aa r X i v : . [ m a t h . QA ] A ug IRREGULAR VERTEX ALGEBRAS
AKISHI IKEDA, YOTA SHAMOTO
Abstract.
We introduce the notion of irregular vertex (operator) algebras. Theirregular versions of fundamental properties, such as Goddard uniqueness theo-rem, associativity, and operator product expansions are formulated and proved.We also give some elementary examples of irregular vertex operator algebras.
Contents
1. Introduction 12. Coherent state modules 83. Irregular vertex operator algebras 134. Associativity and operator product expansions 205. Irregular Heisenberg vertex operator algebras 256. Irregular Virasoro vertex operator algebras 32Concluding remarks 36References 361.
Introduction
The vertex algebras, the definition of which was introduced by Borcherds [Bor86]and the foundation of the theory of which was developed by Frenkel-Lepowsky-Meurman [FLM88], may be seen as a mathematical language of the two-dimensionalconformal field theory initiated by Belavin-Polyakov-Zamolodchikov [BPZ84].Recently, several people [Gai13, GT12, JNS08, NS10, Nag15, Nag18] study irreg-ular singularities in conformal field theory. They are mainly motivated by Alday-Gaiotto-Tachikawa (AGT) correspondence [AGT10] and their applications. Wewould like to note that the notion of coherent states plays a fundamental role inthese studies.In the present paper, we shall initiate an attempt to give a mathematical languageof irregular singularities in conformal field theory by introducing the notions of coherent state modules and irregular vertex ( operator ) algebras . The main result ofthis paper is to formulate and prove the irregular versions of fundamental propertiesof irregular vertex algebras. We also give some elementary examples of coherentstate modules and irregular vertex algebras. In this introduction, we shall explain these notions and examples. We will explainthe notion of coherent state module in Section 1.1, irregular vertex algebras and theirfundamental properties in Section 1.2, and the examples of irregular vertex algebrasin Section 1.3.1.1.
Coherent states and irregular singularities in conformal field theory.
In conformal field theory on a Riemann sphere, Belavin-Polyakov-Zamolodchikov[BPZ84] and Knizhnik-Zamolodchikov [KZ84] found that the chiral correlation func-tions (conformal blocks) of vertex operators corresponding to highest weight vectorsin minimal models and in Wess-Zumino-Witten models respectively, satisfy certainsystems of differential equations with regular singularities. These equations areknown as the BPZ equation and the KZ equation respectively.Mathematically, these vertex operators can be interpreted as intertwining opera-tors [FHL93] associated with highest weight vectors of modules over vertex algebras.The conformal block is then given by the composition of intertwining operators[Hua03].In [AGT10], Alday-Gaiotto-Tachikawa found the relationship between Virasoroconformal blocks and Nekrasov partition functions of N = 2 superconformal gaugetheories. The relation is now known as the AGT correspondence. In order to gen-eralize the AGT correspondence to Nekrasov partition functions of asymptoticallyfree gauge theories, Gaiotto [Gai13] introduced irregular conformal blocks as thecounter parts in CFT side.In his construction of the irregular conformal block, a coherent state, which isa simultaneous eigenvector of some positive modes of the Virasoro algebra, playsa central role. He found that the coherent state corresponds to a state creatingan irregular singularity of the stress-energy tensor (irregular state), while a highestweight vector corresponds to a state creating a regular singularity (regular state).Correlation functions of vertex operators corresponding to irregular states arecalled irregular conformal blocks. More general studies of such irregular states forthe Virasoro algebra were given in [BMT12, GT12]. In particular, Gaiotto-Teschner[GT12] constructed an irregular state as a certain collision limit (confluence) ofregular states and characterized this state as an element of a D -module. Irregularstates for the affine Lie algebras and the W -algebra were studied in [GLP] and[KMST13].We note that in the side of mathematics, the idea that non-highest weight statesof the affine Lie algebra create irregular singularities of the KZ equation was alreadyappeared in [FFTL10] and [JNS08] before [Gai13].In Section 2, based on these various studies of coherent states and irregularsingularities in conformal field theory, we introduce the notion of a coherent statemodule over a vertex algebra (Definition 2.4). Let V be a vertex algebra and S RREGULAR VERTEX ALGEBRAS 3 be a positively graded vector space over C . Denote by D S the ring of differentialoperators on S . The coherent state V -module M on S is a D S -module with D S -linear vertex operators Y M : V −→ End D S ( M )[[ z ± ]] , A Y M ( A, z ) = X n ∈ Z A M ( n ) z − n − and a distinguished vector | coh i ∈ M , called a coherent state, together with someaxioms. We call S the space of internal parameters. The definition is motivated bythe description of coherent states of the Virasoro algebra in [GT12] as follows: in theprocess of the confluence of ( r +1) regular states, the resulting irregular state obtains r new parameters and the action of positive modes of the Virasoro algebra is givenby differential operators of these r parameters. We call them internal parametersof the coherent state.Conformal structures, the action of stress-energy tensors, on vertex algebras andtheir modules give coordinate change rules of vertex operators. They play importantroles in coordinate-free approach for various concepts and theory on higher genusRiemann surfaces (see [FBZ04]). In Section 2.3, we give the definition of a conformalstructure on a coherent state module. The appearance of internal parameters makesthe definition a little more complicated than the usual modules since we also needto consider coordinate changes of internal parameters.In the subsequent, we will study irregular conformal blocks as the dual spaceof coinvariants associated to conformal coherent state modules. As an application,the confluent KZ equation [JNS08] is described as an integrable connection on theirregular conformal block associated with coherent state modules over the vertexalgebra V k ( sl ). In the future work, we will also discuss the relationship between 3points irregular conformal blocks of coherent state modules and the irregular typeintertwining operators (see also Section 1.2).1.2. Irregular vertex algebras.
The definition of a vertex algebras was intro-duced by Borcherds [Bor86] and a vertex operator algebra (vertex algebra with aconformal structure) was defined by Frenkel-Lepowsky-Meurman [FLM88]. Nowa-days, various equivalent definitions of vertex algebras are known. We shall adoptthe axioms in [FKRW95] (see also [FBZ04, Kac98]) known as Goddard’s axioms[God89] since our definition of irregular vertex algebra is a generalization of them.The main point is the definition of vertex operators. For a vertex algebra V (seeDefinition 2.3), vertex operators are given by the state-field correspondence Y : V −→ End( V )[[ z ± ]] , A Y ( A, z ) = X n ∈ Z A ( n ) z − n − and they become fields, namely Y ( A, z ) B ∈ V (( z )) for all A, B ∈ V . They alsosatisfy the locality axiom ( z − w ) N [ Y ( A, z ) , Y ( B, w )] = 0for sufficiently large N . As a consequence of the axioms, we have the operatorproduct expansion (OPE) Y ( A, z ) Y ( B, w ) = N X n =0 Y ( C n , w )( z − w ) n +1 + ◦◦ Y ( A, z ) Y ( B, w ) ◦◦ where C n ∈ V are some states and ◦◦ Y ( A, z ) Y ( B, w ) ◦◦ is the normally ordered prod-uct, which is smooth along with z = w . Thus in usual vertex algebras, all singu-larities are poles and this is the reason why correlation functions of usual vertexoperators only have regular singularities.Our irregular vertex algebras are constructed on a particular coherent state mod-ules, called envelopes of vertex algebras. A coherent state module U on a space ofinternal parameter S is called an envelope of a vertex algebra V if it contains V in the fiber U on the origin 0 ∈ S and satisfies some compatibility conditions (seeDefinition 3.11). To consider irregular vertex operators for irregular states in U ,we also need to consider the singular locus H ⊂ S since the composition of twoirregular vertex operators may have singularities not only on z = w but also onsome divisor H ⊂ S . Therefore we need to introduce a D S -module U ◦ satisfies U ⊂ U ◦ ⊂ U ( ∗ H ) with some good properties where U ( ∗ H ) is a localization of U along H (see Definition 3.1).We can generalize the notion of field to have exponential type essential singular-ities. Denote by U µ the fiber of U on µ ∈ S . An irregular field with an irregularity f ( z ; λ, µ ) and an internal parameter λ ∈ S is a Hom( U ◦ µ , U ◦ λ + µ )-valued formal powerseries A λ ( z ) = X n ∈ Z A λ,n z − n − ∈ Hom( U ◦ µ , U ◦ λ + µ )[[ z ± ]]satisfies the condition A λ ( z ) B µ ∈ e f ( z ; λ,µ ) U ◦ λ + µ (( z ))for B µ ∈ U ◦ µ where f ( z ; λ, µ ) = P rk =1 c l ( λ, µ ) z − k is a polynomial of z − with coeffi-cients c k ( λ, µ ) ∈ O S and U ◦ λ + µ is a certain completion of U ◦ λ + µ . Thus the irregularfield A λ ( z ) has an exponential type essential singularity at z = 0, but after dividingthe factor e f ( z ; λ,µ ) it becomes a usual field. We also note that an irregular field withan internal parameter λ shifts internal parameters of states by λ . Since the productof e f ( z ; λ,µ ) and an element in U ◦ λ + µ (( z )) has infinite sums in U ◦ λ + µ , we consider thecompletion U ◦ λ + µ and regard A λ ( z ) as an element in Hom( U ◦ µ , U ◦ λ + µ )[[ z ± ]]. This RREGULAR VERTEX ALGEBRAS 5 condition for irregular fields are first considered in [Nag15] as a part of character-ization of irregular vertex operators for the Virasoro algebra. In the definition ofirregular vertex algebras, we assume that irregular vertex operators given by thestate-field correspondence Y : U ◦ λ −→ Hom (cid:16) U ◦ µ , U ◦ λ + µ (cid:17) [[ z ± ]] , become irregular fields with a fixed irregularity f ( z ; λ, µ ).We also require that these irregular fields satisfy the irregular locality axiom( z − w ) N (cid:16) e − f ( z − w ; λ,µ ) | z | > | w | Y ( A λ , z ) Y ( B µ , w ) − e − f ( z − w ; λ,µ ) | w | > | z | Y ( B µ , w ) Y ( A λ , z ) (cid:17) = 0for A λ ∈ U ◦ λ , B µ ∈ U ◦ µ and sufficiently large N . Here, e − f ( z − w ; λ,µ ) | z | > | w | and e − f ( z − w ; λ,µ ) | w | > | z | denote the expansions of e f ( z − w ; λ,µ ) to their respective domains (see Section 3.4 formore detail). As a consequence, we can show the following: Theorem 1.1 (Theorem 4.10. OPE for irregular vertex operators) . Y ( A λ , z ) Y ( B µ , w ) = e f ( z − w ; λ,ν ) N X n =0 Y ( C λ + µ,n , w )( z − w ) n +1 + ◦◦ Y ( A λ , z ) Y ( B µ , w ) ◦◦ ! where C λ + µ,n ∈ U ◦ λ + µ are some states with an internal parameter λ + µ . We need to note that the definition of normally ordered product for irregularfields is a littele complicated (see Definition 4.7).Another new feature of irregular vertex operators is that the state-field corre-spondence is a D -module homomorphism, which implies that it satisfies[ ∂ λ , Y ( A λ , z )] = Y ( ∂ λ A λ , z )(1.1)for a vector field ∂ λ along the direction of the parameter λ (See Remark 3.13 formore precise). This yields additional integrable differential equations on the spaceof internal parameters S for correlation functions.The irregular vertex operator given in the present paper is a prototype of a moregeneral object, the irregular type intertwining operator among general conformalcoherent state modules. (See [FHL93],[Hua05, Hua03] for regular type intertwiningoperators.) Actually, Nagoya [Nag15] considered the irregular type intertwiningoperator among two coherent state modules and one highest weight module. In thefuture work, we will give the definition of the irregular type intertwining operator.1.3. Examples of irregular vertex algebras.
We will give two classes of ele-mentary examples of irregular vertex algebras in Section 5 and Section 6. We seebasic ideas for the easiest case.
RREGULAR VERTEX ALGEBRAS 6
In Section 5, we define irregular version of Heisenberg vertex algebra F ( r ) . Theconstruction is based on the ideas of [NS10]. In the following, we consider the case r = 1. Recall the Heisenberg Lie algebraHeis = M n ∈ Z C a n ⊕ C with the relations [ a n , a m ] = n δ m + n, and [ a n , ] = 0. Let | i be the vacuumcharacterized by a n | i = 0 for n ≥ F generated by | i over Heis. Then, the Fock space F = L C a − n · · · a − n k | i has the vertex algebrastructure by the state-field correspondence Y ( a − n a − n · · · a − n k | i , z ) := ◦◦ ∂ ( n − z a ( z ) ∂ ( n − z a ( z ) · · · ∂ ( n k − z a ( z ) ◦◦ where a ( z ) = P n ∈ Z a n z − n − is the bosonic current and ∂ ( n ) z = ( n !) − ∂ nz . Bosoniccurrents a ( z ) and a ( w ) have the OPE a ( z ) a ( w ) = ( z − w ) + ◦◦ a ( z ) a ( w ) ◦◦ . Now we consider the coherent state | λ i characterized by a | λ i = λ | λ i and a n | λ i = 0for n = 0 and n ≥
2. We can realize this coherent state as | λ i = e λa − | i in a certaincompletion of F ⊗ C [ λ ] with deg λ = −
1. We set S λ = Spec C [ λ ] and assume it thespace of internal parameters. Consider a family of Fock spaces on S λ generated by | λ i over Heis and denote it by F (1) λ . Then F (1) λ = L C [ λ ] a − n · · · a − n k | λ i . The space F (1) λ is called the envelope of the Heisenberg vertex algebra F . Let us consider thevertex operators corresponding to states in F (1) λ . For the coherent state | λ i , since | λ i = P ∞ k =0 a k − | i k ! , it is natural to assume Y ( | λ i , z ) = ∞ X k =0 Y ( a k − | i , z ) k ! = ∞ X k =0 ◦◦ a ( z ) k ◦◦ k ! = ◦◦ e λa ( z ) ◦◦ where ◦◦ e λa ( z ) ◦◦ = e λa + ( z ) e λa − ( z ) . We can easily check that Y ( | λ i , z ) | i| z =0 = | λ i .The vertex operator Y ( | λ i , z ) is not a field in the usual sense but an irregularfiled with the irregularity f ( z ; λ, µ ) = λµ/z since the direct computation by usingthe Baker-Campbell-Hausdorff formula gives Y ( | λ i , z ) | µ i = ◦◦ e λa ( z ) ◦◦ e µa − | i = e λµ/z ( | λ + µ i + o ( z ))where o ( z ) is positive powers of z . So we call Y ( | λ i , z ) the irregular vertex operator.For a general state a − n · · · a − n k | λ i , we can define irregular vertex operators as Y ( a − n · · · a − n k | λ i , z ) = ◦◦ ∂ ( n − z a ( z ) · · · ∂ ( n k − z a ( z ) e λa ( z ) ◦◦ . RREGULAR VERTEX ALGEBRAS 7
The property that the vertex operation Y is a D -module homomorphism implies[ ∂ λ , Y ( | λ i , z )] = Y ( ∂ λ | λ i , z ) = Y ( a − | λ i , z ) = ◦◦ a ( z ) Y ( | λ i , z ) ◦◦ . Actually, Y ( | λ i , z ) = ◦◦ e λa ( z ) ◦◦ is the solution of the differential equation [ ∂ λ , Y ( | λ i , z )] = ◦◦ a ( z ) Y ( | λ i , z ) ◦◦ under the initial condition Y ( | i , z ) = id.Thus, we can define irregular vertex operators Y : F (1) λ → Hom( F (1) µ , F (1) λ + µ )[[ z ± ]].Finally, we see an example of the OPE for irregular vertex operators. For Y ( | λ i , z )and Y ( a − | µ i , z ), we have Y ( | λ i , z ) Y ( a − | µ i , w ) = e λµ/ ( z − w ) | z | > | w | (cid:18) λY ( | λ + µ i , w )( z − w ) + λY ( a − | λ + µ i , w ) z − w + ◦◦ Y ( | λ i , z ) Y ( a − | µ i , w ) ◦◦ ) . It follows from the computation Y ( | λ i , z ) a − | µ i = e λµ/z (cid:18) λ | λ + µ i z + λa − | λ + µ i z + o (1) (cid:19) . In Section 6, we will define the irregular version Vir ( r ) c of the Virasoro vertexalgebra Vir c for r ∈ Z > . The space of internal parameter of Vir ( r ) c will be thesame as that of F ( r ) . The construction will be given via the free field realizationof Virasoro vertex algebra to the Heisenberg vertex algebra. The main difference isthat Vir ( r ) c have singularity in the space of internal parameters. In the case r = 1 thesingular locus will be λ = 0. This kind of singularity also appears in the definitionof irregular vertex operators for Virasoro Verma modules in [Nag15].1.4. Notations.
Throughout this paper, the term “grading” refers to the Z -grading.For a graded module M = L k ∈ Z M k over a graded C -algebra R = L k ∈ Z R k , thesymbol M [[ z ± ]] = L ℓ M [[ z ± ]] ℓ denote the graded R -module whose degree ℓ part M [[ z ± ]] ℓ is M [[ z ± ]] ℓ := Y n ∈ Z M ℓ + n z n . In other words, we set deg z = − M [[ z ]] (resp. M (( z ))) of positiveformal power series (resp. formal Laurent series) with coefficients in M in a similarway.For two graded R -modules M, N and an integer n , Hom R ( M, N ) n denotes the C -vector space of R -linear morphisms of degree n . We then putHom R ( M, N ) = M n ∈ Z Hom R ( M, N ) n RREGULAR VERTEX ALGEBRAS 8 and regard it as a graded R -module. We also set End R ( M ) := Hom R ( M, M ).The derivation with respect to a (local, formal,...) coordinate x is denoted by ∂ x or ∂∂x . We also set ∂ ( n ) x := ( n !) − ∂ nx for n ∈ Z ≥ .For an affine scheme X = Spec A , we identify the structure sheaf O X with thering A of global sections as an abuse of the notation. Sheaves of modules over O X are also identified with the A -modules of their global sections. In this paper, we onlyconsider the case X = Spec C [ x , . . . , x n ] for some n . Then we denote C [ x , . . . , x n ]by O X . We also set Θ X := L nj =1 O X ∂ x j and D X = C h x i , ∂ x j | i, j = 1 , . . . , n i withthe relation [ ∂ x i , x j ] = δ i,j . Acknowledgement.
The first author would like to thank to Hajime Nagoya forteaching basic ideas of his various works on irregular conformal blocks. The secondauthor would like to thank Takuro Mochizuki and Jeng-Daw Yu for their encourage-ment. This work was supported by World Premier International Research CenterInitiative (WPI), MEXT, Japan. The first author is supported by JSPS KAKENHIGrant Number 16K17588 and 16H06337. The second author is supported by JSPSKAKENHI Grant Number JP18H05829.2.
Coherent state modules
In this section, we introduce the notion of coherent state modules.2.1.
Space of internal parameters.
Let C [ λ j ] j ∈ J be the graded ring of poly-nomials with variables λ j indexed by a finite set J . We assume that the degree d j := deg λ j are all negative. We call the spectrum S := Spec C [ λ j ] j ∈ J a space ofinternal parameters . We set O S := C [ λ j ] j ∈ J to simplify the notation (See Section1.4). We consider S as an additive algebraic group in a natural way. The additionmorphism is denoted by σ : S × S −→ S. The grading defines a C ∗ -action on S naturally. The addition σ is C ∗ -equivariant.Let D S = C h λ j , ∂ λ j i j ∈ J denote the ring of differential operators on O S . Thisis a graded algebra such that deg ∂ λ j = − d j . We also set Θ S := L j ∈ J O S ∂ λ j ⊂D S , which is naturally equipped with the structure of a graded Lie algebra. LetDer( C [[ t ]]) denote the Lie algebra C [[ t ]] ∂ t with the usual Lie bracket, i.e.[ t k +1 ∂ t , t ℓ +1 ∂ t ] = ( ℓ − k ) t k + ℓ +1 ∂ t ( k, ℓ ∈ Z ≥ ) . The grading is given by deg t = −
1, deg ∂ t = 1. Let Der ( C [[ t ]]) denote the Liesubalgebra t C [[ t ]] ∂ t of Der( C [[ t ]]). RREGULAR VERTEX ALGEBRAS 9
Definition 2.1.
A Der ( C [[ t ]]) -structure on S is a grade preserving Lie algebrahomomorphism ρ S : Der ( C [[ t ]]) −→ Θ S , t k +1 ∂ t D k such that [ D , f ] = ℓf, and[ ρ S ( t k +1 ∂ t ) , σ ∗ f ] = σ ∗ [ D k , f ]for f ∈ O S, − ℓ , k ∈ Z ≥
0, where ρ S : Der ( C [[ t ]]) → Θ S denotes the diagonal actioninduced from ρ S . Example 2.2 (see [Ike18, Lemma 4.5]) . Fix r ∈ Z > . Let S := Spec C [ λ j ] rj =1 be aspace of internal parameter with deg λ j = − j . Set D k := r − k X j =1 jλ j + k ∂∂λ j ( k = 0 , . . . , r − D k = 0 for k ≥ r . The morphism ρ S : Der ( C [[ t ]]) −→ Θ S , t k +1 ∂ t D k definesthe Der ( C [[ t ]])-structure on S .2.2. Coherent state modules.
We firstly recall the definition of vertex algebras.In this paper, we only consider Z -graded vertex algebras: Definition 2.3.
A (graded) vertex algebra is a tuple V = ( V, | i , T, Y ) of a gradedvector space V = L n ∈ Z V n , a non-zero vector | i ∈ V , a degree one endomorphism T on V , and a grade-preserving homomorphism Y : V −→ End( V )[[ z ± ]] , A Y ( A, z ) = X n ∈ Z A ( n ) z − n − with the following properties: • (vacuum axiom) Y ( | i , z ) = id V and Y ( A, z ) | i ∈ A + zV [[ z ]] for any A ∈ V . • (translation axiom) [ T, Y ( A, z )] = ∂ z Y ( A, z ) for any A ∈ V and T | i = 0. • (field axiom) Y ( A, z ) B ∈ V (( z )) for any A, B ∈ V . • (locality axiom) For any A, B ∈ V , there exists a positive integer N suchthat ( z − w ) N [ Y ( A, z ) , Y ( B, w )] = 0in End( V )[[ z ± , w ± ]]. Here, w denotes a copy of z with deg w = − V . Definition 2.4 (Coherent state modules) . Let V be a vertex algebra. Let S bea space of internal parameters (Section 2.1). A coherent state V -module on S is atriple ( M , Y M , | coh i ) of a C -graded D S -module M , a degree zero C -linear map Y M : V −→ End D S ( M )[[ z ± ]] , A Y M ( A, z ) = X n ∈ Z A M ( n ) z − n − , (2.1) RREGULAR VERTEX ALGEBRAS 10 and a homogeneous global section | coh i of M (called a coherent state of M ) suchthat the following properties hold:(1) Y M ( | i , z ) = id M .(2) Y M ( A, z ) B is in M (( z )) for any A ∈ V and B ∈ M .(3) For any
A, B ∈ V and C ∈ M , the three elements Y M ( A, z ) Y M ( B, w ) C ∈ M (( z ))(( w )) Y M ( B, w ) Y M ( A, z ) C ∈ M (( w ))(( z )) Y M ( Y ( A, z − w ) B, w ) C ∈ M (( w ))(( z − w ))are the expansions of the same element in M [[ z, w ]][ z − , w − , ( z − w ) − ] totheir respective domains.(4) M is generated by | coh i over V and D S , i.e. if a D S -submodule M ′ ⊂ M contains | coh i and is closed under operations A M ( n ) for any A ∈ V and n ∈ Z ,then we have M ′ = M .(5) Let M O denote the smallest graded O S -submodule of M such that | coh i is contained in M O and closed under operations A M ( n ) for any A ∈ V and n ∈ Z . Then, the support of the quotient module M / M O is either an emptyset or co-dimension one subvariety in S . Remark 2.5.
Let e U ( V ) be the complete topological associative algebra associatedto V (see [FBZ04, Definition 4.3.1]). By the same argument as [FBZ04, Theorem5.1.6], the condition (1), (2), and (3) is equivalent to the condition that M is asmooth e U ( V )-module. By definition, the action of e U ( V ) is compatible with theaction of D S , and hence we may consider M as a module over e U ( V ) ⊗ C D S . Thecondition (4) is then equivalent to the condition M = ( e U ( V ) ⊗ C D S ) | coh i . Themodule M O in the condition (5) is expressed as M O = ( e U ( V ) ⊗ C O S ) | coh i . Definition 2.6 (Singular locus of coherent state modules) . Let M be a coherentstate V -module. Let M O be the O S -submodule defined in (5) of Definition 2.4.The singular locus H of the coherent state module M is the support of M / M O : H := Supp( M / M O ) , which is by assumption a C ∗ -invariant Zariski closed subvariety in S . The coherentstate module M is called non-singular along S if H is empty.2.3. Conformal coherent state modules.
Recall that the
Virasoro algebra is aLie algebra Vir = ( L n ∈ Z C L n ) ⊕ C C whose Lie bracket is defined as follows:[ L m , L n ] = ( m − n ) L m + n + 112 ( m − m ) δ m + n, C, [ L n , C ] = 0 ( m, n ∈ Z ) . (2.2) RREGULAR VERTEX ALGEBRAS 11
Fix a complex number c ∈ C . Let C c be an one dimensional representation ofthe Lie subalgebra Vir > − := L n ≥− C L n ⊕ C C defined by L n n ≥ − C · c
1. Take a induced representationVir c := U (Vir) ⊗ U (Vir > − ) C c ≃ M n ≥···≥ n k > C L − n · · · L − n k v c where U (Vir) (resp. U (Vir > − )) denotes the universal enveloping algebra of Vir(resp. Vir > − ), and v c denotes the image of 1 ⊗
1. Define the Z -gradation on Vir c by the formulas deg L − n = n and deg v c = 0. The power series T ( z ) := X n ∈ Z L n z − n − defines a field on Vir c .A Virasoro vertex algebra with central charge c is a vertex algebraVir c := (Vir c , v c , L − , Y ( · , z ))where Y ( · , z ) : Vir c → End(Vir c )[[ z ± ]] is defined as follows: Y ( L − n · · · L − n k v c , z ) := ◦◦ ∂ ( n − z T ( z ) · · · ∂ ( n k − z T ( z ) ◦◦ where ◦◦ · ◦◦ denotes the normally ordered product.A conformal structure of central charge c ∈ C on a vertex algebra V is a degree twonon-zero vector ω ∈ V called a conformal vector such that the Fourier coefficients L Vn of the corresponding vertex operator Y ( ω, z ) = X n ∈ Z L Vn z − n − satisfy L V − = T and L V ω = c | i . A vertex algebra with a conformal structureis called a vertex operator algebra , or a conformal vertex algebra . The followingproperties of vertex operator algebra is well known (see [FBZ04, Lemma 3.4.5]): • { L Vn } n satisfies the relation (2.2) replacing C by c · id V . • L V = n · id V n on V n for every n ∈ Z . • There exists a unique morphism Vir c → V of vertex algebras such that v c
7→ | i , L − v c ω . Definition 2.7.
Let M = ( M , Y M , | coh i ) be a coherent state V -module over S .Assume that V has a conformal vector ω , and S has a Der ( C [[ t ]])-structure ρ S : Der C [[ t ]] → Θ S , t k +1 ∂ t D k (see Definition 2.1). Let L M n be the Fourier coefficients of the action Y M ( ω, z ) = X n ∈ Z L M n z − n − . RREGULAR VERTEX ALGEBRAS 12
Then, M is called conformal if there exist differential operators L k = h k + D k ∈O S ⊕ Θ S such that the following properties hold:(1) L M k | coh i = L k | coh i for k ∈ Z ≥ .(2) The map ρ M : Der C [[ t ]] → End C ( M ) , t k +1 ∂ t
7→ − ( L M k − L k )is a Lie algebra homomorphism.(3) L M − D is the grading operator on M (i.e. ( L M − D ) m = deg( m ) m forany homogeneous section m ∈ M ) and L M k − L k is locally finite for k > m ∈ M , there exists N > L M k − L k ) N m = 0).2.4. An example of conformal coherent state module.
We shall give an ex-ample of conformal coherent state module, following the idea of Gaiotto-Teschner[GT12, Section 2.1.2]. Take the space of internal parameter S as in Example 2.2 forfixed r >
0. Fix complex numbers ρ and λ . Set h k ( λ ) := P kj =0 λ j λ k − j − ρ ( k +1) λ k for k = 0 , . . . , r , where we put λ ℓ = 0 for ℓ > r . Put h := h = 2 − λ ( λ − ρ ).Put L k := h k + D k for k = 0 , . . . , r and L m := 0 for m > r . Lemma 2.8.
For k, ℓ ∈ Z ≥ , we have [ L k , L ℓ ] = ( ℓ − k ) L k + ℓ . Let Vir > denote the Lie subalgebra L n ≥ C L n ⊕ C C of Vir. By Lemma 2.8, L n · P ( λ, ∂ λ ) := P ( λ, ∂ λ ) L n , C · P ( λ, ∂ λ ) := cP ( λ, ∂ λ ) ( n ≥ , P ( λ, ∂ λ ) ∈ D S )defines left U (Vir > )-module structure on D S , where U (Vir > ) denotes the universalenveloping algebra of Vir > . Definition 2.9.
We set M ( r ) c,h := U (Vir) ⊗ U (Vir > ) D S . (2.3)Set | coh i := 1 ⊗ ∈ M ( r ) c,h . We define the C -grading of M ( r ) c,h so that deg | coh i = h . Proposition 2.10.
The pair ( M ( r ) c,h , | coh i ) naturally equips with the structure ofconformal coherent state Vir c -module.Proof. By construction, M ( r ) c,h is a Vir-module with the property that for any section s ∈ M ( r ) c,h there exists N > L n , s ] = 0 ( n > N ). It follows from thisfact that M ( r ) c,h is a smooth e U (Vir c )-module, which implies the conditions (1), (2),(3) in Definition 2.4 (see Remark 2.5 and [FBZ04, Section 5.1.8]). RREGULAR VERTEX ALGEBRAS 13
Again by the construction, we have the expression M ( r ) c,h = M m ,...,m r ∈ Z ≥ n ≥ n ≥···≥ n k > O S L − n · · · L − n k ⊗ ∂ m λ · · · ∂ m r λ r | coh i . (2.4)This expression implies (4). Since ∂ λ r is not in ( M ( r ) c,h ) O , the quotient M ( r ) c,h / ( M ( r ) c,h ) O is non-empty. The (proof of) Lemma 4.3 in [Ike18] implies that the singularity of M ( r ) c,h is H = { λ r = 0 } . Hence we obtain (5). The conformality is trivial byconstruction. (cid:3) Remark 2.11.
At each point λ o = ( λ o , . . . , λ or ) with λ or = 0, the fiber M ( r ) c,h | λ o of M ( r ) c,h have PBW basis M ( r ) c,h | λ = M n ∈P r C L n | λ o i , where P r denote the set of non-decreasing finite sequence n = ( n , . . . , n ℓ ) , n · · · n ℓ < r, of integers and L n = L n · · · L n ℓ (we also set L ∅ = 1). Hence M ( r ) c,h | λ is a universalWhittaker module of type h r ( λ o ) , . . . , h r ( λ o ) in the sense of [FJK12, Definition2.2]. In particular, M ( r ) c,h | λ o is a simple module ([FJK12, Corollary 2.2], see also[LGZ11, Theorem 7] and [NS10, Remark 2.2]).3. Irregular vertex operator algebras
In this section, we introduce the notion of irregular vertex algebras.3.1.
Notations on pullbacks.
We shall fix the notation for pull backs of D -modules over the space of internal parameters. Let S = Spec ( C [ λ j ] j ∈ J ) be a spaceof internal parameters. To specify coordinate functions on S , we use the notation S λ = Spec ( C [ λ j ] j ∈ J ), O S λ := C [ λ j ] j ∈ J , and D S λ = C h λ j , ∂ λ j i j ∈ J instead of S , O S , and D S . In the rest of this paper, we often use the direct product of two orthree copies of S . To distinguish each component together with specified coordinatefunctions in the direct product, we write S λ × S µ × S ν = Spec ( C [ λ j , µ j , ν j ] j ∈ J )instead of S = Spec ( C [ λ j ] ⊗ j ∈ J ). We also use the notation S λ,µ := S λ × S µ and S λ,µ,ν := S λ × S µ × S ν .Let σ λ + µ : S λ,µ → S ξ be the addition. For a D S ξ -module M , we denote the pullback of M w.r.t. σ λ + µ by M λ + µ := σ ∗ λ + µ M . Here we can explicitly write as M λ + µ = O S λ,µ ⊗ O Sξ M RREGULAR VERTEX ALGEBRAS 14 where the tensor product is given through the morphism O S ξ → O S λ,µ , ξ j λ j + µ j . The action of D S λ,µ = C h λ j , ∂ λ j , µ j , ∂ µ j i j ∈ J on M λ + µ is given by ∂ λ j ( ψ ( λ, µ ) ⊗ s ) = [ ∂ λ j , ψ ( λ, µ )] ⊗ s + ψ ( λ, µ ) ⊗ ∂ ξ j s∂ µ j ( ψ ( λ, µ ) ⊗ s ) = [ ∂ µ j , ψ ( λ, µ )] ⊗ s + ψ ( λ, µ ) ⊗ ∂ ξ j s. Since D S λ and D S µ are subalgebras of D S λ,µ , the module M λ + µ naturally has both D S λ -module and D S µ -module structures. Similarly, we can define D S λ,µ,ν -module M λ + µ + ν by using σ λ + µ + ν : S λ,µ,ν → S ξ .3.2. Filtered small lattices and exponential twists.
Let S be a space of inter-nal parameters. Let V be a vertex algebra. Let M be a coherent state V -moduleover S . As discussed in Section 2.2, M has the singular divisor H = Supp( M / M O ).We shall consider the localization of M along H . In other words, we consider M ( ∗ H ) := M ⊗ O S ( ∗ H ), where O S ( ∗ H ) denotes the ring of rational functions on S with poles in H . By definition, we have M ( ∗ H ) = M O ⊗ O S ( ∗ H ). Definition 3.1. A filtered small lattice of M ( ∗ H ) is a pair ( M ◦ , F • ) of a D S -submodule M ◦ ⊂ M ( ∗ H ) and a decreasing filtration F • ( M ◦ ) of O S -submodulesindexed by Z with the following properties:(L) We have M ⊂ M ◦ ⊂ M ( ∗ H ) and hence M ◦ ( ∗ H ) = M ( ∗ H ).(F1) There exists an integer N such that we have F N ( M ◦ ) = M ◦ .(F2) We have T m ∈ Z F m ( M ◦ ) = 0.(F3) For k, ℓ ∈ Z and m ∈ C , we have D S,k · F ℓ ( M ◦ m ) ⊂ F ℓ − k ( M ◦ m + k ).(F4) For any A ∈ V , n ∈ Z , A M ( n ) F k ( M ◦ ) ⊂ F k ( M ◦ ).Let M ◦ denote the completion of U ◦ with respect to F • ( U ◦ ) in the categoryof Z -graded O S -modules. We have a natural isomorphism M ◦ k ≃ Q ℓ ∈ Z Gr ℓF ( M ◦ k ),where Gr ℓF ( M ◦ k ) := F ℓ ( M ◦ k ) /F ℓ +1 ( M ◦ k ). By condition (F3), M ◦ naturally equipswith the structure of D S -module. By condition (F4), A M ( n ) acts on M ◦ and M ◦ .However, we do not assume that M ◦ (or, M ◦ ) is a coherent state V -module.A coherent state module M is called small if it admits a filtered small lat-tice. In this section, we assume that M is small and fix a filtered small lattice( M ◦ , F • ). We note that M ◦ λ + µ is the completion of M ◦ λ + µ with respect to thefiltration F • ( M ◦ λ + µ ) := σ ∗ λ + µ F • ( M ◦ ).We shall define the product O S λ,µ [[ z − ]] ⊗ M ◦ λ + µ (( z )) −→ M ◦ λ + µ [[ z ± ]](3.1) RREGULAR VERTEX ALGEBRAS 15 as follows: Take a power series f ( z ; λ, µ ) = X n ≥ f n ( λ, µ ) z − n ∈ O S λ,µ [[ z − ]] with f n ( λ, µ ) ∈ O S λ,µ , − n , and the homogeneous series u ( z ; λ, µ ) = X m ≥ p u m ( λ, µ ) z m ∈ M ◦ λ + µ (( z )) d with p ∈ Z , d = deg u ( z ; λ, µ ) ∈ C , u m ( λ, µ ) ∈ M ◦ λ + µ,m + d .Take the biggest integer N such that F N M ◦ = M ◦ (such N exists by (F1)and (F2) in Definition 3.1). Then, by the condition (F3), f n ( λ ; µ ) u m ( λ, µ ) is in F N + n ( M ◦ λ + µ,m − n ). It follows that the infinite sum c k ( λ, µ ) := X m − n = kn ≥ ,m ≥ p f n ( λ, µ ) u m ( λ, µ )converges in M ◦ λ + µ . Hence, we can define (3.1) by f ( z ; λ, µ ) ⊗ u ( z ; λ, µ ) X k ∈ Z c k ( λ, µ ) z k . (3.2)Here, we note that M ◦ λ + µ (( z )) = L d ∈ C M ◦ λ + µ (( z )) d in our notation (see Section 1.4).Let z − O S [ z − ] denote the ring of degree zero sections of z − O S [ z − ]. Let O S [[ z − ]] × denote the abelian group of degree zero invertible elements in O S [[ z − ]].Then the map e • : z − O S [ z − ] −→ O S [[ z − ]] × , f ( z ; λ, µ ) e f ( z ; λ,µ ) := X n ≥ n ! f ( z ; λ, µ ) n defines a morphism of abelian groups. Definition 3.2.
For f ( z ; λ, µ ) ∈ z − O S [ z − ] , let e f ( z ; λ,µ ) M ◦ λ + µ (( z )) denote theimage of { e f ( z ; λ,µ ) ⊗ u ( z ) | u ( z ) ∈ M ◦ λ + µ (( z )) } by the morphism (3.1) (see also (3.2)).3.3. Irregular fields and their compositions.Definition 3.3 (Irregular fields) . Let M µ be a coherent state module on S µ witha filtered small lattice ( M ◦ µ , F • ). An irregular field with an irregularity f ( z ; λ, µ ) ∈ z − O S λ,µ [ z − ] on M µ (with respect to ( M ◦ µ , F • )) is an element A λ ( z ) inHom D Sµ (cid:16) M ◦ µ , M ◦ λ + µ (cid:17) [[ z ± ]] RREGULAR VERTEX ALGEBRAS 16 with the following property: for any B µ ∈ M ◦ µ , we have A λ ( z ) B µ ∈ e f ( z ; λ,µ ) M ◦ λ + µ (( z )) . Remark 3.4.
We also call an element A λ + µ ( z ) ∈ Hom D Sν ( M ◦ ν , M ◦ λ + µ + ν )[[ z ± ]]irregular field with an irregularity f ( z ; λ + µ, ν ) if we have A λ + µ ( z ) C ν ∈ e f ( z ; λ + µ,ν ) M ◦ λ + µ + ν (( z ))for any C ν ∈ M ◦ ν . We can also generalize the notion of irregular fields in a similarway.Similarly to Section 3.2, we can naturally define the product O S [[ z − , w − ]] ⊗ M ◦ λ + µ + ν (( z ))(( w )) −→ M ◦ λ + µ + ν [[ z ± , w ± ]] . (3.3)Then, we define e f ( z ; λ,µ + ν )+ f ( w ; µ,ν ) M ◦ (( z ))(( w )), e f ( z ; λ,ν )+ f ( w ; µ,ν ) M ◦ (( z ))(( w )), andso on in a way similar to Definition 3.2. Definition 3.5.
For irregular fields A λ ( z ) and B µ ( w ) with irregularity f , we definethe composition A λ ( z ) B µ ( w ), which is an element inHom D Sν ( M ◦ ν , M ◦ λ + µ + ν )[[ z ± , w ± ]] , as follows: Since A λ ( z ) and B µ ( w ) are irregular fields, we have expansions A λ ( z ) = e f ( z ; λ,η ) X m ∈ Z A ′ λ ( η ) ( m ) z − m − , B µ ( w ) = e f ( w ; µ,ν ) X n ∈ Z B ′ µ ( ν ) ( n ) w − n − where A ′ λ ( η ) ( m ) ( m ∈ Z ) and B ′ µ ( ν ) ( n ) ( n ∈ Z ) are in Hom O Sη ( M ◦ η , M ◦ λ + η ) andHom O Sν ( M ◦ ν , M ◦ µ + ν ), respectively. We then define A λ ( z ) B µ ( w ) := e f ( z ; λ,µ + ν )+ f ( w ; µ,ν ) X m,n ∈ Z A ′ λ ( µ + ν ) ( m ) B ′ µ ( ν ) ( n ) z − m − w − n − . We can easily check that the composition A λ ( z ) B µ ( w ) is independent of theparameters ν i . For each C ν ∈ M ◦ ν , we have A λ ( z ) B µ ( w ) C ν ∈ e f ( z ; λ,µ + ν )+ f ( w ; µ,ν ) M ◦ λ + µ + ν (( z ))(( w )) ⊂ M ◦ λ + µ + ν [[ z ± , w ± ]] . Exponentially twisted Lie bracket.Lemma 3.6.
Let e − f ( z − w ) | z | > | w | denote the expansion of e − f ( z − w ) ∈ O S [[( z − w ) − ]] in O S [[ z − , w ]] . Then, we have e − f ( z − w ) | z | > | w | e f ( z ) ∈ O S [ z − ][[ w ]] .Proof. A priori, e − f ( z − w ) | z | > | w | e f ( z ) is in O S [[ z − , w ]]. Consider the Taylor expansion e − f ( z − w ) | z | > | w | e f ( z ) = ∞ X k =0 c k ( z ) w k , (3.4) RREGULAR VERTEX ALGEBRAS 17 where each coefficient c k ( z ) is the restriction of1 k ! (cid:18) ∂∂w (cid:19) k e − f ( z − w ) | z | > | w | e f ( z ) to w = 0. The left hand side of (3.4) is 1 when when we restrict it to w = 0. Hencewe have c ( z ) = 1. For general k ∈ Z ≥ , we have1 k ! (cid:18) ∂∂w (cid:19) k e − f ( z − w ) | z | > | w | e f ( z ) = 1( k − (cid:18) ∂∂w (cid:19) k − (cid:18) − k ∂ f ( z − w ) ∂w (cid:19) e − f ( z − w ) | z | > | w | e f ( z ) . Hence we obtain that c k ( z ) ∈ O S [ z − ] by the induction on k . (cid:3) Definition 3.7.
For an element f ( z ; λ, µ ) of z − O S [ z − ] , consider the followingproperties: • (skew symmetry) f ( z ; λ, µ ) = f ( − z ; µ, λ ). • (additivity) f ( z ; λ + µ, ν ) = f ( z ; λ, ν ) + f ( z ; µ, ν ).The set of sections of z − O S [ z − ] with these properties is denoted by Irr( S ). Corollary 3.8.
For two irregular fields A λ ( z ) and B µ ( w ) with an irregularity f in Irr( S ) and an element C ν ∈ M ν , we have e − f ( z − w ; λ,µ ) | z | > | w | A λ ( z ) B µ ( w ) C ν ∈ e f ( z ; λ,ν )+ f ( w ; µ,ν ) M ◦ λ + µ + ν (( z ))(( w )) . Definition 3.9 (Exponentially twisted Lie bracket) . For two irregular fields A λ ( z ),and B µ ( w ) with an irregularity f ∈ Irr( S ), the exponentially twisted Lie bracket [ A λ ( z ) , B µ ( w )] f is defined as e − f ( z − w ; λ,µ ) | z | > | w | A λ ( z ) B µ ( w ) − e − f ( z − w ; λ,µ ) | w | > | z | B µ ( w ) A λ ( z ) . The two irregular fields A λ ( z ) and B µ ( w ) are called mutually f -local if there existsan integer N such that ( z − w ) N [ A λ ( z ) , B µ ( w )] f = 0 . Lemma 3.10.
For mutually f -local irregular fields A λ ( z ) and B µ ( w ) , and for any C ν ∈ M ◦ ν , the two elements e f ( z − w ; λ,µ ) | z | > | w | A λ ( z ) B µ ( w ) C ν ∈ e f ( z ; λ,ν )+ f ( w ; µ,ν ) M ◦ λ + µ + ν (( z ))(( w )) e f ( z − w ; λ,µ ) | w | > | z | B µ ( w ) A λ ( z ) C ν ∈ e f ( z ; λ,ν )+ f ( w ; µ,ν ) M ◦ λ + µ + ν (( w ))(( z )) are the expansions of the same element in e f ( z ; λ,ν )+ f ( w ; µ,ν ) M ◦ λ + µ + ν [[ z, w ]][ z − , w − , ( z − w ) − ] to their respective domains. RREGULAR VERTEX ALGEBRAS 18
Envelopes of vertex algebras.
Let V be a vertex algebra and S be a spaceof internal parameters. Let ( U , Y U , | coh i ) be a Z -graded coherent state V -moduleover S with deg | coh i = 0. We consider the morphismΨ : V −→ U O , A A U ( − | coh i , (3.5)where A U ( − is defined in (2.1).Assume moreover that we have a filtered small lattice ( U ◦ , F • ) of U . Consider U O as submodules of U ◦ . Let U O | ◦ denote the fiber of U O at the origin as a submoduleof U ◦ . In other words, we put U O | ◦ := U O / ( U O ∩ m S, U ◦ ) , where m S, denote the maximal ideal of O S corresponding to the origin of S . Wenote that U O ∩ m S, U ◦ is a V -submodule of U O . Hence U O | ◦ is equipped with thestructure of V -module. Definition 3.11.
A coherent state V -module ( U , Y U , | coh i ) on S together withthe filtered small lattice ( U ◦ , F • ) is called an envelope of V if the morphism Ψ isinjective, and the composition Ψ : V Ψ U O ։ U O | ◦ of Ψ and the quotient map is an isomorphism of V -modules.For an envelope U of V , we identify U O | ◦ and V via Ψ.3.6. Definition of irregular vertex algebras.
For an envelope U = ( U , Y U , | coh i , ( U ◦ , F • ))of V on S λ (resp. S µ ), write U λ := U and | λ i := | coh i (resp. U µ := U and | µ i := | coh i ). We also use the notation Ψ λ ( v ) instead of Ψ( v ) for v ∈ V and so on.Set | λ + µ i := σ ∗ λ + µ | coh i ∈ U λ + µ .We note that an irregular field A λ ( z ) on U with irregularity f ∈ Irr( S ) definesan element in Hom( V, U ◦ λ )[[ z ± ]] in the following way: For a vector v ∈ V consider A λ ( z )Ψ µ ( v ) ∈ e f ( z ; λ,µ ) U ◦ λ + µ (( z )), and take the restriction to µ = 0. Since f ( z ; λ, µ )is a degree zero element in z − O S [ z − ] and satisfies the skew symmetry, we have f ( z ; λ,
0) = 0. Hence we have A λ ( z )Ψ µ ( v ) | µ =0 ∈ U ◦ λ (( z )), which is denoted by A λ ( z ) v for short. In particular, we can define A λ ( z ) | i .For an endomorphism T U ∈ End( U ◦ ) and an irregular field A λ ( z ), we define theLie bracket [ T U , A λ ( z )] by ( σ ∗ λ + µ T U ) A λ ( z ) − A λ ( z ) T U . RREGULAR VERTEX ALGEBRAS 19
Consider Hom D Sµ (cid:16) U ◦ µ , U ◦ λ + µ (cid:17) [[ z ± ]] as a D S λ -module by the D S λ -module struc-ture on U ◦ λ + µ and [ ∂ λ j , z n ] = 0 for n ∈ Z . We shall define the notion of irregularvertex algebra as follows: Definition 3.12 (Irregular vertex algebra) . Let V be a vertex algebra. Let S bea space of internal parameters. An irregular vertex algebra for V on S is a tuple( U , Y, f , T U ) of an envelope U = ( U , Y U , | coh i , ( U ◦ , F • )) of V on S , a grade preserving D S λ -module morphism Y : U ◦ λ −→ Hom D Sµ (cid:16) U ◦ µ , U ◦ λ + µ (cid:17) [[ z ± ]] , an element f ∈ Irr( S ), and an endomorphism T U ∈ End D S ( U ◦ ) with the followingproperties: • (irregular field axiom) For every A λ ∈ U ◦ λ , the series Y ( A λ , z ) is an irregularfield with the irregularity f ( z ; λ, µ ). • (irregular locality axiom) For any A λ ∈ U ◦ λ , B µ ∈ U ◦ µ , two irregular fields Y ( A λ , z ) and Y ( B µ , w ) are mutually f -local. • (vacuum axiom) For any A λ ∈ U ◦ λ , we have Y ( A λ , z ) | i ∈ U ◦ λ [[ z ]], and Y ( A λ , z ) | i| z =0 = A λ . • (coherent state axiom) We have Y ( | λ i , z ) | µ i ∈ e f ( z ; λ,µ ) U ◦ λ + µ [[ z ]], and e − f ( z ; λ,µ ) Y ( | λ i , z ) | µ i| z =0 = | λ + µ i . • (translation axiom) We have [ T U , Y ( A λ , z )] = ∂ z Y ( A λ , z ) for any A λ ∈ U ◦ λ . • (compatibility condition) For any A ∈ V , B µ ∈ U µ , the restriction of Y (Ψ λ ( A ) , z ) B µ to λ = 0 is Y U ( A, z ) B µ . We also have T U (Ψ( A )) | ◦ = Ψ( T A ) , where ∗| ◦ denotes the restriction as a section of U ◦ . Remark 3.13.
The endomorphism T U will be denoted by T if it is not confus-ing. The condition that Y ( · , z ) is a morphism of D S λ -modules and takes values inHom D Sµ (cid:16) U ◦ µ , U ◦ λ + µ (cid:17) [[ z ± ]] implies that for A λ ∈ U ◦ ,[ ∂ λ j , Y ( A λ , z )] = Y ( ∂ λ j A λ , z ) , [ ∂ µ j , Y ( A λ , z )] = 0for any j ∈ J . Thus, Fourier coefficients A λ, ( n ) of the irregular vertex operator Y ( A λ , z ) = P n ∈ Z A λ, ( n ) z − n − are independent of the parameters µ j . A irregularvertex algebra whose coherent state module is non-singular is called a non-singularirregular vertex algebra .Examples of irregular vertex algebras will be given in Section 5 and Section 6. RREGULAR VERTEX ALGEBRAS 20
Conformal structures.
We shall define the notion of conformal structure onirregular vertex algebras i.e. irregular vertex operator algebras . Let V be a vertexoperator algebra, i.e. a vertex algebra V together with the conformal vector ω (seeSection 2.3). Let S be a space of internal parameters with a conformal structure ρ S : Der ( C [[ t ]]) −→ Θ S , t j +1 ∂ t
7→ − D j . Define vector fields D λj ∈ Θ S λ (resp. D µj ∈ Θ S µ ) for j = 0 , . . . , r − − t j +1 ∂ t via the above map ρ S λ (resp. ρ S µ ). We consider the action of D λj and D µj on O S λ,µ . Let f ( z ; λ, µ ) be an irregularity on S . Since f ( z ; λ, µ ) is degree zero,i.e. f ( z ; λ, µ ) ∈ z − O S λ,µ [ z − ] , we have (cid:16) D λ + D µ + z∂ z (cid:17) f ( z ; λ, µ ) = 0 . Definition 3.14.
An irregularity f ( z ; λ, µ ) is called conformal if it satisfies thedifferential equations D µj + X ≤ m ≤ j (cid:16) ∂ ( m ) z z j +1 (cid:17) D λm + z j +1 ∂ z f ( z ; λ, µ ) = 0 mod O S λ,µ [[ z ]](3.6)for any non-negative integer j . An irregular vertex algebra U is called an irregularvertex operator algebra if U and f are conformal and T U = L U− .4. Associativity and operator product expansions
In this section, we shall prove the three fundamental properties of irregular ver-tex algebras: Goddard uniqueness theorem, associativity, and operator productexpansions. The proofs are almost parallel to the classical ones under suitableformulations.4.1.
Goddard Uniqueness theorem.
Let ( U , ( U ◦ , F • ) , Y, f , T ) be an irregularvertex algebra for a vertex algebra V on a space S of internal parameters. Thefollowing is an analog of Goddard Uniqueness theorem: Theorem 4.1.
Let A λ ( z ) be an irregular field on U with the irregularity f . If (1) for any b µ ∈ U ◦ µ , irregular fields A λ ( z ) and Y ( b µ , w ) are mutually f -local, (2) for an element a λ ∈ U ◦ λ , we have A λ ( z ) | i = Y ( a λ , z ) | i ,then we have A λ ( z ) = Y ( a λ , z ) . RREGULAR VERTEX ALGEBRAS 21
Proof.
Let b µ be an element in U ◦ µ . By the assumptions and the irregular localityaxiom (Definition 3.12 (2)), there exists a positive integer N such that( z − w ) N e − f ( z − w ; λ,µ ) | z | > | w | A λ ( z ) Y ( b µ , w ) | i =( z − w ) N e − f ( z − w ; λ,µ ) | w | > | z | Y ( b µ , w ) A λ ( z ) | i (assumption (1))=( z − w ) N e − f ( z − w ; λ,µ ) | w | > | z | Y ( b µ , w ) Y ( a λ , z ) | i (assumption (2))=( z − w ) N e − f ( z − w ; λ,µ ) | z | > | w | Y ( a λ , z ) Y ( b µ , w ) | i (Irregular locality axiom) . Therefore, we obtain( z − w ) N e − f ( z − w ; λ,µ ) | z | > | w | A λ ( z ) Y ( b µ , w ) | i = ( z − w ) N e − f ( z − w ; λ,µ ) | z | > | w | Y ( a λ , z ) Y ( b µ , w ) | i Since we can restrict both sides to w = 0, we get z N e − f ( z ; λ,µ ) Y ( a λ , z ) b µ = z N e − f ( z ; λ,µ ) A λ ( z ) b µ . This implies the theorem. (cid:3)
Lemma 4.2.
For any
A ∈ U ◦ , we have Y ( A , z ) | i = e zT A .Proof. Take the expansion Y ( A , z ) = P n ∈ Z A ( n ) z − n − , A ( n ) ∈ Hom( U ◦ µ , U ◦ λ + µ ).By the vacuum axiom in Definition 3.12, we have A ( n ) | i = 0 for n ≥
0, and A ( − | i = A . By the translation axiom, we have ∂ z Y ( A , z ) = T A ( z ) | i . Hencewe obtain n A ( − n − | i = T A ( − n ) . Therefore, we obtain A ( − n − | i = ( n !) − T n A .This implies the lemma. (cid:3) Corollary 4.3.
Assume that an irregular field A λ ( z ) ∈ Hom( U ◦ µ , U ◦ λ + µ )[[ z ± ]] and Y ( B µ , w ) are mutually f -local for any B µ ∈ U ◦ µ , A λ ( z ) | i − a λ ∈ z U ◦ λ [[ z ]] for some a λ ∈ U ◦ λ , and ∂ z A λ ( z ) | i = T A λ ( z ) | i . Then we obtain A λ ( z ) = Y ( a λ , z ) . Lemma 4.4.
For any A λ ∈ U ◦ λ , we have e wT Y ( A λ ) e − wT = Y ( A λ , z + w ) in Hom( U ◦ µ , U ◦ λ + µ )[[ z ± ]] , where ( z + w ) − is expanded in C (( z ))(( w )) . Proposition 4.5 (skew symmetry) . For A λ ∈ U ◦ λ , B µ ∈ U ◦ µ , we have Y ( A λ , z ) B µ = e zT Y ( B µ , − z ) A λ in e f ( z ; λ,µ ) U ◦ λ + µ (( z )) . RREGULAR VERTEX ALGEBRAS 22
Proof.
For sufficiently large N ∈ Z , we have( z − w ) N e − f ( z − w ; λ,µ ) | z | > | w | Y ( A λ , z ) Y ( B µ , w ) | i =( z − w ) N e − f ( z − w ; λ,µ ) | w | > | z | Y ( B µ , w ) Y ( A λ , z ) | i ( f -locality)=( z − w ) N e − f ( z − w ; λ,µ ) | w | > | z | Y ( B µ , w ) e zT A λ (Lemma 4.2)=( z − w ) N e − f ( z − w ; λ,µ ) | w | > | z | e zT Y ( B µ , w − z ) A λ (Lemma 4.4) . If we take N sufficiently large, any term in these equalities are in U ◦ λ + µ [[ z, w ]]. Hencewe can restrict them to w = 0 and obtain z N e − f ( z ; λ,µ ) Y ( A λ , z ) B µ = z N e − f ( z ; λ,µ ) e zT Y ( B µ , − z ) A λ . This implies the proposition. (cid:3)
Associativity.
The following theorem is a generalization of the associativityto irregular vertex algebras:
Theorem 4.6.
For any A λ ∈ U ◦ λ , B µ ∈ U ◦ µ , and C ν ∈ U ◦ ν , the three elements e − f ( z ; λ,ν ) e − f ( z − w ; λ,µ ) | z | > | w | Y ( A λ , z ) Y ( B µ , w ) C ν ∈ e f ( w ; µ,ν ) U ◦ λ + µ + ν (( z ))(( w )) e − f ( z ; λ,ν ) e − f ( z − w ; λ,µ ) | w | > | z | Y ( B µ , w ) Y ( A λ , z ) C ν ∈ e f ( w ; µ,ν ) U ◦ λ + µ + ν (( w ))(( z )) e − f ( z ; λ,ν ) | w | > | z − w | e − f ( z − w ; λ,µ ) Y ( Y ( A λ , z − w ) B µ , w ) C ν ∈ e f ( w ; µ,ν ) U ◦ λ + µ + ν (( w ))(( z − w )) are the expansions of the same element in e f ( w ; µ,ν ) U ◦ λ + µ + ν [[ z, w ]][ z − , w − , ( z − w ) − ] to their respective domains. Take an expansion e − f ( z − w ; λ,µ ) Y ( A λ , z − w ) = X n ∈ Z A ′ λ ( µ ) ( n ) ( z − w ) − n − , where A ′ λ ( µ ) ( n ) ∈ Hom( U ◦ µ , U ◦ λ + µ ) for each n ∈ Z . Since e − f ( z − w ; λ,µ ) Y ( A λ , z − w ) B µ is in U ◦ λ + µ (( z − w )), we have the expansion e − f ( z − w ; λ,µ ) Y ( A λ , z − w ) B µ = X n ≤ N A ′ λ ( µ ) ( n ) B µ ( z − w ) n +1 for sufficiently large N .Then, the composition e − f ( z ; λ,ν ) | w | > | z − w | e − f ( z − w ; λ,µ ) Y ( Y ( A λ , z − w ) B µ , w ) C ν is defined as e − f ( z ; λ,ν ) | w | > | z − w | X n ≤ N Y ( A ′ λ ( µ ) ( n ) B µ , w ) C ν ( z − w ) n +1 . RREGULAR VERTEX ALGEBRAS 23
Here, note that we have e − f ( z ; λ,ν ) | w | > | z − w | e f ( w ; λ,ν ) ∈ O S λ,ν [ w − ][[ z − w ]] by Lemma 3.6. Hence e − f ( z ; λ,ν ) | w | > | z − w | Y ( A ′ λ ( µ ) ( n ) B µ , w ) C ν is in e f ( w ; µ,ν ) U ◦ λ + µ + ν (( w ))[[ z − w ]]. Proof of Theorem 4.6.
By the skew symmetry (Proposition 4.5), we have e − f ( z − w ; λ,µ ) | z | > | w | Y ( A λ , z ) Y ( B µ , w ) C ν = e − f ( z − w ; λ,µ ) | z | > | w | Y ( A λ , z ) e wT Y ( C ν , − w ) B µ = e − f ( z − w ; λ,µ ) | z | > | w | e wT (cid:0) e − wT Y ( A λ , z ) e wT (cid:1) Y ( C ν , − w ) B µ Since e − f ( z − w ; λ,µ ) | z | > | w | Y ( A λ , z ) e wT Y ( C ν , − w ) B µ is in e f ( z ; λ,ν )+ f ( w ; µ,ν ) U ◦ λ + µ + ν (( z ))(( w )), thelast equality makes sense. Then, again by Lemma 4.4, this equals to e − f ( z − w ; λ,µ ) | z | > | w | e wT Y ( A λ , z − w ) Y ( C ν , − w ) B µ . Therefore, the two elements e f ( z ; λ,ν ) e − f ( z − w ; λ,µ ) | z | > | w | Y ( A λ , z ) Y ( B µ , w ) C ν and e f ( z ; λ,ν ) | z − w | > | w | e − f ( z − w ; λ,µ ) e wT Y ( A λ , z − w ) Y ( C ν , − w ) B µ (4.1)are the expansions of the same element in e f ( w ; µ,ν ) U ◦ λ + µ + ν [[ z, w ]][ z − , w − , ( z − w ) − ]to the modules e f ( w ; µ,ν ) U ◦ λ + µ + ν (( z ))(( w )) and e f ( w ; µ,ν ) U ◦ λ + µ + ν (( z − w ))(( w )) respectively.By the skew symmetry, we have Y ( A ′ λ ( µ ) ( n ) B µ , w ) C ν = e wT Y ( C ν , − w ) A ′ λ ( µ ) ( n ) B µ . Hence, we obtain e f ( z ; λ,ν ) | w | > | z − w | e − f ( z − w ; λ,µ ) Y ( Y ( A λ , z − w ) B µ , w ) C ν = e f ( z ; λ,ν ) | w | > | z − w | e − f ( z − w ; λ,µ ) e wT Y ( C ν , − w ) Y ( A λ , z − w ) B µ (4.2)in U λ + µ + ν [[ w ± , ( z − w ) ± ]]. By Lemma 3.10, (4.1) and (4.2) are the expansion ofthe same element. This proves the theorem. (cid:3) Normally ordered product and operator product expansion.
We shalldefine the normally ordered product for irregular fields:
Definition 4.7 (normally ordered product) . For an irregular field A λ ( z ) with anexpansion A λ ( z ) = e f ( z ; λ,ν ) P n ∈ Z A ′ λ ( ν ) ( n ) z − n − , set A ′ λ ( z ; ν ) := e − f ( z ; λ,ν ) A λ ( z ) = X n ∈ Z A ′ λ ( ν ) ( n ) z − n − and A ′ λ ( z ; ν ) + := X n< A ′ λ ( ν ) ( n ) z − n − , A ′ λ ( z ; ν ) − := X n ≥ A ′ λ ( ν ) ( n ) z − n − . Let B µ ( w ) be an irregular field with irregularity f ( w ; µ, ν ) and set B ′ µ ( w ; ν ) := e − f ( w ; µ,ν ) B µ ( w ). The normally ordered product ◦◦ A λ ( z ) B µ ( w ) ◦◦ of A λ ( z ) and B µ ( w )is defined by ◦◦ A λ ( z ) B µ ( w ) ◦◦ := e f ( z ; λ,ν )+ f ( w ; µ,ν ) (cid:0) A ′ λ ( z ; µ + ν ) + B ′ µ ( w ; ν ) + B ′ µ ( w ; λ + ν ) A ′ λ ( z ; ν ) − (cid:1) . The restriction of ◦◦ A λ ( z ) B µ ( w ) ◦◦ to z = w is well defined and is denoted by ◦◦ A λ ( z ) B µ ( z ) ◦◦ . Note that ◦◦ A λ ( z ) B µ ( z ) ◦◦ is again an irregular field with irregularity f ( z ; λ + µ, ν ). Actually, we can check that ◦◦ A λ ( z ) B µ ( z ) ◦◦ does not depend on theparameters ν i by using the presentation in Lemma 4.8 below.The following two lemmas can be proved by the same way as the classical case: Lemma 4.8.
The restriction ◦◦ A λ ( w ) B µ ( w ) ◦◦ equals to e f ( w ; λ + µ,ν ) times Res z =0 (cid:2) δ ( z − w ) − A ′ λ ( z ; µ + ν ) B ′ µ ( w ; ν ) + δ ( z − w ) + B ′ µ ( w ; λ + ν ) A ′ λ ( z ; ν ) (cid:3) dz, where δ ( z − w ) − := P ∞ n =0 w n /z n +1 and δ ( z − w ) + := P n> z n − /w n . Lemma 4.9 (Dong’s lemma) . Let A λ ( z ) , B µ ( w ) , C ν ( u ) be irregular fields with anirregularity f . Assume that each two of three fields are mutually f -local. Then, thenormally ordered product ◦◦ A λ ( z ) B µ ( z ) ◦◦ and C ν ( w ) are mutually f -local. Theorem 4.10 (Operator product expansion) . For any A λ ∈ U ◦ λ and B µ ∈ U ◦ µ ,there is an equality Y ( A λ , z ) Y ( B µ , w ) = e f ( z − w ; λ,µ ) ∞ X n =0 Y ( A ′ λ ( µ ) ( n ) B µ , w )( z − w ) n +1 + ◦◦ Y ( A λ , z ) Y ( B µ , w ) ◦◦ ! where Y ( A λ , z ) = e f ( z ; λ,µ ) P n ∈ Z A ′ λ ( µ ) ( n ) z − n − and both sides are expanded in thedomain | z | > | w | .Proof. By the associativity, it remains to show that Y ( A ′ λ ( µ ) ( − n − B µ , w ) = ◦◦ (cid:16) ∂ ( n ) w Y ( A λ , w ) (cid:17) Y ( B µ , w ) ◦◦ (4.3)for every non-negative integer n . By the direct computation, we have ◦◦ (cid:16) ∂ ( n ) w Y ( A λ , w ) (cid:17) Y ( B µ , w ) ◦◦ | i| w =0 = A ′ λ ( µ ) ( − n − B µ . The irregular field ◦◦ (cid:16) ∂ ( n ) w Y ( A λ , w ) (cid:17) Y ( B µ , w ) ◦◦ and Y ( C ν , z ) are mutually f -local forevery C ν by the Dong’s lemma (Lemma 4.9). Then, by the Goddard uniquenesstheorem (Theorem 4.1), we obtain (4.3). (cid:3) RREGULAR VERTEX ALGEBRAS 25
As an easy consequence, we obtain the following:
Corollary 4.11.
The composition e − f ( z − w ; λ,µ ) | z | > | w | Y ( | λ i , z ) Y ( | µ i , w ) can be restrictedto z = w . Moreover, the restriction equals to ◦◦ Y ( | λ i , z ) Y ( | µ i , z ) ◦◦ = Y ( | λ + µ i , z ) . Irregular Heisenberg vertex operator algebras
In this section, following the ideas of [NS10], we shall define the irregular vertexalgebras for the Heisenberg vertex algebra.5.1.
Heisenberg vertex algebra.
Let us briefly recall the definition of Heisenbergvertex algebra to fix the notation. Let F denote the Z ≥ -graded vector space ofgraded polynomial ring C [ x n ] n> of variables x n with deg x n = n ∈ Z > . Let | i ∈ F denote the unit of C [ x n ] n> .Define an endomorphism T as the derivation on C [ x n ] n> with T x n = nx n +1 .Fix a non-zero complex number κ . Let a − n (resp. a n ) denote the multiplication of x n , (resp. the derivation 2 κn∂ x n ) for n >
0. Set a := 0 ∈ End( F ). The powerseries a ( z ) = P n ∈ Z a n z − n − defines an field on F .Define Y ( · , z ) : F →
End( F )[[ z ± ]] by Y ( a − n a − n · · · a − n k | i , z ) := ◦◦ ∂ ( n − z a ( z ) · · · ∂ ( n k − z a ( z ) ◦◦ (5.1)for n , . . . , n k ∈ Z > , k >
0. We also set Y ( | i , z ) = id. Then, the tuple F κ := ( F , | i , T, Y ( · , z ))is known to be a vertex algebra called Heisenberg vertex algebra .5.2.
Coherent state module.
Fix a positive integer r . Let S := Spec ( C [ λ j ] rj =1 )be the space of internal parameters with deg λ j = − j .Consider the completion F := Q n ≥ F n , where F n is the degree n -part of F . Thetensor product F S := F ⊗ C O S is a Z -graded O S -module whose degree n -part isgiven by F S,n = Y j ≥ F n + j ⊗ O S, − j , where F n + j = 0 for n + j <
0. It is also considered as a D S -module in an obviousway. We also set End( F ) S,m := Y k ≥ End( F ) m + k ⊗ O S, − k for m ∈ Z and End( F ) S := L m End( F ) S,m . Lemma 5.1.
Every element in
End( F ) S defines an endomorphism on F S . RREGULAR VERTEX ALGEBRAS 26
Proof.
Fix n, m ∈ Z . We have an natural morphism from F S,n ⊗ End( F ) S,m to F S,n + m as follows: F S,n ⊗ End( F ) S,m ≃ ∞ Y ℓ =0 M j,k ≥ j + k = ℓ ( F n + j ⊗ End( F ) m + k ) ⊗ ( O S, − j ⊗ O S, − k ) → ∞ Y ℓ =0 F n + m + ℓ ⊗ O S, − ℓ = F n + m This gives the conclusion. (cid:3)
Set ϕ ( r ) λ := 12 κ r X j =1 λ j a − j j ∈ End( F ) S, . Corollary 5.2.
The exponential Φ ( r ) λ := exp (cid:16) ϕ ( r ) λ (cid:17) = ∞ X n =0 n ! (cid:16) ϕ ( r ) λ (cid:17) n defines an automorphism on F S . Definition 5.3.
We set F ( r ) := Φ ( r ) λ ( F ⊗ O S ) ⊂ F S , and | coh i := Φ ( r ) λ ( | i ⊗ F ( r ) , | coh i ) is equipped with the structure of non-singular coherent state F κ -module over S . Lemma 5.4.
We have h κn∂ λ n , Φ ( r ) λ i = a − n Φ ( r ) λ for n = 1 , . . . , r . By this lemma, we obtain the following.
Corollary 5.5. F ( r ) is a D S -submodule of F S . Lemma 5.6.
As endomorphisms on F S , we have the commutation relation h a n , Φ ( r ) λ i = ( λ n (0 < n ≤ r )0 ( otherwise ) . Proof.
By definition, we have h ϕ ( r ) λ , a n i = − r X j =1 λ j δ j,n id F S . Hence, we obtain Φ ( r ) λ a n (cid:16) Φ ( r ) λ (cid:17) − = ∞ X k =0 k ! (cid:16) ad ϕ ( r ) λ (cid:17) k a n = a n − r X j =1 λ j δ j,n id F S RREGULAR VERTEX ALGEBRAS 27
This implies the lemma. (cid:3)
Corollary 5.7.
We have F ( r ) = M n ≥ n ≥···≥ n k > O S a − n · · · a − n k | coh i as an O S -module. It also follows from Lemma 5.6 that a n ( n ∈ Z ) acts on F ( r ) . We can define Y F ( r ) ( · , z ) : F −→
End D S ( F ( r ) )[[ z ± ]]in a way similar to (5.1). Proposition 5.8.
The tuple F ( r ) κ := (cid:0) F ( r ) , Y F ( r ) ( · , z ) , | coh i (cid:1) is a non-singular co-herent state F -module, which is an envelope of F κ . Irregular vertex algebra structure.Lemma 5.9.
Let F • ( F ( r ) ) be the decreasing filtration on F ( r ) defined by F k ( F ( r ) n ) := Φ ( r ) λ M j ≥ k F n + j ⊗ O S, − j . Then, ( F ( r ) , F • ) is a filtered small lattice of F ( r ) i.e. satisfies the conditions inDefinition 3.1.Proof. The condition (L) requires nothing since H is empty. Since F ( F ( r ) ) equalsto F ( r ) , condition (F1) holds. The conditions (F2), (F4) (resp. (F3), (F5)) are thecorollaries of Lemma 5.6 (resp. Lemma 5.4). (cid:3) Lemma 5.10.
We have an isomorphism
End( F ) S [[ z ± ]] n ≃ Y k ≥ End( F )[[ z ± ]] k + n ⊗ O S, − k . Proof.
We haveEnd( F ) S [[ z ± ]] n ≃ Y ℓ ∈ Z End( F ) S,n + ℓ z ℓ ≃ Y ℓ ∈ Z Y k ≥ End( F n + k + ℓ ) ⊗ O S, − k z ℓ ≃ Y k ≥ Y ℓ ∈ Z End( F n + k + ℓ ) z ℓ ! ⊗ O S, − k This proves the lemma. (cid:3)
RREGULAR VERTEX ALGEBRAS 28
Define F S and End( F ) S by replacing S with S in the definition of F S andEnd( F ) S , respectively. We can replace S with S in Lemma 5.1 and Lemma 5.10.We shall consider the following extension of Y ( · , z ): Definition 5.11.
Define the morphism of D S -modules Y ( · , z ) : F S −→ End( F ) S [[ z ± ]](5.2)by Y ( P ∞ k =0 A n + k ⊗ P − k ( λ, µ ) , z ) := P ∞ k =0 Y ( A k + n , z ) ⊗ P − k ( λ, µ ) for each homo-geneous element P ∞ k =0 A n + k ⊗ P − k ( λ, µ ) ∈ F S ,n .We identify F S λ with a subspace of F S λ,µ along with the S λ -axis. Then D S λ -module structures on F S λ and F S λ,µ are compatible. We also regards the coherentstate module F ( r ) λ ⊂ F S λ defined in Definition 5.3 as a subspace of F S λ,µ throughthe above identification. We do the same thing above for µ .We note that the pull back F ( r ) λ + µ := σ ∗ λ + µ F ( r ) (see Section 3.1) is a D S λ,µ -submodule of F S λ,µ by definition. The completion F ( r ) λ + µ of F ( r ) λ + µ is naturally iden-tified with F S λ,µ . By Lemma 5.1, we can restrict Y ( · , z ) to F ( r ) λ : Y ( · , z ) : F ( r ) λ −→ Hom D Sµ (cid:16) F ( r ) µ , F ( r ) λ + µ (cid:17) [[ z ± ]](5.3)and Y ( · , z ) is a D S λ -module morphism by definition.Set ϕ ( r ) λ ( z ) ± = 12 κ r X n =1 λ n ∂ ( n − z a ( z ) ± n . Lemma 5.12.
There are equalities Y ( | λ i , z ) = exp (cid:16) ϕ ( r ) λ ( z ) + (cid:17) exp (cid:16) ϕ ( r ) λ ( z ) − (cid:17) , and Y ( a − n · · · a − n k | λ i , z ) = ◦◦ Y F ( r ) ( a − n , z ) · · · Y F ( r ) ( a − n k , z ) Y ( | λ i , z ) ◦◦ for k, n , . . . , n k ∈ Z > .Proof. By definition, we have | λ i = ∞ X j =0 j ! κ r X k =1 λ k a − k k ! j | i = X ( j k ) rk =1 ∈ Z ⊕ r ≥ r Y k =1 κ ) j k λ j k k a j k − k j k ! k j k ! | i RREGULAR VERTEX ALGEBRAS 29 and hence Y ( | λ i , z ) = X ( j k ) rk =1 ∈ Z ⊕ r ≥ r Y k =1 κ ) j k λ j k k j k ! ! Y r Y k =1 a j k − k k j k | i , z ! = X ( j k ) rk =1 ∈ Z ⊕ r ≥ r Y k =1 λ j k k j k ! ! ◦◦ r Y k =1 (cid:18) κ Y (cid:16) a − k k , z (cid:17)(cid:19) j k ◦◦ = exp (cid:16) ϕ ( r ) λ ( z ) + (cid:17) exp (cid:16) ϕ ( r ) λ ( z ) − (cid:17) . This proves the first equality. The second equality can also be proved similarly. (cid:3)
Note that we have ϕ ( r ) λ ( z ) + | z =0 = ϕ ( r ) λ and ϕ ( r ) λ ( z ) + | i = 0. Corollary 5.13.
For any A λ ∈ F ( r ) λ , we have Y ( A λ , z ) | i ∈ F ( r ) λ [[ z ]] , and Y ( A λ , z ) | i| z =0 = A λ . Lemma 5.14.
Put f κ ( z ; λ, µ ) := 12 κ X ≤ p,q ≤ r (cid:18) p + qp (cid:19) ( − p +1 p + q λ p µ q z p + q . We have e − f κ ( z ; λ,µ ) Y ( | λ i , z ) | µ i ∈ F ( r ) λ + µ [[ z ]] , and e − f κ ( z ; λ,µ ) Y ( | λ i , z ) | µ i| z =0 = | λ + µ i .Moreover, for any A λ ∈ F ( r ) λ , Y ( A λ , z ) is an irregular field on F ( r ) µ with theirregularity f κ ( z ; λ, µ ) .Proof. Since we have [ ∂ ( p − z a ( z ) + , a − q ] = 0 and h ∂ ( p − z a ( z ) − , a − q i = 2 κ ( p + q − p − q − − p − z p + q id(5.4)for p, q ∈ Z > , we have [ ϕ ( r ) λ ( z ) + , ϕ ( r ) µ ] = 0 and h ϕ ( r ) λ ( z ) − , ϕ ( r ) µ i = f κ ( z ; λ, µ ) · id F S . Using this equality, we obtain that Y ( | λ i , z ) | µ i = exp (cid:16) ϕ ( r ) λ ( z ) + (cid:17) exp (cid:16) ϕ ( r ) λ ( z ) − (cid:17) e ϕ ( r ) µ | i = e f κ ( z ; λ,µ ) e ϕ ( r ) µ exp (cid:16) ϕ ( r ) λ ( z ) + (cid:17) | i We obtain the first statement by Corollary 5.13.The latter statement can be proved by using Corollary 5.7, Lemma 5.12 and(5.4). (cid:3)
RREGULAR VERTEX ALGEBRAS 30
Lemma 5.15.
There is an equality e − f κ ( z − w ; λ,µ ) | z | > | w | Y ( | λ i , z ) Y ( | µ i , w )= exp (cid:16) ϕ ( r ) λ ( z ) + + ϕ ( r ) µ ( w ) + (cid:17) exp (cid:16) ϕ ( r ) λ ( z ) − + ϕ ( r ) µ ( w ) − (cid:17) . Proof.
We have [ ∂ ( p − z a ( z ) + , ∂ ( q − w a ( w ) + ] = [ ∂ p − z a ( z ) − , ∂ ( q − w a ( w ) − ] = 0,[ ∂ ( p − z a ( z ) − , ∂ ( q − w a ( w ) + ] = 2 κ ( p + q − p − q − − p − ( z − w ) p + q | | z | > | w | idwhere ( z − w ) − p − q | | z | > | w | denotes the expansion in positive powers of w/z . Hencewe obtain h ϕ ( r ) λ ( z ) − , ϕ ( r ) µ ( w ) + i = f κ ( z − w ; λ, µ ) | | z | > | w | . By the Baker-Campbell-Hausdorff formula, we obtain the lemma. (cid:3)
Since [ T, Φ ( r ) λ ] = P rj =1 λ j a − j − Φ ( r ) λ , T naturally acts on F ( r ) . Theorem 5.16.
The tuple (cid:16) F ( r ) κ , ( F ( r ) , F • ) , Y ( · , z ) , T, f κ (cid:17) is an irregular vertexalgebra for F κ .Proof. Let us check the axioms in Definition 3.12. The translation axiom and thecompatibility condition are trivial by the construction. The vacuum axiom is provedin Corollary 5.13. The irregular field axiom and coherent state axiom are proved inLemma 5.14. It remains to prove the irregular locality axiom.By Lemma 5.15, Y ( | λ i , z ) and Y ( | µ i , w ) are mutually f κ -local. Then, by thecompatibility, we can apply the Dong’s lemma (Lemma 4.9), to obtain the f κ -localityin general. (cid:3) Conformal structures.
In this subsection, we shall show that F ( r ) is anirregular vertex operator algebra if κ = 1 /
2. Recall that the space S = Spec C [ λ j ] rj =1 is equipped with the Der ( C [[ t ]])-structure as explained in Example 2.2. We firstlyprove the following: Lemma 5.17.
The irregularity f κ ( z ; λ, µ ) is conformal (see Definition 3.14).Proof. We have[ D µj , f κ ( z ; λ, µ )] = 12 κ X k,ℓ> ( − k − (cid:18) k + ℓ − ℓ − (cid:19) λ k µ ℓ + j z k + ℓ . Since ∂ ( m +1) z z j +1 = (cid:0) j +1 m +1 (cid:1) z j − m for m ≥ −
1, we have ∂ ( m +1) z z j +1 [ D λm , f κ ( z ; λ, µ )]= 12 κ X p,q> ( − p − (cid:18) p + q − p − (cid:19)(cid:18) j + 1 m + 1 (cid:19) λ p + m µ q z p + q − j + m . RREGULAR VERTEX ALGEBRAS 31
Hence the coefficient of λ u µ v /z w ( u, v, w >
0) with u + v = j + w in (3.6) is givenby ( − u − (cid:18) w − v − j − (cid:19) + u X s =0 ( − s (cid:18) v + ss (cid:19)(cid:18) j + 1 u − s (cid:19) . Hence we need to show( − u (cid:18) w − u (cid:19) = u X s =0 ( − s (cid:18) v + ss (cid:19)(cid:18) j + 1 u − s (cid:19) . The left hand side of this equation is the coefficients of x u in (1 + x ) − ( w − u ) , andthe right hand side is that of x u in (1 + x ) − ( v +1) (1 + x ) j +1 = (1 + x ) j − v . Hence weobtain the lemma. (cid:3) Let the complex number κ be 1 /
2. Take a complex number ρ , and put c =1 − ρ . It is known that ω ρ := a − + ρa − is a conformal vector of the Heisenbergvertex algebra F . Let h k ( k = 0 , . . . , r ) (and hence L k ) as in Section 2.4 with λ = 0. For the simplicity of the notation, we denote Y F ( r ) ( ω ρ , z ) = X n ∈ Z L n z − n − . Lemma 5.18.
For s ≥ , we have h L s − L s , Φ ( r ) λ i = h s Φ ( r ) λ + s X k =0 λ k Φ ( r ) λ a s − k . Proof.
By Lemma 5.6, we have h L s , Φ ( r ) λ i = s X k =0 a k a s − k + X p> a − p a s + p − ρ ( s + 1) a s , Φ ( r ) λ = 12 s X k =0 [ a k a s − k , Φ ( r ) λ ] + X p> a − p [ a s + p , Φ ( r ) λ ] − ρ ( s + 1)[ a s , Φ ( r ) λ ]= h s Φ ( r ) λ + s X k =0 λ k Φ ( r ) λ a s − k + X p> a − p λ s + p Φ ( r ) λ On the other hand, by Lemma 5.4 we have h L s , Φ ( r ) λ i = X p> pλ s + p ∂ λ p , Φ ( r ) λ = X p> λ s + p a − p Φ ( r ) λ . This proves the lemma. (cid:3)
RREGULAR VERTEX ALGEBRAS 32
Corollary 5.19.
For k ∈ Z ≥ , we have L k | λ i = L k | λ i (5.5) L − k | λ i = r X j =1 λ j a − j − k | λ i + ρ ( k + 1) a − k | λ i + 12 k X ℓ =1 a − ℓ a − k + ℓ | λ i . (5.6) Proposition 5.20.
The irregular Heisenberg vertex algebra F ( r ) for κ = 1 / is anirregular vertex operator algebra.Proof. Lemma 2.8 shows that ρ F ( r ) ; t k +1 ∂ t
7→ − ( L k − L k ) is a Lie algebra homo-morphism.For v k + d ∈ F k + d and f k ( λ ) ∈ O S, − k , by Lemma 5.18, we have( L − L )(Φ ( r ) λ ( v k + d ⊗ f k ( λ )))= Φ ( r ) λ ( L v k + d ⊗ f k ( λ )) − Φ ( r ) λ ( v k + d ⊗ L f k ( λ ))= ( k + d )Φ ( r ) λ ( v k + d ⊗ f k ( λ )) − k Φ ( r ) λ ( v k + d ⊗ f k ( λ ))= d Φ ( r ) λ ( v k + d ⊗ f k ( λ ))This proves that L − L acts as the grading operator on F ( r ) .By Lemma 5.18, for any v λ ∈ F ⊗ O S , we have( L s − L s )Φ ( r ) λ ( v λ ) = [ L s − L s , Φ ( r ) λ ] v λ + Φ ( r ) λ ( L s ( v λ ) − L s ( v λ ))= Φ ( r ) λ s − X k =1 λ s − k a k + L s − D s ! v λ . Since s − X k =1 λ s − k a k + L s − D s is locally nilpotent on F ⊗ O S , we can deduce that L s − L s is locally nilpotent. (cid:3) Irregular Virasoro vertex operator algebras
We shall give a definition of irregular Virasoro vertex algebra via the free fieldrealization.6.1.
Saturated vertex subalgebras.
Let ( V , Y, f ) be a non-singular irregular ver-tex operator algebra for a vertex operator algebra V on S . In particular, a filtration F • ( V ) with the properties (F1)-(F4) is fixed. Let W ⊂ V be a vertex operator sub-algebra of V . Let U ⊂ V be the smallest D S -submodule which contains | coh i ∈ V and is closed under the operation A V ( n ) for every A ∈ W and every n ∈ Z . Definition 6.1.
The vertex operator subalgebra W ⊂ V is saturated with respectto V if U is a coherent state W -module with singularity H and U ( ∗ H ) = V ( ∗ H ). RREGULAR VERTEX ALGEBRAS 33 If W is saturated, set U ◦ := V ⊂ U ( ∗ H ) and F • ( U ◦ ) := F • ( V ). Then we havethe following lemma: Lemma 6.2. (cid:0) U , ( U ◦ , F • ) , Y, f (cid:1) is an irregular vertex operator algebra for W .Proof. The only non-trivial point is that ( U , Y U , ( U ◦ , F • )) is an envelope of W . Inother words, we need to show that the morphismΨ W : W −→ U O | ◦ = U O / ( U O ∩ m S, V ) , defined in Definition 3.11, is an isomorphism. Since V is non-singular, we have anisomorphism Ψ V : V −→ V / m S, V . Since Ψ W is the restriction of Ψ V , it is injective.It remains to prove that Ψ W is surjective. Since Ψ V is an isomorphism of V -modules, we have A V ( n ) | coh i ∈ m S, V (6.1)for A ∈ V and n ≥
0. By the construction, a section of U O can be expressed as an O S -linear combination of the sections of the form A , ( n ) · · · A k, ( n k ) | coh i (6.2)for some A , . . . , A k ∈ W , and n ≤ · · · ≤ n k ∈ Z . If n k ≤ −
1, then (6.2) is theimage of A , ( n ) · · · A k, ( n k ) | i by Ψ W : W → U O , A A ( − | coh i . If n k ≥
0, then by (6.1), the class of (6.2) in U O | ◦ is zero. Hence we obtain the lemma. (cid:3) Irregular Virasoro vertex algebra via free field realization.
Recall thatVir c denotes the Virasoro vertex algebra (Section 2.3). Let F be the Heisenberg ver-tex algebra with κ := 1 /
2. The irregular Heisenberg algebra F ( r ) is also consideredin the case κ = 1 / c → F given by the conformal vector ω = a − + ρa − for a complex number ρ and c = 1 − ρ . We assume that c is generic so that wehave Vir c ⊂ F . Definition 6.3.
Let Vir ( r ) c denote the smallest D S -submodule of F ( r ) which isclosed under all operations of the form A F ( r ) ( n ) for A ∈ Vir c and n ∈ Z .Let M ( r ) c, denote the coherent state module defined in Section 2.4 with λ = 0.The relation between Vir ( r ) c and M ( r ) c, is given by the following proposition: RREGULAR VERTEX ALGEBRAS 34
Proposition 6.4.
We have a unique morphism M ( r ) c, −→ F ( r ) (6.3) of D S ⊗ C U (Vir) -modules such that the coherent state of M ( r ) c, maps to that of F ( r ) .The morphism (6 . is injective and the image of (6 . coincides with Vir ( r ) c .Proof. By Corollary 5.5, there is a unique morphism D S ⊗ C U (Vir) −→ F ( r ) whichsends 1 ⊗ | λ i . By (5.5), the above morphism uniquely induces the morphism(6.3). On the one hand, by Remark 2.11, the morphism (6.3) is injective at eachpoint λ o = ( λ o , . . . , λ or ) with λ or = 0. On the other hand, since M ( r ) c, is a free O S -module (see (2.4)), the kernel of (6.3) should be torsion free. Hence we have thatthe kernel is the zero module, which means that (6.3) is injective. The coincidenceof the image with Vir ( r ) c follows from the minimality in the definition of Vir ( r ) c . (cid:3) Corollary 6.5.
Vir ( r ) c is a coherent state Vir c -module with singularity { λ r = 0 } . The following theorem is the main theorem of this section.
Theorem 6.6.
The Virasoro vertex algebra
Vir c is saturated in the Heisenbergvertex algebra F with respect to the irregular vertex algebra F ( r ) .Proof. We have already checked that Vir ( r ) c is coherent state Vir c -module with sin-gular divisor H = { λ r = 0 } . Hence it remains to show that Vir ( r ) c ( ∗ H ) = F ( r ) ( ∗ H ).We shall prove this by showing the following proposition inductively on n :( P n ) Every section a − n · · · a − n k | λ i with n j > ≤ j ≤ k ) and P kj =1 n k ≤ n isin Vir ( r ) c ( ∗ H ).The proposition ( P ) is trivial since | λ i ∈ Vir ( r ) c . Assume that ( P n − ) holds. Let a − n · · · a − n k | λ i be an arbitrary section with P kj =1 n j = n . If 0 < n ≤ r , then a − n · · · a − n k | λ i = n ∂∂λ n a − n · · · a − n k | λ i ∈ Vir ( r ) c ( ∗ H ) . RREGULAR VERTEX ALGEBRAS 35 If n > r , consider the action of L − n + r on a − n · · · a − n k | λ i : L − n + r a − n · · · a − n k | λ i = k X j =2 a − n · · · a − n j − [ L − n + r , a − n j ] a − n j +1 · · · a − n k | λ i + a − n · · · a − n k L − n + r | λ i = k X j =2 a − n · · · a − n j − n j a − n j − n + r a − n j +1 · · · a − n k | λ i + r X ℓ =1 λ ℓ a − n + r − ℓ + ρ ( n − r − a − n + r ! a − n · · · a − n k | λ i + 12 n − r X i =1 a − n · · · a − n k a − i a − n + r + i | λ i . Each term other than λ r a − n a − n · · · a − n k | λ i is in Vir ( r ) c ( ∗ H ) by ( P n − ). We alsohave L − n + r a − n · · · a − n k | λ i ∈ Vir ( r ) c ( ∗ H ). Hence we obtain that the element a − n · · · a − n k | λ i is in Vir ( r ) c ( ∗ H ). This proves ( P n ) and hence the theorem. (cid:3) By the Lemma 6.2, we can define the irregular Virasoro vertex algebra:
Definition 6.7.
We set V ir ( r ) c := F ( r ) , which is considered as a filtered small latticeof Vir ( r ) c ( ∗ H ). The irregular vertex operator algebraVir ( r ) c := (cid:16) Vir ( r ) c , (cid:16) V ir ( r ) c , F • (cid:17) , Y, f (cid:17) is called an irregular Virasoro vertex algebra . Remark 6.8.
The quotient M c,h = ( M ( r ) c,h ) O / (cid:16) ( M ( r ) c,h ) O ∩ m S, M ( r ) c,h (cid:17) is isomorphic to the usual Verma module for the Virasoro algebra, while the quotient(Vir ( r ) c ) O | ◦ = (Vir ( r ) c ) O / (cid:16) (Vir ( r ) c ) O ∩ m S, F ( r ) (cid:17) is isomorphic to Vir c via Ψ Vir c . Remark 6.9.
By the Theorem 6.6, at least theoretically, we can describe the vertexoperators Y ( A λ , z ) for A λ ∈ V ir ( r ) c only in terms of the Virasoro algebra and thecoherent states after the localization, although the computation is very complicated RREGULAR VERTEX ALGEBRAS 36 in practice. For example, in the case r = 1 and ρ = 0, we have Y ( | λ i , z ) | µ i = e λµ/z (cid:18) λa − z + (cid:18) λ a −
2! + λa − (cid:19) z + · · · (cid:19) | λ + µ i = e λµ/z (cid:18) λλ + µ L − z ++ 1( λ + µ ) (cid:18) λ L −
2! + λµ (cid:18) L − − L − L λ + µ ) (cid:19)(cid:19) z + · · · (cid:19) | λ + µ i , where we put λ = λ and µ = µ . Concluding remarks
The key point of our construction in Section 5 was the Baker-Campbell-Hausdorffformula used in Lemma 5.15. We expect that our way of constructing an irregularvertex algebra used in Section 5 can be easily generalized to the vertex algebrasgenerated by finitely many free fields in the sense of [Kac98, Definition] althoughwe have only treated the case where the vertex algebra is generated by a free field,for simplicity (and to give a canonical conformal structure).Then, we also expect that Lemma 6.2 in Section 6 (or its generalization) wouldbe useful in the construction of irregular versions of Kac-Moody vertex algebrasand W -algebras. In other words, we expect that we may define irregular versionsof V k ( g ) and W ( g ) via their respective free field realizations. We shall refer [Nag15,Nag18] and [GLP, KMST13] as studies on irregular conformal blocks for V k ( sl )and W -algebra, respectively. The details of these expectations would be given inthe subsequent studies. References [AGT10] L. F. Alday, D. Gaiotto, and Y. Tachikawa. Liouville correlation functions from four-dimensional gauge theories.
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Department of Mathematics, Graduate School of Science, Osaka University, Toy-onaka, Osaka 560-0043, Japan
E-mail address : [email protected] RREGULAR VERTEX ALGEBRAS 38
Kavli Institute for the Physics and Mathematics of the Universe (WPI),The Uni-versity of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa,Chiba 277-8583, Japan
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