ ) − ) , and there is a decomposition M = M Osp ( h ′ ) ⊕ M ′ where M ′ = ⊕ λ M λ is the direct sum of non-trivial irreducible Osp ( h ′ )-modules. Therefore, M Osp ( h ′ ) is orthogonal to M ′ . The non-degeneracy of the invariant bilinear form on M implies that it is also non-degenerate on M Osp ( h ′ ) , and therefore the fixed pointVOA is simple by [Li94].For type A case the result is similar with a slight difference. Theorem 3.
For r ∈ Z =0 , ¯ V J A ,r ≃ S ( \ ( h ⊕ h ∗ ) ⊗ h ′ ) − ) Osp ( h ′ ) × C × . Here the C × -action is the one induced by (2.2). This theorem is proved by the corresponding analogues of Proposition 2 andProposition 3. The proof is similar by considering the C × -action on h ⊕ h ∗ , andwe omit the details. ¯ V J ,r In this section we apply the simple VOAs ¯ V J X ,r , r ∈ Z =0 to compute thecorrelation functions of the generating fields in the VOAs V J X ,r .We introduce some notations and conventions. For a VOA V , v ∈ V , wewrite v ( z ) := Y ( a, z )11or short, and suppose the mode expasion is v ( z ) = X k v ( k ) z − k − , we set v ( z ) + := X k ≥ v ( k ) z − k − , v ( z ) − := X k< v ( k ) z − k − . In this section, all VO(S)As V are assumed to be of CFT type, and we use 1 ′ to denote the unique element in the restricted dual V ∗ such that h ′ , i = 1and h ′ , v i = 0for all v ∈ V i , i > v , · · · , v n ∈ V , we call h ′ , v ( z ) · · · v n ( z n )1 i (5.1)the correlation function (or genus zero n -point function) of the fields v ( z ) , · · · , v n ( z n ).It can be viewed either as an element in C (( z )) · · · (( z n )), or as a rationalfunction of ( z i − z j ) , i = j . As a formal power series can be obtained by thecorresponding multivariable function on the domain | z | > · · · | z n | . It is also well known that for any permutation τ ∈ S n (5.1) equals h ′ , v τ (1) ( z τ (1) ) · · · v τ ( n ) ( z τ ( n ) )1 i as multivariable complex functions after doing analytic continuations.The main result of this section is to compute (5.1) for v i = ( v i = L a i ,b i , X = B, C,v i = L a i ,b ∗ i , X = A. The type B case has been computed in [Zha16], but the method we use here isdifferent.We first recall the following recursion formula for CFT type VOAs, which isa variant of Lemma 2.2.1 in [Zhu96] Lemma 3.
Suppose v i ∈ V k , k ≥ , and n ≥ , then h ′ , v ( z ) · · · v n ( z n )1 i = X j =2 , ··· ,n,k ≥ ι z ,z j ( z − z j ) − k − h ′ , v ( z ) · · · ( v ( k ) v j )( z j ) · · · v n ( z n )1 i . (5.2)12 roof. We note that because V is of CFT type, h ′ , v ( z ) − · · · v n ( z n )1 i = 0by counting the degree, hence h ′ , v ( z ) · · · v n ( z n )1 i = h ′ , v ( z ) + · · · v n ( z n )1 i . We also recall that[ v s ( z ) + , v t ( w )] = X j ≥ ( v s ( j ) v t )( w ) ι z,w ( z − w ) − j − , and v ( z ) + . Therefore the recursion formula by obtained by moving v ( z ) + to the rightmostposition. Lemma 4.
As a complex function the correlation function h ′ , L a ,b ( z ) · · · L a n ,b n ( z n )1 i has the form C [ r ][( z i − z j ) − | i < j ] and the constant coefficients only depend on a i , b i . Moreover as a polynomial in r the degree of the correlation function is less than n . To prove the lemma we need to prove a slightly stronger version which allowsthat v i are elements in C X ⊆ V ert r ( C X ) ≃ V J X ,r , as C X is closed under the λ -brackets: Suppose v i = L a i ,b i ( − m i , − n i )1, then the conclusion of Lemma 4 stillholds. We prove this by doing induction on n . When n = 0 , , n = k . thenwe note that for all l ≥ i, j = 1 , · · · , nv i ( l ) v j = v + rv ′ for some v, v ′ ∈ C X which can be determined, therefore we conclude the proofby the induction hypothesis and formula (5.2).As another corollary of the recursion formula, we have Lemma 5.
Suppose There is a VOA homomorphism V → V ′ , v i ∈ V , and v ′ i are the homomorphic images of v i in V , then the correlation function h ′ , v ( z ) · · · v n ( z n )1 i equals the correlation function h ′ , v ′ ( z ) · · · v ′ n ( z n )1 i . V J X ,r → ¯ V J X ,r and when r ∈ Z =0 the simple VOAs ¯ V J X ,r are realized using free fields and dualpair type constructions, therefore by Lemma 6, h ′ , L a ,b ( z ) · · · L a n ,b n ( z n )1 i = h ′ , L ra ,b ( z ) · · · L ra n ,b n ( z n )1 i for r ∈ Z ≥ . Because a complex coefficient polynomial of degree less than n is uniquely determined by its values at n different points and we can view thecorrelation functions as complex functions, therefore by Lemma 5 we only needto calculate the correlation functions ¯ V J X ,r , for n different non-zero integer valueof r . In particular it is sufficient to compute h ′ , L ra ,b ( z ) · · · L ra n ,b n ( z n )1 i for all positive integers r ∈ Z > .We need the following Theorem for free fields. Proposition 4 (Wick’s Theorem, [Kac98], Theorem 3.3) . Let a ( z ) , · · · , a m ( z ) and b ( w ) , · · · , b n ( w ) be a collections of free fields. Then : a ( z ) · · · a m ( z ) :: b ( w ) · · · b n ( w ) := min ( m,n ) X s =0 X ( i ,j ) , ··· , ( i s ,j s ) ( − ǫ ( i ,j ; ··· ; i s ,j s ) [ a i ( z ) + , b j ( w ) − ] · · · [ a i s ( z ) + , b j s ( w ) − ]: a ( z ) · · · a m ( z ) b ( w ) · · · b n ( w ) : ( i ,j ; ··· ; i s ,j s ) . Here the subscript ( i , j ; · · · ; i s , j s ) in the last line means that the fields a i ( z ) , b j ( w ) , · · · a i s ( z ) , b j s ( w ) are removed, and ǫ ( · ) = ± is the sign obtained by the superrule: each permutation of the adjacent odd fields changes the sign. For two free field a ( z ) and b ( w ) we call[ a ( z ) + , b ( w ) − ]the (Wick’s) contraction of a ( z ) and b ( w ). In particular, for free fields a ( z ), b ( w ) in the Heisenberg and symplectic Fermion VOSAs, it is clear that[ a ( z ) + , b ( w )] = [ a ( z ) + , b ( w ) − ] = ι z,w ( a, b )( z − w ) . The following lemma is obtained by counting the degree of a CFT type VOA.
Lemma 6. h ′ , : u ( z ) · · · u m ( z m ) : 1 i = 0 . h ′ , L ra ,b ( z ) · · · L ra n ,b n ( z n )1 i . Because L ra,b ( z ) = 12 X i (( a ⊗ e i )( − b ⊗ e i ))( z )= 12 X i : ( a ⊗ e i )( z )( b ⊗ e i )( z ) :by definition, we can use Proposition 4 repeatedly.Before doing the computation we briefly recall some notations in [Zha16] andwe generalize them slightly. For a sequence of fields L a ,b ( z ) , · · · , L a n ,b n ( z )in the cases X = B, C , we have a corresponding sequence, denoted by T = ( a , b ) · · · ( a n , b n ) . A BC -type diagram over the sequence T = ( a , b ) · · · ( a n , b n ) is a graph, withthe vertex set V = { a , b , · · · , a n , b n } , and edge set E consisting of unorderedpairs { u, v } , u, v ∈ V satisfying: (1). { a i , b i } , { a i , a i } , { b i , b i } / ∈ E for all i = 1 , · · · , n . (2). Any two edges have no common point. (3). | E | = n .Denote the set of all BC -type diagrams over T by D ( T ). Similarly in the A -typecase, a sequence of fields L a ,b ∗ ( z ) , · · · , L a n ,b ∗ n ( z ) gives rise to a sequence T = ( a , b ∗ ) · · · ( a n , b ∗ n )and a diagram over T is a BC -type diagram over T satisfying one more condi-tion: (1)*. { a i , a j } , { b ∗ i , b ∗ j } / ∈ E for all 1 ≤ i < j ≤ n .We also denote the set of A -type diagram over T by D ( T ) if there is no ambi-guity.As an example, for T = ( a , b )( a , b )( a , b )( a , b ), the following( a b ) ( a b ) ( a b ) ( a b )is a BC -type diagram (over T ), and it corresponds to the following contractionwhich gives the term12 X i ,i ,i ,i [( a ⊗ e i )( z ) + , ( b ⊗ e i )( z ) − ][( b ⊗ e i )( z ) + , ( a ⊗ e i )( z ) − ]15( b ⊗ e i )( z ) + , ( b ⊗ e i )( z ) − ][( a ⊗ e i )( z ) + , ( a ⊗ e i )( z ) − ] h ′ , i = r ( a , b )( a , b )( b , b )( a , a )16( z − z ) ( z − z ) ( z − z ) ( z − z ) . (5.3)On the other hand, the following diagram( a b ∗ ) ( a b ∗ ) ( a b ∗ ) ( a b ∗ )is not a A -type diagram over T = ( a , b ∗ )( a , b ∗ )( a , b ∗ )( a , b ∗ ), but( a b ∗ ) ( a b ∗ ) ( a b ∗ ) ( a b ∗ )is A -type diagram over T = ( a , b ∗ )( a , b ∗ )( a , b ∗ )( a , b ∗ ).We note that for each diagram D , We have a corresponding n -vertex graph σ D by collapsing a i , b i to a single vertex labelled by i . More precisely, σ D hasvertex set { , · · · , n } and if ( a i , a j ), ( a i , b j ), or ( b i , b j ) is an edge of D , then( i, j ) is an edge of σ D . It is clear that σ D is a disjoint union of cycles.By Lemma 7, it is sufficient to consider the terms in the summation suchthat all free fields are contracted, and each of these terms corresponds exactlyto a diagram D ∈ D ( T ). More precisely, a ( z i ) is contracted with b ( z j ) if andonly if e := ( a, b ) is an edge of D , and in this case we define K ( e, Z ) := 1( z i − z j ) . We also define Γ( D ) := X { a,b }∈ E ( a, b ) , then by a direct computation similar to (5.3), a single contraction term corre-sponding to a diagram D equalsΓ( D ) r c ( σ D ) Y e ∈ E K ( e, Z ) , where c ( · ) denotes the number of cycles.Therefore by taking the summation over D we have h ′ , L ra ,b ( z ) · · · L ra n ,b n ( z n )1 i = X D ∈ D ( T ) Γ( D ) r c ( σ D ) Y e ∈ E K ( e, Z )16 σ ∈ DR ( T ) Γ( σ, T )Γ( σ ; Z ) r c ( σ ) , Here DR ( T ) denote the set of permutations in n elements such that each blockhas size ≥
2, and Γ( σ, T ), Γ( σ, Z ) means the following: Assume that σ =( C ) · · · ( C s ) = ( k · · · k t ) · · · ( k s · · · k st s ), thenΓ BC ( σ, T ) def = 2 − s − n s Y i =1 T r ( L a ki ,b ki · · · L a kiti ,b kiti ) , Γ( σ ; Z ) def = n Y i =1 z i − z σ ( i ) ) , where L a,b are understood as elements in End ( h ) in the sense of (3.3). Thisresult coincides with [Zha16], Theorem 1. We note that in the type C case,although we need to take into account of the sign, the formula is the same.The formula for the type A case is slightly different. For σ = ( C ) · · · ( C s ) =( k · · · k t ) · · · ( k s · · · k st s ) we defineΓ A ( σ, T ) def = 2 − n s Y i =1 T r ( L a ki ,b ∗ ki · · · L a kiti ,b ∗ kiti ) . where L a,b ∗ are understood as elements in End ( h ) in the sense of (3.2), and wejust need to note that a term corresponds to non-type A diagram D equals zero.Then we have h ′ , L a ,b ∗ ( z ) · · · L a n ,b ∗ n ( z n )1 i X σ ∈ DR ( T ) Γ A ( σ, T )Γ( σ ; Z ) r c ( σ ) . In Summary we have
Theorem 4.
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