(G, χ ϕ ) -equivariant ϕ -coordinated quasi modules for nonlocal vertex algebras
aa r X i v : . [ m a t h . QA ] A ug ( G, χ φ ) -EQUIVARIANT φ -COORDINATED QUASI MODULES FORNONLOCAL VERTEX ALGEBRAS NAIHUAN JING , FEI KONG , HAISHENG LI , AND SHAOBIN TAN Abstract.
In this paper, we study (
G, χ φ )-equivariant φ -coordinated quasi mod-ules for nonlocal vertex algebras. Among the main results, we establish severalconceptual results, including a generalized commutator formula and a generalconstruction of weak quantum vertex algebras and their ( G, χ φ )-equivariant φ -coordinated quasi modules. As an application, we also construct (equivariant) φ -coordinated quasi modules for lattice vertex algebras by using Lepowsky’s workon twisted vertex operators. Introduction
In vertex algebra theory, among the most important notions are module andtwisted module, which were introduced in the early development (see [FLM]). Later,a notion of quasi module, generalizing that of module, was introduced in [Li2] inorder to associate vertex algebras to a certain family of infinite-dimensional Liealgebras. Indeed, with this generalization vertex algebras can be associated to amuch wider variety of infinite-dimensional Lie algebras. To study quasi modulesmore effectively, vertex algebras with a certain enhanced structure, called Γ-vertexalgebras, were studied in [Li5] (cf. [Li2]), and meanwhile a notion of equivariantquasi module for a Γ-vertex algebra was introduced. The notion of quasi moduleactually has a close relation with that of twisted module; It was proved in [Li6]that for a general vertex operator algebra V in the sense of [FLM] and [FHL] witha finite order automorphism σ , the category of σ -twisted V -modules is canonicallyisomorphic to a subcategory of equivariant quasi V -modules.In [EK], Etingof and Kazhdan introduced a fundamental notion of quantum ver-tex operator algebra in the sense of formal deformation, where a key axiom is called S -locality. With this as one of the main motivations, a theory of (weak) quan-tum vertex algebras and their quasi modules was developed in [Li3, Li4], wherequantum vertex algebras are natural generalizations of vertex algebras and vertexsuper-algebras. It has been well understood that vertex algebras are analogues andgeneralizations of commutative and associative algebras. What were called nonlo-cal vertex algebras are noncommutative associative algebra counterparts of vertex Mathematics Subject Classification.
Primary 17B69, 17B68; Secondary 17B10, 81R10.
Key words and phrases. vertex algebras, nonlocal vertex algebras, phi -coordinated quasi mod-ules, generalized commutator formula, lattice vertex operator algebras. Partially supported by NSF of China (No.11531004) and Simons Foundation (No.198129). Partially supported by NSF of China (No.11701183). Partially supported by NSF of China (No.11531004). lgebras, which were studied independently in [BK] as field algebras and in [Li1] asaxiomatic G -vertex algebras. A weak quantum vertex algebra is simply a nonlocalvertex algebra that satisfies the S -locality.With the goal to associate quantum vertex algebras in the sense of [Li3] to quan-tum affine algebras, a theory of what were called φ -coordinated quasi modules forquantum vertex algebras was developed in [Li8, Li9]. In this theory, φ is what wascalled an associate of the 1-dimensional additive formal group, where usual quasimodules are simply φ -coordinated quasi modules with φ taken to be the formalgroup itself. By using this new theory with φ taken to be another particular asso-ciate, weak quantum vertex algebras were associated conceptually to quantum affinealgebras. It remains to make this conceptual association explicit.The current paper is one in a series to establish and study explicit associationsof quantum vertex algebras to various algebras such as quantum affine algebras. Inthis paper, we systematically study φ -coordinated quasi modules for general nonlocalvertex algebras and establish certain general results, laying the foundation for thisseries of studies. Among the main results, we obtain certain generalized commutatorrelations for vertex operators on nonlocal vertex algebras and their φ -coordinatedquasi modules, and we give a general construction of weak quantum vertex algebrasand φ -coordinated quasi modules. Furthermore, we construct φ -coordinated quasimodules for vertex algebras associated to nondegenerate even lattices.Now, we continue the introduction with more technical details. Let us start withthe notion of vertex algebra. For a vertex algebra V , among the main ingredients isa linear map Y ( · , x ) : V → (End V )[[ x, x − ]], called the vertex operator map, wherethe main axioms are weak commutativity (namely, locality): For any u, v ∈ V , thereexists a nonnegative integer k such that( x − x ) k Y ( u, x ) Y ( v, x ) = ( x − x ) k Y ( v, x ) Y ( u, x )(1.1)and weak associativity: For any u, v ∈ V, w ∈ W , there exists a nonnegative integer l such that( x + x ) l Y ( u, x + x ) Y ( v, x ) w = ( x + x ) l Y ( Y ( u, x ) v, x ) w. (1.2)Let V be a vertex algebra. For a V -module W with Y W ( · , x ) denoting the vertexoperator map, literally the same locality and the same weak associativity hold. Thatis, for any u, v ∈ V , there exists a nonnegative integer k such that( x − x ) k Y W ( u, x ) Y W ( v, x ) = ( x − x ) k Y W ( v, x ) Y W ( u, x )(1.3)and for any u, v ∈ V, w ∈ W , there exists a nonnegative integer l such that( x + x ) l Y W ( u, x + x ) Y W ( v, x ) w = ( x + x ) l Y W ( Y ( u, x ) v, x ) w. (1.4)Note that in the definition of a module weak associativity alone is sufficient.As for a quasi module ( W, Y W ) for a vertex algebra V , the key difference is thatvertex operators Y W ( v, x ) on W for v ∈ V , instead of being local, are quasi local inthe sense that for u, v ∈ V , there exists a nonzero polynomial p ( x , x ) such that p ( x , x ) Y W ( u, x ) Y W ( v, x ) = p ( x , x ) Y W ( v, x ) Y W ( u, x ) . (1.5) orrespondingly, quasi modules satisfy quasi weak associativity : For any u, v ∈ V, w ∈ W , there exists a nonzero polynomial q ( x , x ) such that q ( x + x , x ) Y W ( u, x + x ) Y W ( v, x ) w = q ( x + x , x ) Y W ( Y ( u, x ) v, x ) w. (1.6)Notice that in the definitions of the notions of module and quasi module (andtwisted module as well) for a vertex algebra, a common phenomenon is the formalsubstitution x = x + x in the weak associativity axiom. Polynomial F ( x, y ) = x + y is known as the 1-dimensional additive formal group (law) and the role of the formalgroup played in vertex algebra theory has long been recognized (cf. [Bo]). Whatis new is a role played by what was called associate of F ( x, y ). By definition (see[Li8]), an associate is a formal series φ ( x, z ) ∈ C (( x ))[[ z ]], satisfying the condition φ ( x,
0) = x, φ ( φ ( x, y ) , z ) = φ ( x, y + z ) . We see that an associate to F ( x, y ) is the same as a G -set to a group G . It wasproved therein that for any p ( x ) ∈ C (( x )), the formal series φ ( x, z ) defined by φ ( x, z ) = e zp ( x ) ddx x is an associate of F ( x, y ) and every associate is of this form.In particular, taking p ( x ) = 1 we get φ ( x, z ) = x + z = F ( x, z ), whereas taking p ( x ) = x , we get φ ( x, z ) = xe z . The essence of [Li8] is that to each associate φ ( x, z ),a theory of φ -coordinated quasi modules for a general nonlocal vertex algebra V isattached. The main defining property for φ -coordinated quasi modules is the weak φ -associativity: For any u, v ∈ V , there exists a nonzero polynomial p ( x , x ) suchthat p ( x , x ) Y W ( u, x ) Y W ( v, x ) ∈ Hom(
W, W (( x , x )))(1.7)and( p ( x , x ) Y W ( u, x ) Y W ( v, x )) | x = φ ( x ,x ) = p ( φ ( x , x ) , x ) Y W ( Y ( u, x ) v, x ) . Let G be a group with a linear character χ . By a ( G, χ ) -module nonlocal vertexalgebra we mean a nonlocal vertex algebra V together with a representation R of G on V such that R ( g ) = and R ( g ) Y ( u, x ) R ( g ) − = Y ( R ( g ) u, χ ( g ) x )on V for g ∈ G, u ∈ V . This notion slightly generalizes that of Γ-vertex algebra,introduced in [Li5] (cf. [Li2]). Note that if χ is the trivial character, G acts on V as an automorphism group. Assume that V is a ( G, χ )-module nonlocal vertexalgebra. Let χ φ be another linear character of G such that φ ( x, χ ( g ) z ) = χ φ ( g ) φ ( χ φ ( g ) − x, z ) for g ∈ G. We define a (
G, χ φ )-equivariant φ -coordinated quasi V -module to be a φ -coordinatedquasi V -module ( W, Y W ) satisfying the conditions that Y W ( R ( g ) u, x ) = Y W ( u, χ φ ( g ) − x ) for g ∈ G, u ∈ V and that for u, v ∈ V , there exists a nonzero polynomial f ( x ) whose zeroes arecontained in χ φ ( G ) such that f ( x /x ) Y W ( u, x ) Y W ( v, x ) ∈ Hom(
W, W (( x , x ))) . (1.8) n this paper, we mostly restrict ourselves to the case with φ ( x, z ) = φ r ( x, z ) := e zx r +1 ddx x where r ∈ Z . This very notion with φ ( x, z ) = φ − ( x, z ) = x + z isreduced to the notion of G -equivariant quasi module studied in [Li5], and with φ ( x, z ) = φ ( x, z ) = xe z is reduced to the notion of G -equivariant φ -coordinatedquasi module studied in [Li9]. As one of the main results of this paper, we obtaina generalized commutator formula for nonlocal vertex algebras generated by vertexoperators on vector spaces.Note that φ r -coordinated modules for vertex algebras have been previously studiedin [BLP], where a Jacobi type identity and a commutator formula for φ r -coordinatedmodules were obtained, and φ r -coordinated modules for vertex algebras associatedto Novikov algebras by Primc were also studied.In vertex algebra theory, vertex algebras V L associated to nondegenerate evenlattices L (see [Bo], [FLM]) play an important role. These vertex algebras have beenwell studied in literature and their irreducible modules as well as twisted moduleshave been constructed and classified (see [L], [FLM], [D1], [D2], [DL1], [DL2]; [LL]).In this paper, using the work of Lepowsky on twisted vertex operators [L], weconstruct ( G, χ φ )-equivariant φ -coordinated quasi V L -modules with G = h b µ i , where b µ is an automorphism of V L , lifted from a certain isometry µ of L . Note that anassociative algebra A ( L ) was introduced and used in [LL] as an essential tool inthe construction of V L -modules. This algebra A ( L ) also plays a crucial role in thecurrent paper for our construction of φ -coordinated quasi V L -modules.This paper is organized as follows. In Section 2, we revisit formal calculus in-volving associates of the additive formal group and delta functions. In Section 3,we establish several general results on equivariant φ -coordinated quasi modules fornonlocal vertex algebras. In Section 4, we give a vertex operator construction of( Z N , χ φ )-equivariant φ -coordinated quasi V L -modules.2. φ -coordinated quasi modules for nonlocal vertex algebras In this section, we revisit φ -coordinated quasi modules for nonlocal vertex alge-bras. As the main results, we establish several technical results, including a gen-eralized commutator formula (Theorem 2.19) and a general construction of weakquantum vertex algebras and φ -coordinated quasi modules (Theorem 2.23).2.1. Associates of formal groups.
In this subsection, we first briefly recall from[Li8] the notion of associate for the additive formal group (law) and some basicresults, and then we present some technical results on delta functions (Lemma 2.5).A one-dimensional formal group (law) over C (cf. [H]) is a formal power series F ( x, y ) ∈ C [[ x, y ]] such that F (0 , y ) = y, F ( x,
0) = x, F ( F ( x, y ) , z ) = F ( x, F ( y, z )) . A typical example of one-dimensional formal group is the additive formal group F a ( x, y ) = x + y , simply denoted by F a . et F ( x, y ) be a one-dimensional formal group over C . An associate of F ( x, y ) isa formal series φ ( x, z ) ∈ C (( x ))[[ z ]], satisfying the condition that φ ( x,
0) = x, φ ( φ ( x, y ) , z ) = φ ( x, F ( y, z )) . The following classification result was obtained in [Li8]:
Proposition 2.1.
Let p ( x ) ∈ C (( x )) . Set φ ( x, z ) = e zp ( x ) ddx x = X n ≥ z n n ! (cid:18) p ( x ) ddx (cid:19) n x ∈ C (( x ))[[ z ]] . Then φ ( x, z ) is an associate of F a ( x, y ) . On the other hand, every associate of F a ( x, y ) is of this form with p ( x ) uniquely determined. Remark 2.2.
Notice that taking p ( x ) = 1 in Proposition 2.1, we get φ ( x, z ) = x + z = F a ( x, z ) (the formal group itself), and taking p ( x ) = x we get φ ( x, z ) = xe z .For convenience, for n ∈ N , set ∂ nz = (cid:18) ∂∂z (cid:19) n and ∂ ( n ) z = 1 n ! (cid:18) ∂∂z (cid:19) n . (2.1)From now on, we assume that φ ( x, z ) is an associate of F a with φ ( x, z ) = x , thatis, φ ( x, z ) = e zp ( x ) ddx x with p ( x ) = 0. Remark 2.3.
We here recall some basic facts and conventions from [Li8, Remark2.6, Lemma 2.7]. First of all, for every f ( x , x ) ∈ C (( x , x )), f ( φ ( x, z ) , x ) exists in C (( x ))[[ z ]]. Furthermore, the correspondence f ( x , x ) f ( φ ( x, z ) , x ) gives a ringembedding π φ : C (( x , x )) → C (( x ))[[ z ]] ⊂ C (( x ))(( z )) . Denote by C ∗ (( x , x )) the fraction field of C (( x , x )). Then π φ naturally extends toa field embedding π ∗ φ : C ∗ (( x , x )) → C (( x ))(( z )) . As a convention, for any f ( x , x ) ∈ C ∗ (( x , x )), we shall view f ( φ ( x, z ) , x ) as anelement of C (( x ))(( z )) through this field embedding π ∗ φ . Remark 2.4.
Note that for any f ( x , x ) ∈ C (( x , x )), f ( φ ( x, y ) , φ ( x, z )) equals X m,n ≥ y m z n ( p ( x ) ∂ x ) m m ! ( p ( x ) ∂ x ) n n ! f ( x , x ) ! | x = x = x , which exists in C (( x ))[[ y, z ]]. Suppose f ( φ ( x, z ) , φ ( x, y )) = 0. Then f ( φ ( x, z ) , x ) = f ( φ ( x, z ) , φ ( x, f ( x , x ) = 0. Therefore, thecorrespondence f ( x , x ) f ( φ ( x, y ) , φ ( x, z ))is a ring embedding of C (( x , x )) into C (( x ))[[ y, z ]]. Just as with other iota-maps,we have field embeddings ι x,y,z : C ∗ (( x , x )) → C (( x ))(( y ))(( z )) , (2.2) ι x,z,y : C ∗ (( x , x )) → C (( x ))(( z ))(( y )) . (2.3) s a convention, for f ( x , x ) ∈ C ∗ (( x , x )), we shall view f ( φ ( x, y ) , φ ( x, z )) as anelement of the fraction field of C (( x ))[[ y, z ]].The following are some results about (formal) delta-functions: Lemma 2.5.
Let c ∈ C × . Set ∆ c ( x , x , z ) = e zp ( x ) ∂ x p ( x ) x − cx ∈ C ∗ (( x , x ))[[ z ]] . Then ( ι x ,x ,z − ι x ,x ,z ) ∆ c ( x , x , z ) = e zp ( x ) ∂ x p ( x ) x − δ (cid:18) cx x (cid:19) , (2.4) ∆ c ( x , φ ( x , w ) , z ) = ∆ c ( x , x , w + z ) , (2.5) ∆ c ( c ′ x , x , z ) ∈ C (( x ))[[ z ]] for c ′ = c, (2.6) and ( ι x,x ,x ,z − ι x,x ,x ,z ) ∆ c ( φ ( x, x ) , φ ( x, x ) , z ) = ( e z∂ x x − δ (cid:16) x x (cid:17) if c = 1 , c = 1 . (2.7) Proof.
Relation (2.4) follows immediately from the fact x − δ (cid:18) cx x (cid:19) = ι x ,x (cid:18) x − cx (cid:19) − ι x ,x (cid:18) x − cx (cid:19) . Note that ∆ c ( x , x , z ) = p ( x ) x − cφ ( x , z ) . Using this we get (2.5) and (2.6). We also have∆ c ( φ ( x, x ) , φ ( x, x ) , z ) = p ( φ ( x, x )) φ ( x, x ) − cφ ( φ ( x, x ) , z ) = p ( φ ( x, x )) φ ( x, x ) − cφ ( x, x + z ) . If c = 1, as φ ( x, − cφ ( x,
0) = (1 − c ) x = 0 , we see that φ ( x, x ) − cφ ( x, x ) is invert-ible in C (( x ))[[ x , x ]]. Consequently, ∆ c ( φ ( x, x ) , φ ( x, x ) , z ) ∈ C (( x ))[[ x , x , z ]]and hence ( ι x,x ,x ,z − ι x,x ,x ,z )∆ c ( φ ( x, x ) , φ ( x, x ) , z ) = 0 . Now, we consider the case c = 1. As φ ( x,
0) = x , we have φ ( x, x ) − φ ( x, x ) = ( x − x ) F ( x, x , x )for some F ( x, x , x ) ∈ C (( x ))[[ x , x ]]. We see that F ( x, ,
0) = p ( x ), so that F ( x, x , x ) is an invertible element of C (( x ))[[ x , x ]]. In view of this, we have ι x,x ,x (cid:18) p ( φ ( x, x )) F ( x, x , x ) (cid:19) = ι x,x ,x (cid:18) p ( φ ( x, x )) F ( x, x , x ) (cid:19) . hen we get ( ι x,x ,x − ι x,x ,x ) p ( φ ( x, x )) φ ( x, x ) − φ ( x, x )= ( ι x,x ,x − ι x,x ,x ) 1 x − x · p ( φ ( x, x )) F ( x, x , x )= (cid:18) ι x ,x x − x − ι x ,x x − x (cid:19) ι x,x ,x (cid:18) p ( φ ( x, x )) F ( x, x , x ) (cid:19) = x − δ (cid:18) x x (cid:19) ι x,x ,x p ( φ ( x, x )) φ ( x,x ) − φ ( x,x ) x − x = x − δ (cid:18) x x (cid:19) ι x,x (cid:18) p ( φ ( x, x )) ∂ x φ ( x, x ) (cid:19) = x − δ (cid:18) x x (cid:19) , noticing that ∂ x φ ( x, x ) = e x p ( x ) ∂ x ( p ( x ) ∂ x ) x = e x p ( x ) ∂ x p ( x ) = p ( φ ( x, x )) . Using this, we get ( ι x,x ,x ,z − ι x,x ,x ,z ) ∆( φ ( x, x ) , φ ( x, x ) , z )= ( ι x,x ,x ,z − ι x,x ,x ,z ) p ( φ ( x, x )) φ ( x, x ) − φ ( x, x + z )= e z∂ x ( ι x,x ,x − ι x,x ,x ) p ( φ ( x, x )) φ ( x, x ) − φ ( x, x )= e z∂ x x − δ (cid:18) x x (cid:19) , as desired. (cid:3) Note that∆( x , x , z ) = e zp ( x ) ∂ x p ( x ) x − x = X k ≥ k ! z k ( p ( x ) ∂ x ) k p ( x ) x − x . As an immediate consequence we have:
Corollary 2.6.
For k ∈ N , set F k ( x , x ) = ( p ( x ) ∂ x ) k p ( x ) x − x ∈ C ∗ (( x , x )) . Then ( ι x ,x − ι x ,x ) F k ( x , x ) = ( p ( x ) ∂ x ) k p ( x ) x − δ (cid:18) x x (cid:19) , ( ι x,x ,x − ι x,x ,x ) F k ( φ ( x, x ) , φ ( x, x )) = ∂ kx x − δ (cid:18) x x (cid:19) . emma 2.7. Let
Γ = { c , . . . , c k } be a finite nonempty subset of C × . Set ∆ Γ ( x , x , z , . . . , z k ) = k Y i =1 ∆ c i ( x , x , z i ) . (2.8) Then ( ι x ,x ,z ,...,z k − ι x ,x ,z ,...,z k ) ∆ Γ ( x , x , z , . . . , z k )= k X i =1 ι x ,z ,...,z k Y j = i ∆ c j ( c i φ ( x , z i ) , x , z j ) p ( x ) x − δ (cid:18) c i φ ( x , z i ) x (cid:19) . (2.9) Furthermore, for ≤ i ≤ k we have ι x ,z ,...,z k Y j = i ∆ c j ( c i φ ( x , z i ) , x , z j ) ∈ C (( x ))[[ z , . . . , z k ]] . (2.10) Proof.
Note that by Lemma 2.5 (see (2.6)), for distinct c, c ′ ∈ C × , ∆ c ( c ′ φ ( x , w ) , x , z )exists in C (( x ))[[ z, w ]], so that ι x ,x ,z,w ∆ c ( x , x , z ) p ( x ) x − δ (cid:18) c ′ φ ( x , w ) x (cid:19) = ι x ,z,w ∆ c ( c ′ φ ( x , w ) , x , z ) e wp ( x ) ∂ x p ( x ) x − δ (cid:18) c ′ x x (cid:19) . (2.11)From (2.4), we have( ι x ,x ,z i − ι x ,x ,z i ) ∆ c i ( x , x , z i ) = p ( x ) x − δ (cid:18) c i φ ( x , z i ) x (cid:19) for 1 ≤ i ≤ k . Therefore,( ι x ,x ,z ,...,z k − ι x ,x ,z ,...,z k ) ∆ Γ ( x , x , z , . . . , z k )= k X i =1 ι x ,x ,z ,...,z k Y j = i ∆ c j ( x , x , z j ) ! p ( x ) x − δ (cid:18) c i φ ( x , z i ) x (cid:19) (2.12) = k X i =1 ι x ,z ,...,z k Y j = i ∆ c j ( c i φ ( x , z i ) , x , z j ) p ( x ) x − δ (cid:18) c i φ ( x , z i ) x (cid:19) , where the existence of (2.12) follows from (2.11). The furthermore statement followsimmediately from (2.6). (cid:3) φ -coordinated quasi modules for nonlocal vertex algebras. Throughoutthis subsection, we assume that φ ( x, z ) = e zp ( x ) ddx x is an associate of the formaladditive group F a such that p ( x ) = 0, or equivalently φ ( x, z ) = x . Note that underthis assumption, we have f ( φ ( x, z ) , x ) = 0 for any nonzero f ( x , x ) ∈ C [[ x , x ]] . We start with the notion of nonlocal vertex algebra (see Remark 2.11). efinition 2.8. A nonlocal vertex algebra is a vector space V equipped with a linearmap Y ( · , x ) : V → (End V )[[ x, x − ]]; v Y ( v, x )and a distinguished vector ∈ V , satisfying the conditions that for u, v ∈ V , Y ( u, x ) v ∈ V (( x )) , (2.13) Y ( , x ) v = v, Y ( v, x ) ∈ V [[ x ]] and lim x → Y ( v, x ) = v (2.14)and that for u, v ∈ V , there exists a nonnegative integer k such that( x − x ) k Y ( u, x ) Y ( v, x ) ∈ Hom(
V, V (( x , x )))(2.15)and (cid:0) ( x − x ) k Y ( u, x ) Y ( v, x ) (cid:1) | x = x + x = x k Y ( Y ( u, x ) v, x ) . (2.16) Remark 2.9.
Let A ( x , x ) ∈ Hom(
W, W (( x , x ))) with W a vector space. Then A ( x , x ) | x = x + x exists in (Hom( W, W (( x )))[[ x ]]and A ( x , x ) | x = x + x exists in (Hom( W, W (( x )))[[ x , x − ]] . We have A ( x + x , x ) = Res x x − δ (cid:18) x + x x (cid:19) A ( x , x ) . (2.17)In view of this, under the condition (2.15), we have (cid:0) ( x − x ) k Y ( u, x ) Y ( v, x ) (cid:1) | x = x + x = Res x x − δ (cid:18) x + x x (cid:19) (cid:0) ( x − x ) k Y ( u, x ) Y ( v, x ) (cid:1) . (2.18) Lemma 2.10.
Let V be a nonlocal vertex algebra. Then the following weak as-sociativity holds: For any u, v, w ∈ V , there exists a nonnegative integer l suchthat ( x + x ) l Y ( u, x + x ) Y ( v, x ) w = ( x + x ) l Y ( Y ( u, x ) v, x ) w. (2.19) Proof.
Let u, v, w ∈ V . From definition, there exists k ∈ N such that (2.15) holds.Then ( x − x ) k Y ( u, x ) Y ( v, x ) w ∈ V (( x , x )) . Hence, there exists l ∈ N such that x l ( x − x ) k Y ( u, x ) Y ( v, x ) w ∈ V [[ x , x ]][ x − ] , involving only nonnegative integer powers of x . Then (cid:0) x l ( x − x ) k Y ( u, x ) Y ( v, x ) w (cid:1) | x = x + x = (cid:0) x l ( x − x ) k Y ( u, x ) Y ( v, x ) w (cid:1) | x = x + x . sing this and (2.16) we get( x + x ) l x k Y ( Y ( u, x ) v, x ) w = (cid:0) x l ( x − x ) k Y ( u, x ) Y ( v, x ) w (cid:1) | x = x + x = (cid:0) x l ( x − x ) k Y ( u, x ) Y ( v, x ) w (cid:1) | x = x + x =( x + x ) l x k Y ( u, x + x ) Y ( v, x ) w, which immediately yields (2.19). (cid:3) Remark 2.11.
Note that a notion of nonlocal vertex algebra was defined in [Li3]in terms of weak associativity (2.19). In view of Lemma 2.10, the notion of nonlocalvertex algebra in the sense of Definition 2.8 is theoretically stronger than the othersame named notion.Let V be a nonlocal vertex algebra. Define a linear operator D by D ( v ) = v − for v ∈ V . Then we have (see [Li3]):[ D , Y ( v, x )] = Y ( D ( v ) , x ) = ddx Y ( v, x ) for v ∈ V. (2.20)What were called weak quantum vertex algebras in [Li3] form a special classof nonlocal vertex algebras, where a weak quantum vertex algebra is defined byreplacing the compatibility and weak associativity conditions in Definition 2.8 withthe condition that for any u, v ∈ V , there exist u ( i ) , v ( i ) ∈ V, f i ( x ) ∈ C (( x )) ( i = 1 , . . . , r )such that x − δ (cid:18) x − x x (cid:19) Y ( u, x ) Y ( v, x ) − x − δ (cid:18) x − x − x (cid:19) r X i =1 f i ( − x + x ) Y ( v ( i ) , x ) Y ( u ( i ) , x )= x − δ (cid:18) x + x x (cid:19) Y ( Y ( u, x ) v, x ) . (2.21) Definition 2.12.
Let V be a nonlocal vertex algebra and let φ be an associate of F a . A φ -coordinated quasi V -module is a vector space W equipped with a linearmap Y W ( · , x ) : V → (End W )[[ x, x − ]]; v Y W ( v, x ) , satisfying the conditions that Y W ( u, x ) w ∈ W (( x )) for u ∈ V, w ∈ W,Y W ( , x ) = 1 W (the identity operator on W ) , and that for any u, v ∈ V , there exists a nonzero f ( x , x ) ∈ C [[ x , x ]] such that f ( x , x ) Y W ( u, x ) Y W ( v, x ) ∈ Hom(
W, W (( x , x ))) , (2.22) ( f ( x , x ) Y W ( u, x ) Y W ( v, x )) | x = φ ( x ,z ) = f ( φ ( x , z ) , x ) Y W ( Y ( u, z ) v, x ) . (2.23) et W be a vector space, which is fixed for the rest of this section. Set E ( W ) = Hom( W, W (( x ))) . More generally, for any positive integer r , set E ( r ) ( W ) = Hom( W, W (( x , x , . . . , x r ))) . (2.24)An ordered sequence ( a ( x ) , . . . , a r ( x )) in E ( W ) is said to be quasi-compatible (see[Li3, Li1]) if there exists a nonzero series f ( x, y ) ∈ C [[ x, y ]] such that Y ≤ i W, W (( x , x ))) (= E (2) ( W )) . (2.27)A quasi-compatible subspace U of E ( W ) is said to be Y φ E -closed if a ( x ) φn b ( x ) ∈ U (2.28)for all a ( x ) , b ( x ) ∈ U, n ∈ Z .The following result was obtained in [Li8]: Theorem 2.13. Let W be a vector space and let V be a Y φ E -closed quasi-compatiblesubspace of E ( W ) with W ∈ V . Then ( V, Y φ E , W ) carries the structure of a nonlocalvertex algebra and W is a φ -coordinated quasi V -module with Y W ( α ( x ) , z ) = α ( z ) for α ( x ) ∈ V . On the other hand, for every quasi-compatible subset U of E ( W ) , thereexists a unique minimal Y φ E -closed quasi-compatible subspace h U i φ that contains W and U . Furthermore, ( h U i φ , Y φ E , W ) carries the structure of a nonlocal vertex algebraand W is a φ -coordinated quasi h U i φ -module. Definition 2.14. In view of Theorem 2.13, we call a Y φ E -closed quasi-compatiblesubspace of E ( W ) containing 1 W a nonlocal vertex subalgebra of E ( W ) . The following is an immediate consequence: emma 2.15. Let ( α ( x ) , β ( x )) be a quasi-compatible pair in E ( W ) . Then for any f ( x ) ∈ C (( x )) , ( f ( x ) α ( x ) , β ( x )) is quasi-compatible and we have Y φ E ( f ( x ) α ( x ) , z ) β ( x ) = f ( φ ( x, z )) Y φ E ( α ( x ) , z ) β ( x ) . We also have the following result (cf. [Li8, Proposition 4.5]): Lemma 2.16. Let V be a nonlocal vertex subalgebra of E ( W ) . Suppose that α i ( x ) , β i ( x ) ∈ V, g i ( x , x ) ∈ C ∗ (( x , x )) ( i = 1 , , . . . , r ) such that r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) ∈ Hom( W, W (( x , x ))) . Then r X i =1 ι x,z ( g i ( φ ( x, z ) , x )) Y φ E ( α i ( x ) , z ) β i ( x )= r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = φ ( x,z ) . (2.29) Proof. It is similar to the proof in [Li8]. As V is a quasi-compatible subspace of E ( W ), there exists a nonzero series g ( x , x ) ∈ C [[ x , x ]] such that g ( x , x ) α i ( x ) β i ( x ) ∈ Hom( W, W (( x , x ))) for i = 1 , , . . . , r. From the definition of Y φ E we have g ( φ ( x, z ) , x ) Y φ E ( α i ( x ) , z ) β i ( x ) = ( g ( x , x ) α i ( x ) β i ( x )) | x = φ ( x,z ) for i = 1 , , . . . , r . Then we get g ( φ ( x, z ) , x ) r X i =1 ι x,z ( g i ( φ ( x, z ) , x )) Y φ E ( α i ( x ) , z ) β i ( x )= r X i =1 ι x,z ( g i ( φ ( x, z ) , x )) ( g ( x , x ) α i ( x ) β i ( x )) | x = φ ( x,z ) = g ( x , x ) r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = φ ( x,z ) = g ( φ ( x, z ) , x ) r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = φ ( x,z ) . Noticing that both P ri =1 ι x,z ( g i ( φ ( x, z ) , x )) Y φ E ( α i ( x ) , z ) β i ( x ) and r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = φ ( x,z ) lie in ( E ( W )) (( z )) , hich is a vector space over C (( x ))(( z )), and that g ( φ ( x, z ) , x ) is a nonzero elementof C (( x ))(( z )), we immediately get (2.29). (cid:3) On the other hand, we have: Lemma 2.17. Let V be a nonlocal vertex subalgebra of E ( W ) . Suppose that α i ( x ) , β i ( x ) ∈ V, g i ( x , x ) ∈ C ∗ (( x , x )) ( i = 1 , , . . . , r ) such that r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) ∈ Hom( W, W (( x , x ))) . Then r X i =1 ι x,x ,x ( g i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) ∈ Hom ( V, V (( x ))(( x , x ))) . Proof. As V is a quasi-compatible subspace of E ( W ), from the definition of Y φ E ,there exists a positive integer k such that x k Y φ E ( α i ( x ) , x ) β i ( x ) ∈ V [[ x ]]for all 1 ≤ i ≤ r . Let θ ( x ) ∈ V be arbitrarily fixed. By the weak associativity of thenonlocal vertex algebra V , we can update k so that we also have( x + x ) k Y φ E ( α i ( x ) , x + x ) Y φ E ( β i ( x ) , x ) θ ( x )=( x + x ) k Y φ E (cid:16) Y φ E ( α i ( x ) , x ) β i ( x ) , x (cid:17) θ ( x ) , (2.30)which implies that the common quantity on both sides lies in V (( x , x )). Then x k ( x − x ) k Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) θ ( x )= (cid:16) ( x + x ) k x k Y φ E (cid:16) Y φ E ( α i ( x ) , x ) β i ( x ) , x (cid:17) θ ( x ) (cid:17) | x = x − x for all 1 ≤ i ≤ r . Hence, we have x k ( x − x ) k r X i =1 ι x,x ,x ( g i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) θ ( x )= x k ( x + x ) k r X i =1 ι x,x ,x ( g i ( φ ( x, x + x ) , φ ( x, x ))) × Y φ E (cid:16) Y φ E ( α i ( x ) , x ) β i ( x ) , x (cid:17) θ ( x ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x − x . (2.31) ecall again that g i ( φ ( x, x ) , x ) exists in C (( x ))(( x )). Then g i ( φ ( φ ( x, x ) , x ) , φ ( x, x ))exists in C (( x ))[[ x ]](( x )) ⊂ C (( x ))(( x , x )). From Lemma 2.15 and (2.30), we have( x + x ) k Y φ E (cid:16) g i ( φ ( x, x ) , x ) Y φ E ( α i ( x ) , x ) β i ( x ) , x (cid:17) θ ( x )=( x + x ) k g i ( φ ( φ ( x, x ) , x ) , φ ( x, x )) Y φ E (cid:16) Y φ E ( α i ( x ) , x ) β i ( x ) , x (cid:17) θ ( x ) , which implies that the common quantity on both sides lies in V (( x ))(( x , x )). Then( x + x ) k Y φ E (cid:16) g i ( φ ( x, x ) , x ) Y φ E ( α i ( x ) , x ) β i ( x ) , x (cid:17) θ ( x )=( x + x ) k g i ( φ ( x, x + x ) , φ ( x, x )) Y φ E (cid:16) Y φ E ( α i ( x ) , x ) β i ( x ) , x (cid:17) . By Lemma 2.16, we have r X i =1 g i ( φ ( x, x ) , x ) Y φ E ( α i ( x ) , x ) β i ( x )= r X i =1 g i ( x , x ) α i ( x ) β i ( x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = φ ( x,x ) ∈ V (( x ))[[ x ]] . Then( x + x ) k Y φ E r X i =1 g i ( φ ( x, x ) , x ) Y φ E ( α i ( x ) , x ) β i ( x ) , x ! θ ( x ) ∈ V (( x ))[[ x ]](( x )) . Consequently, ( x + x ) k Y φ E r X i =1 g i ( φ ( x, x ) , x ) Y φ E ( α i ( x ) , x ) β i ( x ) , x ! θ ( x ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x − x lies in V (( x ))(( x , x )). Combining this with (2.31), we get x k ( x − x ) k r X i =1 g i ( φ ( x, x ) , φ ( x, x )) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) θ ( x )=( x − x ) k ( x + x ) k r X i =1 g i ( φ ( x, x + x ) , φ ( x, x )) × Y φ E (cid:16) Y φ E ( α i ( x ) , x ) β i ( x ) , x (cid:17) θ ( x ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x − x . Hence, x k r X i =1 g i ( φ ( x, x ) , φ ( x, x )) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) θ ( x )= ( x + x ) k Y φ E r X i =1 g i ( φ ( x, x ) , x ) Y φ E ( α i ( x ) , x ) β i ( x ) , x ! θ ( x ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x − x ies in V (( x ))(( x , x )). Therefore, we have r X i =1 g i ( φ ( x, x ) , φ ( x, x )) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) θ ( x ) ∈ V (( x ))(( x , x )) , as desired. (cid:3) We have the following technical result (cf. [Li8, Proposition 5.3]): Proposition 2.18. Let W be a vector space and let V be a nonlocal vertex subalgebraof E ( W ) . Suppose α i ( x ) , β i ( x ) , µ j ( x ) , ν j ( x ) ∈ V, g i ( x , x ) , h j ( x , x ) ∈ C ∗ (( x , x )) for ≤ i ≤ r, ≤ j ≤ s such that the following relation holds on W : r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) = s X j =1 ι x ,x ( h j ( x , x )) µ j ( x ) ν j ( x ) . (2.32) Then the following relation holds on V : r X i =1 ι x,x ,x ( g i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x )= s X j =1 ι x,x ,x ( h i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( µ j ( x ) , x ) Y φ E ( ν j ( x ) , x ) . (2.33) Proof. Recall from Remark 2.4 that ι x,x ,x g i ( φ ( x, x ) , φ ( x, x )) ∈ C (( x ))(( x ))(( x ))and ι x,x ,x h j ( φ ( x, x ) , φ ( x, x )) ∈ C (( x ))(( x ))(( x )). Let θ ( x ) ∈ V . As V is quasi-compatible, there exists nonzero f ( x , x ) ∈ C [[ x , x ]] such that f ( x , x ) β i ( x ) θ ( x ) , f ( x , x ) ν j ( x ) θ ( x ) ∈ E (2) ( W ) ,f ( x , x ) f ( x , x ) f ( x , x ) α i ( x ) β i ( x ) θ ( x ) ∈ E (3) ( W ) ,f ( x , x ) f ( x , x ) f ( x , x ) µ j ( x ) ν j ( x ) θ ( x ) ∈ E (3) ( W )for 1 ≤ i ≤ r, ≤ j ≤ s . From [Li8, Lemma 4.7], we have f ( φ ( x, x ) , φ ( x, x )) f ( φ ( x, x ) , x ) f ( φ ( x, x ) , x ) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) θ ( x )= Y ≤ a
As the main result of this section we have: Theorem 2.19. Let W be a vector space and let V be a nonlocal vertex subalgebraof E ( W ) . Suppose α i ( x ) , β i ( x ) , µ j ( x ) , ν j ( x ) ∈ V, g i ( x , x ) , h j ( x , x ) ∈ C ∗ (( x , x )) or ≤ i ≤ r, ≤ j ≤ s such that the following relation holds on W : r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) − s X j =1 ι x ,x ( h j ( x , x )) µ j ( x ) ν j ( x )= t X i =0 N X k =1 γ k,i ( x ) 1 i ! ( p ( x ) ∂ x ) i p ( x ) x − δ (cid:18) c k x x (cid:19) , (2.35) where N, t ≥ and c , . . . , c N are distinct nonzero complex numbers with c = 1 and γ k,i ( x ) ∈ V for ≤ i ≤ t , ≤ k ≤ N . Then r X i =1 ι x,x ,x ( g i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) − s X j =1 ι x,x ,x ( h j ( φ ( x, x ) , φ ( x, x ))) Y φ E ( µ j ( x ) , x ) Y φ E ( ν j ( x ) , x )= t X i =0 Y φ E ( γ ,i ( x ) , x ) 1 i ! ∂ ix x − δ (cid:18) x x (cid:19) . (2.36) Proof. Notice that we have( x − cx ) i +1 ( p ( x ) ∂ x ) i p ( x ) x − δ (cid:18) cx x (cid:19) = 0for any c ∈ C × and i ∈ N and that( p ( x ) ∂ x ) i p ( x ) x − δ (cid:18) x x (cid:19) = ι x ,x F i ( x , x ) − ι x ,x F i ( x , x ) , where F i ( x , x ) = ( p ( x ) ∂ x ) i ( p ( x ) / ( x − x )) for i ≥ 0. Set f ( x , x ) = N Y k =1 ( x − c k x ) t +1 . Then we have f ( x , x ) t X i =0 N X k =0 γ k,i ( x ) 1 i ! ( p ( x ) ∂ x ) i p ( x ) x − δ (cid:18) c k x x (cid:19) = f ( x , x ) t X i =0 γ ,i ( x ) 1 i ! ( p ( x ) ∂ x ) i p ( x ) x − δ (cid:18) x x (cid:19) = f ( x , x ) t X i =0 i ! ( ι x ,x ( F i ( x , x )) − ι x ,x ( F i ( x , x ))) γ ,i ( x ) . sing this and (2.45) we get f ( x , x ) r X i =1 ι x ,x ( g i ( x , x )) α i ( x ) β i ( x ) − f ( x , x ) t X i =0 i ! ι x ,x F i ( x , x ) γ ,i ( x )= f ( x , x ) s X j =1 ι x ,x ( h j ( x , x )) µ j ( x ) ν j ( x ) − f ( x , x ) t X i =0 i ! ι x ,x ( F i ( x , x )) γ ,i ( x ) . Then by Proposition 2.18, we have r X i =1 ι x,x ,x ( f ( φ ( x, x ) , φ ( x, x )) g i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) − t X i =0 ι x,x ,x f ( φ ( x, x ) , φ ( x, x )) 1 i ! ι x,x ,x F i ( φ ( x, x ) , φ ( x, x )) Y φ E ( γ ,i ( x ) , x )= s X j =1 ι x,x ,x ( f ( φ ( x, x ) , φ ( x, x )) h j ( φ ( x, x ) , φ ( x, x ))) Y φ E ( µ j ( x ) , x ) Y φ E ( ν j ( x ) , x ) − t X i =0 ι x,x ,x f ( φ ( x, x ) , φ ( x, x )) 1 i ! ι x,x ,x F i ( φ ( x, x ) , φ ( x, x )) Y φ E ( γ ,i ( x ) , x ) . Note that with f ( x , x ) ∈ C [ x , x ], we have f ( φ ( x, x ) , φ ( x, x )) ∈ C (( x ))[[ x , x ]].As f ( φ ( x, , φ ( x, f ( x, x ) = x N ( t +1) N Y k =1 (1 − c k ) t +1 = 0 ,f ( φ ( x, x ) , φ ( x, x )) is invertible in C (( x ))[[ x , x ]]. Then by cancellation we get r X i =1 ι x,x ,x ( g i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) − t X i =0 i ! ι x,x ,x ( F i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( γ ,i ( x ) , x )= s X j =1 ι x,x ,x ( h j ( φ ( x, x ) , φ ( x, x ))) Y φ E ( µ j ( x ) , x ) Y φ E ( ν j ( x ) , x ) − t X i =0 i ! ι x,x ,x ( F i ( φ ( x, x ) , φ ( x, x ))) Y φ E ( γ ,i ( x ) , x ) . Combining this relation with Corollary 2.6, we obtain (2.36). (cid:3) Next, we consider a special form of Theorem 2.19. For the rest of this section, weassume Ψ( z , z ) ∈ C [ z ± , z ± ] , ψ ( x ) ∈ C [[ x ]] with ψ ′ (0) = 1(2.37) uch that Ψ( φ ( x, x ) , φ ( x, x )) = ψ ( x − x ) . (2.38)As φ ( x, 0) = x , we have Ψ( φ ( x, z ) , x ) = ψ ( z ) . (2.39)This implies φ ( x, z ) = x as ψ ′ ( z ) = 0 from the assumption above. Remark 2.20. For r ∈ Z , set φ r ( x, z ) = e zx r +1 ddx x. (2.40)From [FHL], we have φ r ( x, z ) = ( x (1 − rzx r ) − r if r = 0 xe z if r = 0 . Set Ψ r ( z , z ) = ( − r ( z − r − z − r ) if r = 0 z /z if r = 0(2.41)and ψ r ( z ) = ( z if r = 0 e z if r = 0 . (2.42)Then we have Ψ r ( z , z ) ∈ C [ z ± , z ± ] and ψ r ( z ) ∈ C [[ z ]] with ψ ′ r (0) = 1 such thatΨ r ( φ r ( x, x ) , φ r ( x, x )) = ψ r ( x − x ) . Recall that C ( x ) is the fraction field of C [ x ]. Let C ∗ ( x ) denote the fraction fieldof C [[ x ]], which is isomorphic to C (( x )). Denote the canonical isomorphism by ι x, : ι x, : C ∗ ( x ) → C (( x )) . (2.43)As ψ ( x ) ∈ C [[ x ]] with ψ ′ ( x ) = 0, we have q ( ψ ( x )) = 0 for any nonzero q ( z ) ∈ C [ z ](since C is algebraically closed and ψ ( x ) / ∈ C ). Consequently, for any Q ( z ) ∈ C ( z ), Q ( ψ ( x )) is a well defined element of C ∗ ( x ). Then we have a field embedding ι z = ψ ( x ) : C ( z ) → C (( x )) , (2.44)which is defined by ι z = ψ ( x ) ( Q ( z )) = ι x, ( Q ( ψ ( x ))) for Q ( z ) ∈ C ( z ) . Note that Q (Ψ( x , x )) ∈ C ( x , x ) for Q ( z ) ∈ C ( z ) . Under this setting with Ψ and ψ satisfying the conditions (2.37) and (2.38), weimmediately have: heorem 2.21. Let W be a vector space and let V be a nonlocal vertex subalgebraof E ( W ) . Suppose α i ( x ) , β i ( x ) , µ j ( x ) , ν j ( x ) ∈ V, f i ( z ) , g j ( z ) ∈ C ( z ) for ≤ i ≤ r, ≤ j ≤ s such that the following relation holds on W : r X i =1 ι x ,x ( f i (Ψ( x , x ))) α i ( x ) β i ( x ) − s X j =1 ι x ,x ( g j (Ψ( x , x ))) µ j ( x ) ν j ( x )= t X i =0 N X k =1 γ k,i ( x ) 1 i ! ( p ( x ) ∂ x ) i p ( x ) x − δ (cid:18) c k x x (cid:19) , (2.45) where N, t ≥ and c , . . . , c N are distinct nonzero complex numbers with c = 1 and γ k,i ( x ) ∈ V for ≤ i ≤ t , ≤ k ≤ N . Then r X i =1 ι x ,x ( f i ( ψ ( x − x ))) Y φ E ( α i ( x ) , x ) Y φ E ( β i ( x ) , x ) − s X j =1 ι x ,x ( g j ( ψ ( x − x ))) Y φ E ( µ j ( x ) , x ) Y φ E ( ν j ( x ) , x )= t X i =0 Y φ E ( γ ,i ( x ) , x ) 1 i ! ∂ ix x − δ (cid:18) x x (cid:19) . (2.46)Let φ, Ψ , ψ be given as before. Motivated by Theorem 2.21 we formulate thefollowing notion: Definition 2.22. Let W be a vector space. A subset U of E ( W ) is said to be quasi ( φ, S ) -local if for any a ( x ) , b ( x ) ∈ U , there exist u i ( x ) , v i ( x ) ∈ U, f i ( z ) ∈ C ( z ) ( i = 1 , . . . , r )and a nonzero polynomial q ( x , x ) ∈ C [ x , x ] such that q ( x , x ) a ( x ) b ( x ) = q ( x , x ) r X i =1 ι x ,x f i (Ψ( x , x )) u i ( x ) v i ( x ) . (2.47)We define a notion of ( φ, S )-local subset (without the prefix “quasi”) by simplychanging the phrase “a nonzero polynomial q ( x , x ) ∈ C [ x , x ]” to “a Laurentpolynomial q ( x , x ) = Ψ( x , x ) k with k ∈ N ”. Theorem 2.23. Let W be a vector space and let U be a quasi ( φ, S ) -local subset of E ( W ) . Then U is quasi compatible and the nonlocal vertex algebra h U i φ generatedin E ( W ) by U is a weak quantum vertex algebra.Proof. Note that quasi ( φ, S )-locality is a special case of quasi S ( x , x )-locality inthe sense of [Li7]. Then by Proposition 4.6 therein, U is a quasi compatible, so thatwe have a nonlocal vertex algebra h U i φ . For a ( x ) , b ( x ), let u i ( x ) , v i ( x ) ∈ U, f i ( z ) ∈ ( z ) ( i = 1 , . . . , r ) and 0 = q ( x , x ) ∈ C [ x , x ] such that (2.47) holds. By Theorem2.19, we have q ( φ ( x, x ) , φ ( x, x )) Y φ E ( a ( x ) , x ) Y φ E ( b ( x ) , x )= q ( φ ( x, x ) , φ ( x, x )) r X i =1 ι x ,x f i ( ψ ( x − x )) Y φ E ( u i ( x ) , x ) Y φ E ( v i ( x ) , x ) . (2.48)As q ( x , x ) = 0, we have q ( x , x ) = ( x − x ) k ¯ q ( x , x ) for some k ∈ N , ¯ q ( x , x ) ∈ C [ x , x ] with ¯ q ( x , x ) = 0. Noticing that φ ( x, z ) = x + p ( x ) z + p ( x ) p ′ ( x ) z + · · · ,we have q ( φ ( x, x ) , φ ( x, x )) = ( x − x ) k Q ( x, x , x )for some Q ( x, x , x ) ∈ C (( x ))[[ x , x ]] with Q ( x, , = 0. Noticing that Q ( x, x , x )is invertible in C (( x ))[[ x , x ]], then from (2.48) by cancellation we get( x − x ) k Y φ E ( a ( x ) , x ) Y φ E ( b ( x ) , x )= ( x − x ) k r X i =1 ι x ,x f i ( ψ ( x − x )) Y φ E ( u i ( x ) , x ) Y φ E ( v i ( x ) , x ) . This shows that { Y φ E ( a ( z ) , x ) | a ( z ) ∈ U } is an S -local set of vertex operators on thenonlocal vertex algebra h U i φ . Since h U i φ as a nonlocal vertex algebra is generatedby U , from [LTW] (Proposition 2.6), h U i φ is a weak quantum vertex algebra. (cid:3) 3. ( G, χ φ ) -equivariant φ -coordinated quasi modules In this section, we study ( G, χ φ )-equivariant φ -coordinated quasi modules for ageneral ( G, χ )-module nonlocal vertex algebra. The main result (Theorem 3.11) isa refinement of Theorem 2.13. Throughout this section, G is a group equipped witha linear character χ .First, we formulate the following notion (cf. [Li2], [Li5]): Definition 3.1. Let G be a group with a linear character χ : G → C × . A ( G, χ ) -module nonlocal vertex algebra is a nonlocal vertex algebra V equipped with a grouphomomorphism R : G → GL( V ) such that R ( g ) = and R ( g ) Y ( v, x ) R ( g ) − = Y ( R ( g ) v, χ ( g ) x ) for g ∈ G, v ∈ V. (3.1)We shall also denote a ( G, χ )-module nonlocal vertex algebra by a pair ( V, R ). Remark 3.2. Notice that in Definition 3.1, for g ∈ G with χ ( g ) = 1, we have R ( g ) ∈ Aut( V ), where Aut( V ) denotes the automorphism group of V as a nonlocalvertex algebra. Thus a ( G, χ )-module nonlocal vertex algebra with χ = 1 (the trivialcharacter) is simply a nonlocal vertex algebra on which G acts as an automorphismgroup. In this case, we call V a G -module nonlocal vertex algebra. For ( G, χ )-module nonlocal vertex algebras ( V, R ) and ( V ′ , R ′ ), a ( G, χ ) -modulenonlocal vertex algebra homomorphism is a nonlocal vertex algebra homomorphism ψ : V → V ′ such that ψ ◦ R ( g ) = R ′ ( g ) ◦ ψ for g ∈ G. (3.2) he following is a convenient technical result: Lemma 3.3. Suppose that V is a nonlocal vertex algebra, ρ : G → Aut( V ) and L : G → GL( V ) are group homomorphisms such that L ( g ) = , ρ ( g ) L ( g ′ ) = L ( g ′ ) ρ ( g ) for g, g ′ ∈ G,L ( g ) Y ( v, x ) L ( g ) − = Y ( L ( g ) v, χ ( g ) x ) for g ∈ G, v ∈ S, where S is a generating subset of V . Define a map R : G → GL( V ) by R ( g ) = ρ ( g ) L ( g ) for g ∈ G . Then ( V, R ) is a ( G, χ ) -module nonlocal vertex algebra.Proof. We first prove that ( V, L ) is a ( G, χ )-module nonlocal vertex algebra. Forthis we need to prove L ( g ) Y ( v, x ) L ( g ) − = Y ( L ( g ) v, χ ( g ) x )for g ∈ G, v ∈ V . Let V ′ consist of those vectors v ∈ V such that the above relationholds for all g ∈ G . Then we must prove V ′ = V . As S ∪ { } ⊂ V ′ from assumption,we need to prove that V ′ is a nonlocal vertex subalgebra of V . It remains to provethat u n v ∈ V ′ for u, v ∈ V ′ , n ∈ Z . Let u, v ∈ V ′ and let g ∈ G . There exists k ∈ N such that ( x − x ) k Y ( u, x ) Y ( v, x ) ∈ Hom( V, V (( x , x ))) , ( x − x ) k Y ( L ( g ) u, x ) Y ( L ( g ) v, x ) ∈ Hom( V, V (( x , x )))and z k Y ( Y ( u, z ) v, x ) = Res x x − δ (cid:18) x + zx (cid:19) (cid:0) ( x − x ) k Y ( u, x ) Y ( v, x ) (cid:1) ,z k Y ( Y ( L ( g ) u, z ) L ( g ) v, x ) = Res x x − δ (cid:18) x + zx (cid:19) (cid:0) ( x − x ) k Y ( L ( g ) u, x ) Y ( L ( g ) v, x ) (cid:1) . Then z k L ( g ) Y ( Y ( u, z ) v, x ) L ( g ) − = Res x x − δ (cid:18) x + zx (cid:19) (cid:0) ( x − x ) k L ( g ) Y ( u, x ) Y ( v, x ) L ( g ) − (cid:1) = Res x x − δ (cid:18) x + zx (cid:19) (cid:0) ( x − x ) k Y ( L ( g ) u, χ ( g ) x ) Y ( L ( g ) v, χ ( g ) x ) (cid:1) = χ ( g ) − k Res χ ( g ) x ( χ ( g ) x ) − δ (cid:18) χ ( g ) x + χ ( g ) zχ ( g ) x (cid:19) (cid:16) ( χ ( g ) x − χ ( g ) x ) k × Y ( L ( g ) u, χ ( g ) x ) Y ( L ( g ) v, χ ( g ) x ) (cid:17) = χ ( g ) − k ( χ ( g ) z ) k Y (cid:16) Y ( L ( g ) u, χ ( g ) z ) L ( g ) v, χ ( g ) x (cid:17) = z k Y (cid:16) L ( g ) Y ( u, z ) v, χ ( g ) x (cid:17) , which gives L ( g ) Y ( Y ( u, z ) v, x ) L ( g ) − = Y (cid:16) L ( g ) Y ( u, z ) v, χ ( g ) x (cid:17) . his proves that u n v ∈ V ′ for u, v ∈ V ′ , n ∈ Z . Thus V ′ = V . Therefore ( V, L )is a ( G, χ )-module nonlocal vertex algebra. Furthermore, since ρ is a group homo-morphism from G to the automorphism group Aut( V ), it is straightforward to showthat ( V, R ) is a ( G, χ )-module nonlocal vertex algebra. (cid:3) For a subgroup Γ of C × , denote by C Γ [ x ] the monoid generated multiplicativelyby polynomials x − α for α ∈ Γ, i.e., the set of all monic polynomials whose rootsare contained in Γ. Definition 3.4. Let ( V, R ) be a ( G, χ )-module nonlocal vertex algebra and let χ φ : G → C × be a linear character of G . A ( G, χ φ ) -equivariant φ -coordinated quasi V -module is a φ -coordinated quasi V -module ( W, Y φW ) satisfying the conditions that Y φW ( R ( g ) v, x ) = Y φW ( v, χ φ ( g ) − x ) for g ∈ G, v ∈ V (3.3)and that for any u, v ∈ V , there exists f ( x ) ∈ C χ φ ( G ) [ x ] such that f ( x /x ) Y φW ( u, x ) Y φW ( v, x ) ∈ Hom( W, W (( x , x ))) . (3.4) Lemma 3.5. Let ( V, R ) be a ( G, χ ) -module nonlocal vertex algebra and let χ φ : G → C × be a linear character of G . Suppose that there exists a ( G, χ φ ) -equivariant φ -coordinated quasi V -module ( W, Y φW ) such that ddx Y φW ( v, x ) = 0 for some v ∈ V .Then φ ( x, χ ( g ) x ) = χ φ ( g ) φ ( χ φ ( g ) − x, x ) for g ∈ G. (3.5) On the other hand, (3.5) is equivalent to p ( χ φ ( g ) x ) = χ ( g ) − χ φ ( g ) p ( x ) for g ∈ G, (3.6) where φ ( x, z ) = e zp ( x ) ∂ x x with p ( x ) ∈ C (( x )) .Proof. We first prove the second assertion. As φ ( x, z ) = e zp ( x ) ∂ x x , we have φ ( x, αx ) = e αx p ( x ) ∂ x x,βφ ( β − x, x ) = βe x p ( β − x ) ∂ β − x ( β − x ) = e x βp ( β − x ) ∂ x x for any α, β ∈ C × . It then follows immediately that the second assertion holds.To prove the first assertion, let v ∈ V such that ddx Y φW ( v, x ) = 0. Recall that Y φW ( , x ) = 1 and R ( g ) = for g ∈ G . Notice that for a, b ∈ V , if h ( x , x ) is anynonzero polynomial such that h ( x , x ) Y φW ( a, x ) Y φW ( b, x ) ∈ Hom( W, W (( x , x ))) , then (cid:16) h ( x , x ) Y φW ( a, x ) Y φW ( b, x ) (cid:17) | x = φ ( x ,x ) = h ( φ ( x , x ) , x ) Y φW ( Y ( a, x ) b, x ) . sing this fact and (3.3), for g ∈ G we get (cid:16) Y φW ( v, x ) Y φW ( , x ) (cid:17) | x = φ ( x ,χ ( g ) x ) = Y φW ( Y ( v, χ ( g ) x ) , x )= Y φW ( R ( g ) Y ( R ( g ) − v, x ) , x )= Y φW ( Y ( R ( g ) − v, x ) , χ φ ( g ) − x )= (cid:16) Y φW ( R ( g ) − v, x ) Y φW ( , χ φ ( g ) − x ) (cid:17) | x = φ ( χ φ ( g ) − x ,x ) = (cid:16) Y φW ( v, χ φ ( g ) x ) Y φW ( , x ) (cid:17) | x = φ ( χ φ ( g ) − x ,x ) = (cid:16) Y φW ( v, x ) Y φW ( , x ) (cid:17) | x = χ φ ( g ) φ ( χ φ ( g ) − x ,x ) . As ddx Y φW ( v, x ) = 0, there exist w ∈ W, α ∈ W ∗ such that ddx h α, Y φW ( v, x ) w i 6 = 0. Set F ( x ) = h α, Y φW ( v, x ) w i ∈ C (( x )). Then we have F ′ ( x ) = 0 and F ( φ ( x , χ ( g ) x )) = F ( χ φ ( g ) φ ( χ φ ( g ) − x , x )) . As F ( φ ( x , χ ( g ) x )) = e χ ( g ) x p ( x ) ∂ x F ( x ) ,F (cid:0) χ φ ( g ) φ ( χ φ ( g ) − x , x ) (cid:1) = e x χ φ ( g ) p ( χ φ ( g ) − x ) ∂ x F ( x ) , by extracting the coefficients of x we obtain (3.5). (cid:3) Remark 3.6. Under the setting of Lemma 3.5, assume that ( W, Y φW ) is a ( G, χ φ )-equivariant φ -coordinated quasi V -module such that ddx Y φW ( v, x ) = 0 for all v ∈ V . Itis straightforward to show that the quotient nonlocal vertex algebra A := V / (ker Y φW )is simply an associative algebra on which G acts trivially and W is a (faithful) A -module.In view of Lemma 3.5 and Remark 3.6, from now on we shall always assume that(3.5) holds for ( G, χ φ ) -equivariant φ -coordinated quasi V -modules unless it is statedotherwise. Remark 3.7. Let Γ be a subgroup of C × and let p ( x ) ∈ C (( x )) nonzero. It isstraightforward to show that there exists a linear character λ : Γ → C × such that p ( cx ) = λ ( c ) p ( x ) for c ∈ Γ(3.7)if and only if p ( x ) = x r for some r ∈ Z when | Γ | = ∞ and p ( x ) = x r p ( x m ) for some r ∈ Z , p ( x ) ∈ C [[ x ]] with p (0) = 0 when | Γ | = m < ∞ . Remark 3.8. Consider the case φ ( x, z ) = x + z with p ( x ) = 1. Then the con-dition (3.6) amounts to χ = χ φ . In this case, the notion of ( G, χ φ )-equivariant φ -coordinated quasi V -module coincides with the notion of G -equivariant quasi V -module introduced in [Li2]. On the other hand, suppose φ ( x, z ) = xe z with p ( x ) = x .From (3.6), we have χ = 1, so that G acts on V as an automorphism group. Inthis case, χ φ can be any linear character of G and the notion of ( G, χ φ )-equivariant -coordinated quasi V -module coincides with the same named notion introduced in[Li9] with χ = χ − φ .The following technical result is analogous to Lemma 3.3: Lemma 3.9. Under the setting of Definition 3.4, assume that all the conditionshold except (3.3), and instead assume Y φW ( R ( g ) v, x ) = Y φW ( v, χ φ ( g ) − x ) for g ∈ G, v ∈ S, (3.8) where S is a generating subset of V . Then W is a ( G, χ φ ) -equivariant φ -coordinatedquasi V -module.Proof. Let V ′ consist of all vectors v ∈ V such that Y φW ( R ( g ) v, x ) = Y φW ( v, χ φ ( g ) − x ) for g ∈ G. Let u, v ∈ V ′ and let g ∈ G . Notice that there exists f ( x ) ∈ C χ φ ( G ) [ x ] such that f ( x /x ) Y φW ( u, x ) Y φW ( v, x ) , f ( x /x ) Y φW ( R ( g ) u, x ) Y φW ( R ( g ) v, x ) ∈ Hom( W, W (( x , x ))) , and f ( φ ( x, z ) /x ) Y φW ( Y ( u, z ) v, x ) = Res x x − δ (cid:18) φ ( x, z ) x (cid:19) (cid:16) f ( x /x ) Y φW ( u, x ) Y φW ( v, x ) (cid:17) ,f ( φ ( x, z ) /x ) Y φW ( Y ( R ( g ) u, z ) R ( g ) v, x )=Res x x − δ (cid:18) φ ( x, z ) x (cid:19) (cid:16) f ( x /x ) Y φW ( R ( g ) u, x ) Y φW ( R ( g ) v, x ) (cid:17) . Using all of these and (3.5) we get f ( φ ( x, χ ( g ) z ) /x ) Y φW ( R ( g ) Y ( u, z ) v, x )= f ( φ ( x, χ ( g ) z ) /x ) Y φW ( Y ( R ( g ) u, χ ( g ) z ) R ( g ) v, x )=Res x x − δ (cid:18) φ ( x, χ ( g ) z ) x (cid:19) (cid:16) f ( x /x ) Y φW ( R ( g ) u, x ) Y φW ( R ( g ) v, x ) (cid:17) =Res x x − δ (cid:18) φ ( x, χ ( g ) z ) x (cid:19) (cid:16) f ( x /x ) Y φW ( u, χ φ ( g ) − x ) Y φW ( v, χ φ ( g ) − x ) (cid:17) =Res x x − δ (cid:18) φ ( χ φ ( g ) − x, z ) χ φ ( g ) − x (cid:19) (cid:16) f ( x /x ) Y φW ( u, χ φ ( g ) − x ) Y φW ( v, χ φ ( g ) − x ) (cid:17) = f ( φ ( χ φ ( g ) − x, z ) /χ φ ( g ) − x ) Y φW ( Y ( u, z ) v, χ φ ( g ) − x )= f ( φ ( x, χ ( g ) z ) /x ) Y φW ( Y ( u, z ) v, χ φ ( g ) − x ) . Multiplying both sides by f ( φ ( x, χ ( g ) z ) /x ) − in C (( x ))(( z )), we get Y φW ( R ( g ) Y ( u, z ) v, x ) = Y φW ( Y ( u, z ) v, χ φ ( g ) − x ) . This proves that u n v ∈ V ′ for u, v ∈ V ′ , n ∈ Z . Hence, V ′ is a nonlocal vertexsubalgebra of V . As S ⊂ V ′ by assumption, we must have V ′ = V . Therefore, W isa ( G, χ φ )-equivariant φ -coordinated quasi V -module. (cid:3) n the following, we shall get a refinement of Theorem 2.13 in terms of ( G, χ )-module nonlocal vertex algebras and ( G, χ φ )-equivariant φ -coordinated quasi mod-ules. Let W be a vector space and let G be a subgroup of C × . For g ∈ G , define alinear automorphism R g of E ( W ) by R g ( a ( x )) = a ( g − x ) for a ( x ) ∈ E ( W ) . (3.9)This gives a group homomorphism R from G to GL( E ( W )).Let φ ( x, z ) = exp( zp ( x ) ddx ) x with p ( x ) = ( x r +1 if | G | = ∞ x r +1 p ( x m ) if | G | = m, (3.10)where r ∈ Z , p ( x ) ∈ C [[ x ]] such that p (0) = 0. Take χ φ to be the naturalembedding of G into C × , i.e., χ φ ( g ) = g for g ∈ G. (3.11)Then using χ φ we define a linear character χ : G → C × by (3.6), i.e., χ ( g ) − = p ( χ φ ( g ) x ) χ φ ( g ) p ( x ) = p ( gx ) gp ( x ) for g ∈ G. (3.12)More explicitly, we have χ ( g ) = g − r for g ∈ G. (3.13)We also have φ ( g − x, x ) = g − φ ( x, χ ( g ) x ) for g ∈ G. (3.14) Definition 3.10. A subset U of E ( W ) is said to be G -quasi compatible if for anyfinite sequence a ( x ) , a ( x ) , . . . , a k ( x ) ∈ U , there exists f ( x ) ∈ C G [ x ] such that Y ≤ i W, W (( x , . . . , x k ))) . From definition, it is clear that any G -quasi compatible subset of E ( W ) is quasicompatible. Let U be a G -quasi compatible subset of E ( W ). In view of Theorem2.13, U generates a nonlocal vertex algebra h U i φ with W as a φ -coordinated quasimodule. The following is a refinement of Theorem 2.13: Theorem 3.11. Let W be a vector space and let G be a subgroup of C × . Assumethat U is a G -quasi compatible subset of E ( W ) , which is G -stable. Then the nonlocalvertex algebra h U i φ together with the group homomorphism R defined in (3.9) is a ( G, χ ) -module nonlocal vertex algebra and W is a ( G, χ φ ) -equivariant φ -coordinatedquasi h U i φ -module with Y W ( a ( x ) , z ) = a ( z ) for a ( x ) ∈ h U i φ .Proof. Let ( a ( x ) , b ( x )) be any quasi-compatible pair in E ( W ). Then for any g, h ∈ G ,( a ( gx ) , b ( hx )) is a quasi-compatible pair in E ( W ). More specifically, assume that f ( x ) is a nonzero polynomial such that f ( x /x ) a ( x ) b ( x ) ∈ Hom( W, W (( x , x ))) . hen f ( gh − x /x ) a ( gx ) b ( hx ) ∈ Hom( W, W (( x , x ))) . From definition we have (cid:0) f ( x /g − x ) a ( x ) b ( g − x ) (cid:1) | x = φ ( g − x,x ) = f ( φ ( g − x, x ) /g − x ) R g (cid:16) Y φ E ( a ( x ) , x ) b ( x ) (cid:17) . (3.15)Using (3.14) and (3.15) we get f ( φ ( x, χ ( g ) x ) /x ) R g (cid:16) Y φ E ( a ( x ) , x ) b ( x ) (cid:17) = f ( φ ( g − x, x ) /g − x ) R g (cid:16) Y φ E ( a ( x ) , x ) b ( x ) (cid:17) = (cid:0) f ( x /g − x ) a ( x ) b ( g − x ) (cid:1) | x = g − φ ( x,χ ( g ) x ) = (cid:0) f ( x /x ) a ( g − x ) b ( g − x ) (cid:1) | x = φ ( x,χ ( g ) x ) = f ( φ ( x, χ ( g ) x ) /x ) Y φ E ( R g ( a ( x )) , χ ( g ) x ) R g ( b ( x )) , which implies R g (cid:16) Y φ E ( a ( x ) , x ) b ( x ) (cid:17) = Y φ E ( R g ( a ( x )) , χ ( g ) x ) R g ( b ( x )) . (3.16)As U is G -stable and h U i φ as a nonlocal vertex algebra is generated by U , it followsfrom (3.16) that R g preserves h U i φ for every g ∈ G . It also follows from (3.16) thatthe nonlocal vertex algebra h U i φ together with the group homomorphism R is a( G, χ )-module nonlocal vertex algebra.From Theorem 2.13, W is naturally a φ -coordinated quasi h U i φ -module. For a ( x ) ∈ h U i φ , g ∈ G , we have Y W ( R g a ( x ) , z ) = Y W ( a ( g − x ) , z ) = a ( g − z ) = Y W ( a ( x ) , g − z ) = Y W ( a ( x ) , χ φ ( g ) − z ) . On the other hand, following the proof of [Li8, Proposition 4.9], we see that h U i φ isalso G -quasi compatible. Therefore, W is a ( G, χ φ )-equivariant φ -coordinated quasi h U i φ -module. (cid:3) φ -coordinated quasi modules for lattice vertex algebras In this section, we shall use the general results in Section 3 and a result of Lep-owsky to construct φ -coordinated quasi modules for lattice vertex algebras.4.1. Lattice vertex algebras. In this subsection, we recall the construction ofvertex algebras associated to non-degenerate even lattices and some results involvingthe associative algebra A ( L ) introduced in [LL].We start by briefly recalling the construction of lattice vertex algebras. (For adetailed exposition, see [LL, Section 6.4–6.5] for example.) Let L be a finite rankeven lattice in the sense that L is a free abelian group of finite rank equipped with asymmetric Z -valued bilinear form h· , ·i such that h α, α i ∈ Z for α ∈ L . We assumethat L is nondegenerate in the obvious sense. Set h = C ⊗ Z L nd extend h· , ·i to a symmetric C -valued bilinear form on h . View h as an abelianLie algebra with h· , ·i as a nondegenerate symmetric invariant bilinear form. Thenwe have an affine Lie algebra b h . By definition, b h = h ⊗ C [ t, t − ] + C k as a vector space, where k is central and[ α ( m ) , β ( n )] = mδ m + n, h α, β i k (4.1)for α, β ∈ h , m, n ∈ Z with α ( m ) denoting α ⊗ t m . Set b h ± = h ⊗ t ± C [ t ± ] , (4.2)which are abelian Lie subalgebras. Identify h with h ⊗ t . Notice that the center of b h equals h + C k . Furthermore, set b h ′ = b h + + b h − + C k , (4.3)which is a Heisenberg algebra. Then b h = b h ′ ⊕ h , which is a direct sum of Lie algebras.Denote by L o the dual lattice of L , i.e., L o = { α ∈ h | h α, β i ∈ Z for β ∈ L } . (4.4)Fix a positive integer s such that s h α, β i ∈ Z for α, β ∈ L o . Furthermore, set ω = e πi/s , the principal primitive 2 s -th root of unity.Let ǫ : L o × L o → C × be a 2-cocycle such that ǫ ( α, β ) ǫ ( β, α ) − = ω s h α,β i ,ǫ ( α, 0) = 1 = ǫ (0 , α )for α, β ∈ L o . Such a 2-cocycle ǫ indeed exists (cf. Remark 6.4.1, [LL]). Note that ω s h α,β i = ( − h α,β i if h α, β i ∈ Z .Let C ǫ [ L o ] be the ǫ -twisted group algebra of L o , which by definition has a desig-nated basis { e α | α ∈ L o } with e α · e β = ǫ ( α, β ) e α + β for α, β ∈ L o . We make C ǫ [ L o ]an b h -module by letting b h ′ act trivially and letting h act by he β = h h, β i e β for h ∈ h , β ∈ L o . (4.5)For α ∈ L , define a linear operator z α : C ǫ [ L o ] → C ǫ [ L o ][ z, z − ] by z α · e β = z h α,β i e β for β ∈ L o . (4.6)Recall that the canonical irreducible module for the Heisenberg algebra b h ′ of level1 is isomorphic to S ( b h − ) as an b h − -module. Then we view S ( b h − ) as an b h -module oflevel 1 with h acting trivially. Set V L o = S ( b h − ) ⊗ C ǫ [ L o ] , (4.7)the tensor product of b h -modules. Then V L o is an b h -module of level 1. More generally,for any subset K of L o , we define V K = X α ∈ K S ( b h − ) ⊗ C e α ⊂ V L o , hich is an b h -submodule of V L o .Set = e ∈ C ǫ [ L o ] ⊂ V L o . Identify h and C ǫ [ L o ] as subspaces of V L o via a a ( − ⊗ a ∈ h ) and e α ⊗ e α ( α ∈ L o ). For α ∈ h , set α ( z ) = X n ∈ Z α ( n ) z − n − ∈ E ( V L o ) . (4.8)For any a ( x ) , b ( x ) ∈ E ( V L o ), define normally ordered product ◦◦ a ( x ) b ( x ) ◦◦ = a ( x ) + b ( x ) + b ( x ) a ( x ) − , (4.9)where a ( x ) + = P n< a n x − n − and a ( x ) − = P n ≥ a n x − n − . Define a linear map Y ( · , z ) : V L → (End V L o )[[ z, z − ]](4.10)by Y ( α, z ) = α ( z ) ,Y ( e α , z ) = exp X n> α ( − n ) n z n ! exp − X n> α ( n ) n z − n ! e α z α ,Y ( α (1) − n − · · · α ( r ) − n r − e α , z ) = ◦◦ ∂ ( n ) z α (1) ( z ) · · · ∂ ( n r ) z α ( r ) ( z ) Y ( e α , z ) ◦◦ , where r ≥ n , . . . , n r ≥ α (1) , . . . , α ( r ) , α ∈ L . Then ( V L , Y, ) carries thestructure of a vertex algebra (see [Bo], [FLM]). Set ω = d X i =1 u ( i ) ( − u ( i ) ( − ∈ V L , (4.11)where u (1) , . . . , u ( d ) is an orthonormal basis of h , and furthermore, set Y ( ω, x ) = X n ∈ Z L ( n ) x − n − . Then [ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + d 12 ( m − m ) δ m + n, (4.12)for m, n ∈ Z , V L is Z -graded by the eigenvalues of L (0), called the conformal weights ,and Y ( L ( − v, x ) = ddx Y ( v, x ) for v ∈ V L . Especially, we have L (0) h = h, L (0) e α = 12 h α, α i e α (4.13)for h ∈ h , α ∈ L . On the other hand, V L o is a V L -module and for any λ ∈ L o , V λ + L is an irreducible V L -module.Furthermore, we have (see [D1], [DLM, Theorem 3.16]): roposition 4.1. Let L be a nondegenerate even lattice. Then every V L -module iscompletely reducible and any irreducible V L -module is isomorphic to V λ + L for some λ ∈ L o . Furthermore, for each α ∈ L , α (0) acts semi-simply with only integereigenvalues on every V L -module. Now, we recall the associative algebra A ( L ) associated to V L (see [LL, Section6.5]). By definition, A ( L ) is the associative algebra with unit 1 generated by { h [ n ] , e α [ n ] | h ∈ h , α ∈ L, n ∈ Z } , subject to a set of relations written in terms ofgenerating functions: h [ z ] = X n ∈ Z h [ n ] z − n − , e α [ z ] = X n ∈ Z e α [ n ] z − n − . The relations are(AL1) e [ z ] = 1 , (AL2) [ h [ z ] , h ′ [ w ]] = h h, h ′ i ∂∂w z − δ (cid:16) wz (cid:17) , (AL3) [ h [ z ] , e α [ w ]] = h α, h i e α [ w ] z − δ (cid:16) wz (cid:17) , (AL4) [ e α [ z ] , e β [ w ]] = 0 if h α, β i ≥ , (AL5) ( z − w ) −h α,β i [ e α [ z ] , e β [ w ]] = 0 if h α, β i < , for h, h ′ ∈ h , α, β ∈ L . An A ( L )-module W is said to be restricted if for every w ∈ W , we have h [ z ] w, e α [ z ] w ∈ W (( z )) for all h ∈ h , α ∈ L .The following result gives a connection between V L -modules and certain A ( L )-modules (see [LL, Section 6.5]): Proposition 4.2. Let ( W, Y W ) be any V L -module. Then W is a restricted A ( L ) -module with h [ z ] = Y W ( h, z ) , e α [ z ] = Y W ( e α , z ) for h ∈ h , α ∈ L . Furthermore, the following relations hold on W for α, β ∈ L : (AL6) ∂ z e α [ z ] = ◦◦ α [ z ] e α [ z ] ◦◦ , (AL7) Res x (cid:0) ( x − z ) −h α,β i− e α [ x ] e β [ z ] − ( − z + x ) −h α,β i− e β [ z ] e α [ x ] (cid:1) = ε ( α, β ) e α + β [ z ] . On the other hand, let W be a restricted A ( L ) -module such that (AL6) and (AL7)hold. Then W admits a V L -module structure which is uniquely determined by Y W ( h, z ) = h [ z ] , Y W ( e α , z ) = e α [ z ] for h ∈ h , α ∈ L, where the module structure is explicitly given by Y W ( h (1) − n − · · · h ( r ) − n r − e α , z ) = ◦◦ ∂ ( n ) z h (1) [ z ] · · · ∂ ( n r ) z h ( r ) [ z ] e α [ z ] ◦◦ for r ≥ , h ( i ) ∈ h , n i ≥ and α ∈ L . On the other hand, we have the following universal property of V L : roposition 4.3. Let V be a nonlocal vertex algebra and let ψ : h ⊕ C ε [ L ] → V bea linear map such that ψ ( e ) = , the relations (AL1-3) and (AL6) hold with h [ z ] = Y ( ψ ( h ) , z ) , e α [ z ] = Y ( ψ ( e α ) , z ) for h ∈ h , α ∈ L, and such that the following relation holds for α, β ∈ L : ( x − z ) −h α,β i− e α [ x ] e β [ z ] − ( − z + x ) −h α,β i− e β [ z ] e α [ x ]= ε ( α, β ) e α + β [ z ] x − δ (cid:16) zx (cid:17) . (4.14) Then ψ can be extended uniquely to a nonlocal vertex algebra homomorphism from V L to V .Proof. From Proposition 4.2, V L is a restricted A ( L )-module with h [ z ] = Y ( h, z )and e α [ z ] = Y ( e α , z ) for h ∈ h , α ∈ L . On the other hand, we deduce from relation(4.14) that (AL4), (AL5) and (AL7) holds on V . Then V is an A ( L )-module suchthat (AL6) and (AL7) hold. For h ∈ h , we have h [ z ] V = Y V ( ψ ( h ) , z ) V ∈ V [[ z ]] , which implies h [ n ] V = 0 for n ≥ 0. It follows from [LL, Proposition 6.5.17] that V induces an A ( L )-module isomorphism ψ from V L to A ( L ) V . Notice that ψ ( h ) = ψ ( h [ − ) = h [ − ψ ( ) = ψ ( h ) − V = ψ ( h ) ,ψ ( e α ) = ψ ( e α [ − ) = e α [ − ψ ( ) = ψ ( e α ) − V = ψ ( e α )for h ∈ h , α ∈ L . Thus ψ extends ψ and we have ψ ( Y ( u, x ) v ) = ψ ( u [ x ] v ) = Y V ( ψ ( u ) , x ) ψ ( v ) for u ∈ h ⊕ C ǫ [ L ] , v ∈ V L . As h + C ǫ [ L ] generates V L as a vertex algebra, it follows that ψ is a nonlocal ver-tex algebra homomorphism. Therefore, ψ is an injective nonlocal vertex algebrahomomorphism, which extends ψ . The uniqueness is clear. (cid:3) φ -coordinated quasi-modules for V L . In this subsection, we first generalizethe twisted vertex operators constructed in [L] and then construct φ -coordinatedquasi-modules for V L .Let µ be an isometry of the lattice L with a period N , which are fixed throughoutthis subsection. As in [L], assume X k ∈ Z N h µ k ( α ) , α i ∈ Z for α ∈ L. (4.15)Recall h = C ⊗ Z L . We then view µ naturally as a linear automorphism of h , so that h µ ( h ) , µ ( h ′ ) i = h h, h ′ i for h, h ′ ∈ h .Notice that for α, β ∈ L , we have ǫ ( µ ( α ) , µ ( β )) ǫ ( µ ( β ) , µ ( α )) − = ( − h µ ( α ) ,µ ( β ) i = ( − h α,β i = ǫ ( α, β ) ǫ ( β, α ) − . rom [FLM, Proposition 5.4.1], µ can be lifted to an automorphism of C ǫ [ L ] ofperiod N such that µ ( e α ) ∈ h ± i e µ ( α ) for α ∈ L . From [L, Section 5], we canassume that µ ( e α ) = e α if µ ( α ) = α. (4.16)The following is straightforward (cf. Proposition 4.3): Lemma 4.4. There exists an automorphism b µ of V L which is uniquely determinedby b µ ( h ) = µ ( h ) , b µ ( e α ) = µ ( e α ) for h ∈ h , α ∈ L. (4.17)Identify Z N with h b µ i as an automorphism group of the vertex algebra V L . Set ξ = exp(2 πi/N ) , (4.18)the principal primitive N -th root of unity. Definition 4.5. Define a map ǫ µ ( · , · ) : L × L −→ C × by ǫ µ ( α, β ) = ǫ ( α, β ) Y = k ∈ Z N (1 − ξ k ) −h α,µ k ( β ) i . (4.19)It is straightforward to show that ǫ µ is a 2-cocycle with ǫ µ ( α, 0) = 1 = ǫ µ (0 , α ).Furthermore, the commutator map of ǫ µ is given by c µ ( α, β ) = ( − h α,β i Y = k ∈ Z N (1 − ξ k ) h β,µ k α i−h α,µ k β i = Y k ∈ Z N ( − ξ k ) h µ k α,β i = Y k ∈ Z N ( − ξ k ) −h α,µ k β i . (4.20)Associated to the 2-cocycle ǫ µ , we have the twisted group algebra C ǫ µ [ L ] whichhas a basis { e µα | α ∈ L } with e µα · e µβ = ǫ µ ( α, β ) e µα + β for α, β ∈ L. Notice that c µ ( µ ( α ) , µ ( β )) = c µ ( α, β ) for α, β ∈ L . Then µ can also be lifted to anautomorphism b µ on C ǫ µ [ L ] (see [L, Section 5]) such that for α ∈ L , b µ ( e µα ) ∈ h ξ ′ i e µµ ( α ) and b µ ( e µα ) = e µα if µ ( α ) = α. (4.21)For the rest of this subsection, we assume p ( x ) = x r +1 p ( x N ) , (4.22)where r ∈ Z and p ( x ) ∈ C [ x ] with p (0) = 0. Set φ ( x, z ) = exp( zp ( x ) ∂ x ) x .Recall (cf. [LL, Section 6]) that V L is a conformal vertex algebra, which is Z -graded by the conformal weight. Especially, we have z L (0) Y ( u, x ) z − L (0) = Y ( z L (0) u, zx ) , (4.23) L (0) h = h, L (0) e α = 12 h α, α i e α (4.24)for u ∈ V L , h ∈ h , α ∈ L . On the other hand, b µ preserves the conformal vector, sothat [ L (0) , b µ ] = 0. Set L φ = ξ − rL (0) , (4.25) ecalling that r is the integer in (4.22). Then we have L φ Y ( u, x ) v = Y (cid:0) L φ u, ξ − r x (cid:1) L φ v for u, v ∈ V L . (4.26)We now fix a linear character χ φ : Z N → C × with χ φ ([ k ]) = ξ k . (4.27)Set χ = χ − rφ , another linear character of Z N . More specifically, we have χ ([ k ]) = ξ − rk for [ k ] ∈ Z N . Furthermore, we define a group homomorphism R : Z N → GL( V L )by R ( k ) = b µ k χ ( k ) L (0) = b µ k ξ − krL (0) . (4.28)Then ( V L , R ) is a ( Z N , χ )-module vertex algebra.In the following, we are going to construct ( Z N , χ φ )-equivariant φ -coordinatedquasi V L -modules. For h ∈ h , n ∈ Z , set h ( n ) = N X k =1 µ k ( h ) ξ − nk ∈ h . (4.29)We have µh ( n ) = ξ n h ( n ) and h = 1 N ( h (1) + · · · + h ( N ) ) . (4.30)Recall that for n ∈ Z , we have P Nk =1 ξ nk = 0 if n / ∈ N Z . Set h (0) = { h (0) | h ∈ h } , L (0) = { α (0) | α ∈ L } , (4.31)where h (0) is a subspace of h and L (0) is a subgroup of L . Definition 4.6. Let T denote the category of L (0) -graded C ǫ µ [ L ]-modules T = ⊕ γ ∈ L (0) T γ such that e µα w ∈ T ( α + β ) (0) , b µ ( e µα ) w = ξ −h α (0) ,β i−h α (0) ,α i / e µα w (4.32)for α, β ∈ L, w ∈ T β (0) .Note that all irreducible objects in category T were classified and constructedexplicitly in [L, Section 6]. Definition 4.7. We denote also by µ the automorphism of Lie algebra b h defined by µ ( k ) = k , µ ( h ( n )) = ξ − n µ ( h )( n ) for h ∈ h , n ∈ Z . (4.33)Denote by b h µ the fixed-point subalgebra of b h under µ . For h ∈ h , n ∈ Z , set h µ ( n ) = h ( n ) ( n ) = X k ∈ Z N ( µ k h )( n ) ξ − nk ∈ b h . (4.34)Then b h µ = span { h µ ( n ) | h ∈ h , n ∈ Z } ⊕ C k , (4.35) here [ a µ ( m ) , b µ ( n )] = mδ m + n, h a ( m ) , b i ( N k )(4.36)for a, b ∈ h , m, n ∈ Z .Set ( b h µ ) ′ = span { h µ ( n ) | h ∈ h , n ∈ Z , n = 0 } ⊕ C k , (4.37)which is a Heisenberg algebra. Identify h (0) as a subspace of b h µ in the obvious way.Then b h µ = ( b h µ ) ′ ⊕ h (0) , a direct sum of Lie algebras. Set b h µ − = span { h µ ( − n ) | h ∈ h , n ≥ } , an abelian subalgebra of ( b h µ ) ′ . It was well known that there is an irreducible ( b h µ ) ′ -module structure of any nonzero level on S ( b h µ − ). We here view S ( b h µ − ) as an b h µ -module of level /N with h (0) acting trivially.Let T be an L (0) -graded C ǫ µ [ L ]-module from the category T . We make T an b h µ -module by letting ( b h µ ) ′ act trivially and letting h (0) act by h (0) w = h h (0) , β i w for h ∈ h , w ∈ T ( β (0) ) , β ∈ L. Set V T = S ( b h µ − ) ⊗ T, (4.38)the tensor product of b h µ -modules, which is of level 1 /N .For α ∈ L , set E µ ± ( α, x ) = exp X n ∈ Z ± α µ ( n ) x − n n , (4.39)where Z ± denote the sets of positive (negative) integers. The following results canbe found in [L]: Lemma 4.8. The following relations hold on V T for α, β ∈ L : E µ ± ( µα, x ) = E µ ± ( α, ξ − x ) , (4.40) E µ + ( α, x ) E µ − ( β, x ) = Y k ∈ Z N (cid:18) − ξ − k x x (cid:19) h µ k α,β i ! E µ − ( β, x ) E µ + ( α, x ) . (4.41) Definition 4.9. Let T be given as before. Define a linear map Y T ( · , x ) : h ⊕ C ǫ [ L ] → (End V T )[[ x, x − ]]by Y T ( h, x ) = p ( x ) X m ∈ Z h µ ( m ) x − m − = p ( x ) X k ∈ Z N ξ k µ k ( h )( ξ k x ) , (4.42) Y T ( e α , x ) = E µ − ( − α, x ) E µ + ( − α, x ) e α x α (0) p ( x ) h α,α i x h α (0) − α,α i (4.43)for h ∈ h , α ∈ L . ote that for α ∈ L , we have α (0) = P k ∈ Z N µ k α . Since12 h α (0) , α i = X k ∈ Z N h µ k α, α i ∈ Z by (4.15) and since h α, α i ∈ Z , we have h α (0) − α, α i ∈ Z . Remark 4.10. We here give some simple facts. For α ∈ L , we have X = k ∈ Z N h µ k ( α ) , α i = X k ∈ Z N h µ k ( α ) , α i − h α, α i = h α (0) − α, α i , (4.44) X = k ∈ Z N − ξ k h µ k ( α ) , α i = 12 X = k ∈ Z N − ξ k h µ k ( α ) , α i + X = k ∈ Z N − ξ − k h µ − k ( α ) , α i ! = 12 X = k ∈ Z N h µ k ( α ) , α i (cid:18) − ξ k + 11 − ξ − k (cid:19) = 12 X = k ∈ Z N h µ k ( α ) , α i = 12 h α (0) − α, α i , (4.45) X = k ∈ Z N ξ k − ξ k h µ k ( α ) , α i = X = k ∈ Z N (cid:18) − ξ k − (cid:19) h µ k ( α ) , α i = 12 h α − α (0) , α i . (4.46)We have: roposition 4.11. The following relations hold on V T for h, h ′ ∈ h , α, β ∈ L : [ Y T ( h, x ) , Y T ( h ′ , x )] = X k ∈ Z N h µ k ( h ) , h ′ i ( p ( x ) ∂ x ) p ( x ) x − δ (cid:18) ξ − k x x (cid:19) , (4.47)[ Y T ( h, x ) , Y T ( e α , x )] = X k ∈ Z N h µ k ( h ) , α i Y T ( e α , x ) p ( x ) x − δ (cid:18) ξ − k x x (cid:19) , (4.48) Y = k ∈ Z N (cid:0) x /x − ξ k (cid:1) −h α,µ k ( β ) i (cid:18) x − x p ( x ) (cid:19) −h α,β i− Y T ( e α , x ) Y T ( e β , x ) − Y = k ∈ Z N (cid:0) − ξ k + x /x (cid:1) −h α,µ k ( β ) i (cid:18) − x + x p ( x ) (cid:19) −h α,β i− Y T ( e β , x ) Y T ( e α , x )= ǫ µ ( α, β ) Y T ( e α + β , x ) p ( x ) x − δ (cid:18) x x (cid:19) . (4.49) Proof. The first relation holds as[ Y T ( h, x ) , Y T ( h ′ , x )]= p ( x ) p ( x ) X ¯ r, ¯ s ∈ Z N N h µ ¯ r ( h ) , µ ¯ s ( h ′ ) i ∂∂x x − δ (cid:18) ξ ¯ s − ¯ r x x (cid:19) = p ( x ) p ( x ) X ¯ r, ¯ s ∈ Z N N h µ ¯ r − ¯ s ( h ) , h ′ i ∂∂x x − δ (cid:18) ξ ¯ s − ¯ r x x (cid:19) = X k ∈ Z N h µ k ( h ) , h ′ i ( p ( x ) ∂ x ) p ( x ) x − δ (cid:18) ξ − k x x (cid:19) . For h ∈ h , set h µ ± ( x ) = X n ∈ Z ± h µ ( n ) x − n − = X n ∈ Z ± X k ∈ Z N ξ − nk ( µ k h )( n ) x − n − , (4.50) h µ (0) = X k ∈ Z N ( µ k h )(0) . (4.51) e have h µ ± ( x ) , X n ∈ Z ∓ α µ ( n ) x − n n = − X m ∈ Z ± X r,s ∈ Z N N h µ r h, µ s α i ξ m ( s − r ) x − m − x m = − X m ∈ Z ± X r,s ∈ Z N N h µ r − s h, α i ξ m ( s − r ) x − m − x m = − X m ∈ Z ± X k ∈ Z N h µ k h, α i ξ − mk x − m − x m = − X k ∈ Z N h µ k h, α i ξ k x − ( x /x ) ± − ξ k . Thus [ h µ + ( x ) , E µ − ( − α, x )] = E µ − ( − α, x ) X k ∈ Z N h µ k ( h ) , α i ξ k x − x /x − ξ k , [ h µ − ( x ) , E µ + ( − α, x )] = E µ + ( − α, x ) X k ∈ Z N h µ k ( h ) , α i ξ k x − x /x − ξ k . We also have " x − p ( x ) X k ∈ Z N µ k ( h )(0) , e α = x − p ( x ) X k ∈ Z N h µ k ( h ) , α i e α . Then we get the second relation by using the identity ξ k x /x − ξ k + ξ k x /x − ξ k + 1 = δ (cid:18) ξ − k x x (cid:19) . As for the last relation, we shall use (4.41). Note that Y k ∈ Z N (cid:18) x x − ξ k (cid:19) −h α,µ k β i = Y k ∈ Z N (cid:18) x x − ξ − k (cid:19) −h µ k α,β i = Y k ∈ Z N (cid:18) − ξ − k x x (cid:19) −h µ k α,β i Y k ∈ Z N (cid:18) x x (cid:19) h µ k α,β i = (cid:18) x x (cid:19) h α (0) ,β i Y k ∈ Z N (cid:18) − ξ − k x x (cid:19) −h µ k α,β i (4.52) nd that x α (0) e µβ = x h α (0) ,β i e µβ x α (0) . Using these and (4.41) we get Y = k ∈ Z N (cid:0) x /x − ξ k (cid:1) −h α,µ k ( β ) i (cid:18) x − x p ( x ) (cid:19) −h α,β i− Y T ( e α , x ) Y T ( e β , x )= Y k ∈ Z N (cid:0) x /x − ξ k (cid:1) −h α,µ k ( β ) i x −h α,β i p ( x ) h α,β i +1 Y T ( e α , x ) Y T ( e β , x )( x − x ) − = ǫ µ ( α, β ) E µ − ( − α, x ) E µ − ( − β, x ) E µ + ( − α, x ) E µ + ( − β, x ) e µα + β x α (0) x β (0) x h α (0) − α,β i × x h α (0) − α,α i x h β (0) − β,β i p ( x ) h α,α i p ( x ) h β,β i p ( x ) h α,β i +1 ( x − x ) − . Similarly, we have Y = k ∈ Z N (cid:0) − ξ k + x /x (cid:1) −h α,µ k ( β ) i (cid:18) − x + x p ( x ) (cid:19) −h α,β i− Y T ( e β , x ) Y T ( e α , x )= Y k ∈ Z N (cid:0) − ξ k + x /x (cid:1) −h α,µ k ( β ) i x −h α,β i p ( x ) h α,β i +1 Y T ( e β , x ) Y T ( e α , x )( − x + x ) − = ǫ µ ( β, α ) Y k ∈ Z N ( − ξ k ) −h α,µ k β i E µ − ( − α, x ) E µ − ( − β, x ) E µ + ( − α, x ) E µ + ( − β, x ) e µα + β x h β (0) − β,α i × x α (0) x β (0) x h α (0) − α,α i x h β (0) − β,β i p ( x ) h α,α i p ( x ) h β,β i p ( x ) h α,β i +1 ( − x + x ) − = ǫ µ ( α, β ) E µ − ( − α, x ) E µ − ( − β, x ) E µ + ( − α, x ) E µ + ( − β, x ) e µα + β x h β (0) − β,α i × x α (0) x β (0) x h α (0) − α,α i x h β (0) − β,β i p ( x ) h α,α i p ( x ) h β,β i p ( x ) h α,β i +1 ( − x + x ) − . Recall that ǫ µ ( β, α ) Y k ∈ Z N ( − ξ k ) −h α,µ k β i = ǫ µ ( α, β ) . Then using the fact ( x − x ) − − ( − x + x ) − = x − δ ( x /x ) and the delta-functionsubstitution we obtain the last relation. (cid:3) Lemma 4.12. The following relations hold on V T for h ∈ h , α ∈ L : Y T ( µ ( h ) , x ) = ξ r Y T ( h, ξ − x ) ,Y T ( b µ ( e α ) , x ) = ξ h α,α i r Y T ( e α , ξ − x ) . Proof. Note that p ( ξ − x ) = ξ − r − p ( x ). Using this we get Y T ( h, ξ − x ) = p ( ξ − x ) X m ∈ Z h ( m ) ( m )( ξ − x ) − m − = ξ − r p ( x ) X m ∈ Z ξ m h ( m ) ( m ) x − m − = ξ − r Y T ( µ ( h ) , x ) , proving the first relation. For the second relation, we first verify that it is true on T . Let w ∈ T ( β (0) ) with β ∈ L . Recall the relation (4.32): b µ ( e µα ) w = ξ −h α (0) ,β i− h α (0) ,α i e µα w. hen we have Y T ( e µα , ξ − x ) w = E µ − ( − α, ξ − x ) E µ + ( − α, ξ − x ) e µα ( ξ − x ) α (0) p ( ξ − x ) h α,α i ( ξ − x ) h α (0) − α,α i w = ξ h α,α i ( − r ) ξ − h α (0) ,α i−h α (0) ,β i E µ − ( − µ ( α ) , x ) E µ + ( − µ ( α ) , x ) e µα x α (0) p ( x ) h α,α i x h α (0) − α,α i w = ξ h α,α i ( − r ) E µ − ( − µ ( α ) , x ) E µ + ( − µ ( α ) , x )ˆ µ ( e µα ) x ( µα ) (0) p ( x ) h µα,µα i x h ( µα ) (0) − µα,µα i w = ξ h α,α i ( − r ) Y T ( b µ ( e µα ) , x ) w, noticing that b µ ( e α ) = λe µ ( α ) with λ ∈ C × and ( µα ) (0) = α (0) . This shows that thesecond relation holds on T . Since V T as an b h µ -module is generated by T , it thenfollows from (4.48) in Proposition 4.11 that the second relation holds on the wholespace V T . (cid:3) Let T be given as before. Set U T = (cid:8) Y T ( u, x ) (cid:12)(cid:12) u ∈ h + C ǫ [ L ] (cid:9) . From the commutation relations in Proposition 4.11 we see that U T is a quasi S -localsubspace of E ( V T ), in particularly, U T is quasi compatible. Then by Theorem 2.13, U T generates a nonlocal vertex algebra h U T i φ naturally with V T as a φ -coordinatedquasi module.Set Γ N = { ξ k | k ∈ Z } ⊂ C × . From Proposition 4.11, we see that U T is Γ N -quasi compatible. On the other hand,from Lemma 4.12, U T is Γ N -stable. Fix a linear character χ φ : Z N → C × with χ φ ( k ) = ξ k , and we define a group homomorphism R : Z N → GL( E ( V T )) by R ( k )( a ( x )) = a ( ξ − k x ) = a ( χ φ ( k ) − x ) for k ∈ Z N , a ( x ) ∈ E ( V T ) . (4.53)Recall that χ : Z N → C × with χ = χ − rφ . With this, by Theorem 3.11 we immediatelyhave: Corollary 4.13. The nonlocal vertex algebra h U T i φ with the map R is a ( Z N , χ ) -module nonlocal vertex algebra and V T is a ( Z N , χ φ ) -equivariant φ -coordinated quasi h U T i φ -module with Y W ( a ( x ) , z ) = a ( z ) for a ( x ) ∈ h U T i φ . Furthermore, we have: emma 4.14. Let h, h ′ ∈ h and let α, β ∈ L . Then [ Y φ E ( Y T ( h, x ) , x ) , Y φ E ( Y T ( h ′ , x ) , x )] = h h, h ′ i ∂∂x x − δ (cid:18) x x (cid:19) , (4.54) [ Y φ E ( Y T ( h, x ) , x ) , Y E ( Y T ( e α , x ) , x )] = h h, α i Y φ E ( Y T ( e α , x ) , x ) x − δ (cid:18) x x (cid:19) , (4.55) ( x − x ) −h α,β i− Y φ E ( Y T ( e α , x ) , x ) Y φ E ( Y T ( e β , x ) , x ) − ( − x + x ) −h α,β i− Y φ E ( Y T ( e β , x ) , x ) Y φ E ( Y T ( e α , x ) , x )= ǫ ( α, β ) Y φ E ( Y T ( e α + β , x ) , x ) x − δ (cid:18) x x (cid:19) . (4.56) Proof. Relations (4.54) and (4.55) follow immediately from (4.47–4.48) and Theorem2.19. On the other hand, with relation (4.49), by Theorem 2.19 we have ι x,x ,x Y = k ∈ Z N (cid:0) φ ( x, x ) /φ ( x, x ) − ξ k (cid:1) −h α,µ k ( β ) i (cid:18) φ ( x, x ) − φ ( x, x ) p ( φ ( x, x )) (cid:19) −h α,β i− × Y φ E ( Y T ( e α , x ) , x ) Y φ E ( Y T ( e β , x ) , x ) − ι x,x ,x Y = k ∈ Z N (cid:0) φ ( x, x ) /φ ( x, x ) − ξ k (cid:1) −h α,µ k ( β ) i (cid:18) − φ ( x, x ) + φ ( x, x ) p ( φ ( x, x )) (cid:19) −h α,β i− × Y φ E ( Y T ( e β , x ) , x ) Y φ E ( Y T ( e α , x ) , x )= ǫ µ ( α, β ) Y φ E ( Y T ( e α + β , x ) , x ) x − δ (cid:18) x x (cid:19) . Noticing that φ ( x, x ) − φ ( x, x )( x − x ) p ( φ ( x, x )) , φ ( x, x ) /φ ( x, x ) − ξ k for k = 0 in Z N are invertible elements of ∈ C (( x ))[[ x , x ]] andlim x → x φ ( x, x ) − φ ( x, x )( x − x ) p ( φ ( x, x )) = 1 and lim x → x ( φ ( x, x ) /φ ( x, x ) − ξ k ) = 1 − ξ k , we have (cid:18) φ ( x, x ) − φ ( x, x )( x − x ) p ( φ ( x, x )) (cid:19) m x − δ (cid:18) x x (cid:19) = x − δ (cid:18) x x (cid:19) , (cid:0) φ ( x, x ) /φ ( x, x ) − ξ k (cid:1) m x − δ (cid:18) x x (cid:19) = (1 − ξ k ) m x − δ (cid:18) x x (cid:19) or m ∈ Z . Then we get( x − x ) −h α,β i− Y φ E ( Y T ( e α , x ) , x ) Y φ E ( Y T ( e β , x ) , x ) − ( − x + x ) −h α,β i− Y φ E ( Y T ( e β , x ) , x ) Y φ E ( Y T ( e α , x ) , x )= Y = k ∈ Z N (1 − ξ k ) h α,µ k ( β ) i ! ǫ µ ( α, β ) Y φ E ( Y T ( e α + β , x ) , x ) x − δ (cid:18) x x (cid:19) = ǫ ( α, β ) Y φ E ( Y T ( e α + β , x ) , x ) x − δ (cid:18) x x (cid:19) , recalling the definition of ǫ µ (see (4.19)). This proves (4.56). (cid:3) We also have: Lemma 4.15. The following relation holds in h U T i φ for α ∈ L : Y T ( α, x ) φ − Y T ( e α , x ) = p ( x ) ddx Y T ( e α , x ) . (4.57) Proof. Set α µ ± ( x ) = p ( x ) P n ∈ Z ± α µ ( n ) x − n − . Notice that[ α µ + ( x ) , E µ − ( − α, x )] = X k ∈ Z N h α, µ k ( α ) i ξ k p ( x ) x − x /x − ξ k ! E µ − ( − α, x )and Q k ∈ Z N ( x /x − ξ k ) = ( x /x ) N − 1. Using these we have(( x /x ) N − Y T ( α, x ) Y T ( e α , x )= (( x /x ) N − α µ − ( x ) Y T ( e α , x ) + Y T ( e α , x ) α µ + ( x ) + Y T ( e α , x ) α µ (0) p ( x ) x − + h α (0) , α i Y T ( e α , x ) p ( x ) x − + X k ∈ Z N Y T ( e α , x ) h α, µ k ( α ) i ξ k p ( x ) x − x /x − ξ k ! , which lies in Hom( V T , V T (( x , x ))). Then Y φ E ( Y T ( α, x ) , z ) Y T ( e α , x )= (cid:0) ( φ ( x, z ) /x ) N − (cid:1) − (cid:2) (( x /x ) N − Y T ( α, x ) Y T ( e α , x ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = φ ( x,z ) = α µ − ( φ ( x, z )) Y T ( e α , x ) + Y T ( e α , x ) α µ + ( φ ( x, z )) + Y T ( e α , x ) α µ (0) p ( φ ( x, z )) φ ( x, z ) − + h α (0) , α i Y T ( e α , x ) p ( φ ( x, z )) φ ( x, z ) − + X k ∈ Z N Y T ( e α , x ) h α, µ k ( α ) i ξ k p ( φ ( x, z )) φ ( x, z ) − φ ( x, z ) /x − ξ k . With φ ( x, z ) an invertible element of C (( x ))(( z )), (cid:0) ( φ ( x, z ) /x ) N − (cid:1) − is viewedas an element of C (( x ))(( z )).) Note thatRes z z − ξ k p ( φ ( x, z )) φ ( x, z ) − φ ( x, z ) /x − ξ k = ( p ′ ( x ) − p ( x ) x − if k = 0 , ξ k − ξ k p ( x ) x − if k = 0 . Then we have Y T ( α, x ) φ − Y T ( e α , x )= Res z z − Y φ E ( Y T ( α, x ) , z ) Y T ( e α , x )= α µ − ( x ) Y T ( e α , x ) + Y T ( e α , x ) α µ + ( x ) + Y T ( e α , x ) α µ (0) p ( x ) x − + Y T ( e α , x ) h α, α i p ′ ( x ) + h α (0) − α, α i p ( x ) x − + X = k ∈ Z N ξ k − ξ k h µ k ( α ) , α i p ( x ) x − ! . Using the simple facts in Remark 4.10, we obtain Y T ( α, x ) φ − Y T ( e α , x )=Res z z − Y φ E ( Y T ( α, x ) , z ) Y T ( e α , x )= α µ − ( x ) Y T ( e α , x ) + Y T ( e α , x ) α µ + ( x ) + Y T ( e α , x ) α µ (0) p ( x ) x − + 12 h α, α i p ′ ( x ) Y T ( e α , x ) + 12 h α (0) − α, α i Y T ( e α , x ) p ( x ) x − = p ( x ) ddx Y T ( e α , x ) , as desired. (cid:3) Now, we are in a position to present our main result of this section. Theorem 4.16. Let T be an L (0) -graded C ǫ µ [ L ] -module from category T . Thenthere exists a ( Z N , χ φ ) -equivariant φ -coordinated quasi V L -module structure Y T ( · , x ) on V T , which is uniquely determined by Y T ( h, x ) = Y T ( h, x ) , Y T ( e α , x ) = Y T ( e α , x ) for h ∈ h , α ∈ L. (4.58) Furthermore, if T is an irreducible L (0) -graded C ǫ µ [ L ] -module, then V T is an irre-ducible φ -coordinated quasi V L -module.Proof. First of all, it follows from the commutation relations in Lemma 4.14 that h U T i φ is in fact a vertex algebra. From Lemma 4.14, we see that the relations(AL1-3) and (4.14) hold on h U T i φ with h [ z ] = Y φ E ( Y T ( h, x ) , z ) , e α [ z ] = Y φ E ( Y T ( e α , x ) , z ) for h ∈ h , α ∈ L. Noticing that p ( x ) ∂ x is the canonical derivation (the D -operator) of the vertex al-gebra h U T i φ , using Lemma 4.15 we get ∂ z Y φ E ( Y T ( e α , x ) , z ) = Y φ E ( p ( x ) ∂ x Y T ( e α , x ) , z ) = Y φ E ( Y T ( α, x ) φ − Y T ( e α , x ) , z )= ◦◦ Y φ E ( Y T ( α, x ) , z ) Y φ E ( Y T ( e α , x ) , z ) ◦◦ . (4.59) ote that Y T ( e , x ) = 1 V T the vacuum vector of h U T i φ . By Proposition 4.3, thereexists a vertex algebra homomorphism ψ from V L to h U T i φ such that ψ ( h ) = Y T ( h, x ) , ψ ( e α ) = Y T ( e α , x ) for h ∈ h , α ∈ L. As V T is a φ -coordinated quasi h U T i φ -module with Y W ( a ( x ) , z ) = a ( z ) for a ( x ) ∈h U T i φ (by Corollary 4.13), it then follows that V T is a φ -coordinated quasi V L -modulewith Y T ( u, z ) = Y W ( ψ ( u ) , z ) for u ∈ V L . For h ∈ h , α ∈ L , we have Y T ( h, z ) = Y W ( ψ ( h ) , z ) = Y W ( Y T ( h, x ) , z ) = Y T ( h, z ) ,Y T ( e α , z ) = Y W ( ψ ( e α ) , z ) = Y W ( Y T ( e α , x ) , z ) = Y T ( e α , z ) . From Lemma 4.12 we have Y T ( R ( k ) h, x ) = Y T ( h, χ φ ( k ) − x ) , Y T ( R ( k ) e α , x ) = Y T ( e α , χ φ ( k ) − x ) . (4.60)Then Y T ( R ( k ) h, z ) = Y T ( h, χ φ ( k ) − z ) , Y T ( R ( k ) e α , z ) = Y T ( e α , χ φ ( k ) − z ) . 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