CHENG Jun-fang,CHU Yan-jun

(1.School of Mathematics and Statistics,Henan University,Kaifeng,475004,China;2.Institute of Contemporary Mathematics,Henan University,Kaifeng,475004,China)

Abstract:The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one.In this paper,we first study the variety of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra,which is a finite set consisting of two nontrivial elements.Based on this property,we also show that the twisted Heisenberg-Virasoro vertex operator algebra is a tensor product of two vertex operator algebras.Moreover,associating to properties of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra,we charaterized twisted Heisenberg-Virasoro vertex operator algebras.This will be used to understand the classifi cation problems of vertex operator algebras whose varieties of semi-conformal vectors are finite sets.

Key words:Twisted Heisenberg-Virasoro algebra;Vertex operator algebra;Semi-conformal vector;Semi-conformal subalgebra

§1.Introduction

The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one,which has been first studied by Arbarello etal,in Ref.[3].It contains the classical Heisenberg algebra and the Virasoro algebra as subalgebras.And they also have established a connection between the second cohomology of certain moduli spaces of curves and the second cohomology of the Lie algebra of differential operators of order at most one.

The representation theory of the twisted Heisenberg-Virasoro algebra is closely related to those of other Lie algebras,such as the Virasoro algebra and toroidal Lie algebras,and has been studied in Refs.[1,4-6,14,19-20,26-27].In Ref.[4],the free field realization of the twisted Heisenberg-Virasoro algebra at level zero is given and its applications can be obtained.In Ref.[2],A.Alexandrov constructed new relations connecting Kontsevich-Witten tau-functions,Hodge integrals and Hurwitz numbers and derived linear constraints for all of them.These constraints as operators form a twisted Heisenberg-Virasoro algebra.

It’s well known that the vertex operator algebra theory provides a rigorous mathematical foundation for two dimensional conformal field theory and string theory from the Hamiltonian point of view(Refs.[23,28]).It follows from Proposition 3.1 in Ref.[6]that the twisted Heisenberg-Virasoro vertex operator algebra has a vertex operator algebra structure which is the tensor product of a Virasoro vertex operator algebra and a Heisenberg vertex operator algebra.In Refs.[8-9],we used their semi-conformal vectors to describe Heisenberg vertex operator algebras and affine vertex operator algebras(began from Refs.[21-22]).[8,Theorem 1.1]tells that the set of all semi-conformal vectors of a vertex operator algebra V= ⊕n∈ZVnforms a Zarisk closed subset(or,an affine algebraic variety)in the weight-two subspace V2.For the twisted Heisenberg-Virasoro vertex operator algebra,we find it has only two nontrivial semiconformal vectors.Thus,we can also see easily that the twisted Heisenberg-Virasoro vertex operator algebra is a tensor product of two vertex operator algebras.Based on the variety of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra,we describe such a class of vertex operator algebras.In general,for a simple CFT-type vertex operator algebra(V,ω),if its variety Sc(V,ω)of semi-conformal vectors contains only finite nontrivial elements with the some conditions,then(V,ω)is isomorphic to a twisted Heisenberg-Virasoro vertex operator algebra.Actually,this result shows a characterization of twisted Heisenberg-Virasoro vertex operator algebras.

In further work,we shall understand the properties of some class of vertex operator algebras whose varieties of semi-conformal vectors are finite sets,which will lead to classifying vertex operator algebras by properties of their varieties of semi-conformal vectors from a geometric viewpoint.

Notation:C is the complex number field;R is the real number field;Z is the set of all integer numbers;N is the set of all non-negative integer numbers;Z+is the set of all positive integer numbers.

§2.The Vertex Operator Algebra Associated to the Twisted Heisenberg-Virasoro Algebra

In the section,we shall review the vertex operator algebra associated to the twisted Heisenberg-Virasoro algebra.You can refer to the Refs.[3,4,6]for more details.

Let C[t±1]be the ring of Laurent polynomials with the variable t.Denoted the Lie algebra of derivations on C[t±]by Der(C[t±1]).Let Ln=-,n ∈ Z.Then Der(C[t±1])has a basis{Ln|n ∈ Z}.Let A be the universal central extension of abelian Lie algebra C[t±]with a basis{tn,Ch|∈Z}.The twisted Heisenberg-Virasoro algebra is the universal central extension of the semi-direct product Lie algebra Der(C[t±1])A,denoted by HV.The Lie algebra HV has a basis

The non-trivial Lie bracket relations are as follows

where m,n∈Z.

For convenience,we write tnas bn,HV has a Z-graded structure

where for n/=0,HV(n)=CLn⊕Cbnfor n/=0 and HV(0)=SpanC{L0,b0,Cv,Ch,C}.So HV has a triangle decomposition

where HV+= ⊕n＞0HV(n);HV-= ⊕n＞0HV(-n).

Let C be a 1-dimensional HV+⊕ HV(′)-module as follows

Then we get the induced HV-module

M(h,h1,cv,ch,c)is Z-graded by eigenvalues of the operator

with M(h,h1,cv,ch,c)n={v∈M(h,h1,cv,ch,c)|L(0)v=(n+h)v}.

Lemma 2.1[2]Let ch=0,and c/=0.

Denoted by V(cv,ch,c)=M(0,0,cv,ch,c).Denoted by 1=1?1.Let I be the HV-submodule of V(cv,ch,c)generated by L-11.Then we consider the quotient module V(cv,ch,c)=V(cv,ch,c)/I.And it has a basis

V(cv,ch,c)has a unique maximal proper submodule,so it has an unique irreducible quotient which is denoted by L(cv,ch,c).We can defines a N-graded structure on V(cv,ch,c)as follows

From above the relations,we have the OPE relations

Corollary 2.2 There are the following OPE relations

Theorem 2.3[4]V(cv,ch,c)is a N-graded vertex operator algebra with the conformal vector L-21 and the central charge cvand are generated strongly by{1,L(z),b(z)}.

According to the Lemma 2.1(b),we have the following results

Corollary 2.4 For ch=0,c/=0,the HV-module M(0,0,ch,0,c)possesses a singular vector L(-1)1 in M(0,0,cv,0,c)1.So the factor-module

is a simple vertex operator algebra.

Let Hchbe the Heisenberg vertex operator algebra with the level chgenerated by{bn,Ch|n∈Z{0}}.It follows from Proposition 3.1 in Ref.[6]that

Proposition 2.5 If ch/=0,the vertex operator algebra V(cv,ch,c)is isomorphic to the tensor product V(c′v,0)? Hchof a Virasoro vertex operator algebras V(c′v,0)with the central charge c′vand Hch,whereis the conformal vector of Hch.

§3.Semi-conformal Vectors of the Vertex Operator Algebra V(cv,ch,c)

In this section,let(V,Y,1,ω)(Abbrev.(V,ω))be a Z-graded vertex operator algebra(Refs.[15,23,28]for details).We shall review basic notions and results associated with semi-conformal vectors for a vertex operator algebra V.This content can be seen in Refs.[14-15]

3.1 First,we review the commutant of a vertex algebra.It’s well-known as the coset construction in conformal field theory(Refs.[17-18]).

Definition 3.1[7,18,23,25]Let W be a vertex algebra,and U be any subset of W.The commutant of U in W is defined by

Remark 3.2 Obviously,1∈CW(U).Furthermore,CW(U)is a vertex subalgebra of W.And we also have CW(U)=CW(

Remark 3.3 In a VOA(V,ω),let(U,ω′)be a subalgebra of V.If CV(Cv(U))=U,we say(U,CV(U))forms a Howe pair in V(Refs.[7,25]).According to the conclusions in Refs.[18,23],a subalgebra U can be realized as a commutant subalgebra of V if and only if(U,CV(U))forms a Howe pair in V.

3.2 For two given vertex algebras(V,YV)and(W,YW)a homomorphism f:V→W of vertex algebras satisfies

If(V,ωV)and(W,ωW)are two VOAs with conformal vectors ωVand ωW,respectively,then f is said to be conformal if f(ωV)= ωW.We say f is semi-conformal if f?LV(n)=LW(n)?f,for all n ≥ -1.Let(V,ωV)be a VOA and a vertex subalgebra of(W,ωW).We say V is a conformal subalgebra(or subVOA)if ωW= ωV,i.e,they have the same conformal vector.If the inclusion from V to W is semi-conformal,then V is called a semi-conformal subalgebra of W and ωVis called a semi-conformal vector of W.

For a VOA(W,ωW)with the conformal vector ωW,let

Lemma 3.4[8]A vector ω′∈ W is a semi-conformal vector of(W,ωW)if and only if it satisfies the following conditions

Let(W,ωW)be a general Z-graded vertex operator algebra.The set Sc(W,ωW)forms an affine algebraic variety([8,Theorem 1.1]).In fact,a semi-conformal vector ω′∈ W can be characterized by algebraic equations of degree at most 2 as described in[8,Proposition 2.2].The algebraic variety Sc(W,ωW)has also a partial order(See[8,definition 2.7]),and this partial order can be characterized by algebraic equations in[8,Proposition 2.8].

Proposition 3.5 If ch/=0,then Sc(V(c,ch,cv),ω)={0,ω′,ω - ω′,ω},where ω′=Moreover,there are two longest partial order chain in Sc(V(c,ch,cv),ω)such as follows

Proof Note that the weight-two subspace of V(c,ch,cv)is spanned by{ω=L(-2)1,b(-1)21,b(-2)1}.Set ω′=xb(-1)21+yb(-2)1+zL(-2)1,where x,y,z ∈ C.According to the Lemma 3.4,we have ω′∈ Sc(V(c,ch,cv),ω)if and only if x,y,z satisfy that

Equivalently,

So we have nontrival solutions:xthere are only two nontrival semi-conformal vectors

With respect to the partial orderof[8,definition 2.7],we have two longest partial order chain in Sc(V(c,ch,cv),ω)such as follows

Remark 3.6 For each ω′∈ Sc(W,ωW),it determines a unique dual pair(CW(CW(< ω′＞)),CW(< ω′＞))as semi-conformal subalgebras of(W,ωW)in the sense of Howe duality in VOA theory.Let(V,ωV)be a semi-conformal subalgebra of(W,ωW).Then(V,ωV)has a unique maximal conformal extension(CW(CW(V)),ωV)in(W,ωW)in the sense that if(V,ωV) ?(U,ωV),then(U,ωV)? (CW(CW(V)),ωV)(see[23,Corollary 3.11.14]).

Lemma 3.7 Let(V,ω)be a N-graded vertex operator algebra with V0=C1 and the conformal vector ω.If ω′∈ Sc(V,ω),then CV(< ω′＞)? CV(CV(< ω′＞))is a conformal subalgbra of V,where< ω′＞ is the Virasoro VOA generated by ω′in V.

Proof We know that L′(n)=0 on CV(< ω′＞)and L(n)=L′(n)on CV(CV(< ω′＞))for n ≥ -1,then CV(< ω′＞)∩CV(CV(< ω′＞))=C1.So CV(< ω′＞)?CV(CV(< ω′＞))is a conformal subalgebra of V.

Theorem 3.8 For ch/=0,the Heisenberg-Virasoro vertex operator algebra V(cv,ch,c)is isomorphic to the tensor product V(c′v,0)? Hchof the simple Virasoro VOA V(c′v,0)and the Heisenberg VOA Hchwith the conformal vector

Proof By Remark 3.6,we note that the maximal semi-conformal subalgebra with the conformal vector ω′is the Heisenberg VOA Hchin V(cv,ch,c),i.e.,CV(cv,ch,c)(CV(cv,ch,c)(< ω′＞))Hch.By Lemma 3.7,we know that CV(cv,ch,c)(< ω′＞))? Hchis a subVOA of V(cv,ch,c).And since< ω-ω′＞ ?Hch? CV(cv,ch,c)(< ω′＞))?Hch,then CV(cv,ch,c)(< ω′＞))?Hchas a subVOA of V(cv,ch,c)has at less two generators{ω - ω′,b(-1)1},where b(-1)1 generates Hch.We know V(cv,ch,c)is also generated by two vectors{b(-1)1,ω}and CV(cv,ch,c)(< ω′＞))∩ Hch=C1,then CV(cv,ch,c)(< ω′＞))=< ω - ω′＞=V(c′v,0)and V(cv,ch,c) ～=V(c′v,0) ? Hch,when

Lemma 3.9[23]Let V be a simple vertex operator algebra and U be any vertex operator subalgebra(with the same conformal vector ω),for example,U=< ω ＞.Then the vertex subalgebra

In particular,

Lemma 3.10[21]Let(V′,Y′,1′,ω′),(V′′,Y′′,1′′,ω′′)be two vertex operator algebras.Then there are

In particular,if V′is simple vertex operator algebra,then

According to above Lemma 3.10,3.11,we have

2)If there exists coprime integers p,q ≥ 2 such thatfor ch/=0,the vertex operator algebra V(cv,ch,c)has a unique simple quotient L(cv,ch,c)=L(c′v,0) ? Hch.

§4.The Characterization of Twisted Heisenberg-Virasoro Vertex Operator Algebras

In this section,according to the properties of twisted Heisenberg-Virasoro vertex operator algebras,we characterize this class of vertex operator algebras by semi-conformal vectors.

Let V be a simple N-graded vertex operator algebra with V0=C1.Such V is also called a simple CFT type vertex operator algebra(Refs.[10-11]).If V satisfies the further condition that L(1)V1=0,it is of strong CFT type.Li has shown(Ref.[24])that such a vertex operator algebra V has a unique non-degenerate invariant bilinear form<,＞up to a multiplication of a nonzero scalar.In particular,the restriction of<,＞to V1endows V1with a non-degenerate symmetric invariant bilinear form

Lemma 4.1 Let(U,ωU)and(V,ωU)be two semi-conformal subalgebras of the VOA(W,ωW).If(U,ωU)is a conformal extension of(V,ωU)in(W,ωW),then

1)

2)

Proof Since(U,ωU)is a conformal extension of(V,ωU)in(W,ωW),then CW(V)is a conformal extension of CW(U)in(W,ωW)and they are both semi-conformal subalgebras with the conformal vector ωW- ωU.According to Refs.[12,16],we know that there is a unique maximal conformal extension for a semi-conformal subalgebra(S,ωS),which is realized as the double commutant(CW(CW(S)))of(S,ωS)in(W,ωW)in the sense that if(S,ωS) ?(T,ωS),then(T,ωS) ? (CW(CW(S)),ωS).So CW(CW(CW(V)))=CW(CW(CW(U))).Since CW(CW(CW(S)))=CW(S)for a general subalgebra S of W,then we have CW(V)=CW(U);

According to the definition of semi-conformal vectors of W,the assert 2)is obvious.

Lemma 4.2 Let(V,ω)be a Z-graded vertex operator algebra and(U,ω′)be a vertex subalgebra of V.Then ω′∈ Sc(V,ω)if and only if Sc(U,ω′)? Sc(V,ω).

Proof Since ω′∈ Sc(V,ω),then(U,ω′)is a semi-conformal subalgebra of V.For any ω′′∈ Sc(U,ω′),we have L′′(n)=L′(n)on W for n ≥ -1,where(W,ω′′)is a semi-conformal subalgebra of U.Since ω′∈ Sc(V,ω),then we have L(n)=L′(n)on U for n ≥ -1.So we have L(n)=L′′(n)on W for n ≥ -1.Hence ω′′∈ Sc(V,ω).

If Sc(U,ω′)? Sc(V,ω),it is obvious that ω′∈ Sc(V,ω).

Lemma 4.3[8]Let(V,ω)be a nondegenerate simple CFT type vertex operator algebra generated by V1.Let(V′,ω′)and(V′′,ω′′)be two vertex operator subalgebras with possible different conformal vectors.Assume that(V,ω)=(V′,ω′) ? (V′′,ω′′)is a tensor product of vertex operator algebras(see[12,Section 3.12]).Then

1)(V′,ω′)and(V′′,ω′′)are semi-conformal subalgebras and both are also non-degenerate simple CFT type;

2)V1=V′1?1′′⊕1′?V′′1,is an orthogonal decomposition with respect to the non-degenerate

symmetric bilinear form 〈·,·〉on V1;

3)[V1′?1′′,1′?V1′′]=0 with the Lie bracket[·,·]on V1;

4)Sc(V′,ω′) ? 1′′,1′? Sc(V′′,ω′′),and Sc(V′,ω′) ? 1′′+1′? Sc(V′′,ω′′)are subsets of Sc(V,ω);

5)For each~ω′∈Sc(V′,ω′),we have

and

Lemma 4.4 For a simple CFT type VOA(V,ω),if V=V1? V2and(V1,ω1)and(V2,ω2)are vertex operator subalgebras of V,then

1)CV(< ω1＞)=CV(CV(<ω2＞))=V2and CV(< ω2＞)=CV(CV(< ω1＞))=V1;

2)When Sc(V,ω)={0,ω1,ω2,ω},we have Sc(V1,ω1)={0,ω1}and Sc(V2,ω2)={0,ω2}.

Proof First,we note that ω = ω1+ω2.Since L1(n)=0 on V2and L2(n)=0 on V1,so L(n)=L1(n)on V1and L(n)=L2(n)on V2for n ∈ Z,that is ω1,ω2∈ Sc(V,ω).

According to Lemma 3.10,we know that CV(V1)=CV(CV(V2))=V2and CV(V2)=CV(CV(V1))=V1.Since there exists a unique maximal semi-conformal subalgebra of V for each ω′∈ Sc(V,ω),which can be realized as the double commutant subalgebra containing ω′as the conformal vector,then we have CV(< ω1＞)=CV(CV(< ω2＞))=V2and CV(< ω2＞)=CV(CV(<ω1＞))=V1.

When Sc(V,ω)={0,ω1,ω2,ω},since V=V1? V2,then V1,V2are both semi-conformal subalgebras of V.By Lemma 4.2,we know that Sc(V1,ω1)={0,ω1}and Sc(V2,ω2)={0,ω2}.

For a CFT-type VOA(V,ω),we know that V1forms a Lie algebra with the bracket operation[u,v]=u(0)v for u,v∈V1.

Lemma 4.5 For a non-degenerate CFT-type vertex operator algebra V=V1?V2,where(V1,ω1)and(V2,ω2)are subVOAs of V,if Sc(V,ω)={0,ω1,ω2,ω},then either V11=0 or V12=0.

Proof Since V=V1?V2,by Lemma 4.3 1),we have V1=V11⊕V12and V11is orthogonal to V12in V1.If V11/=0 and V12/=0,we take h1∈V11,h2∈V12such that

Theorem 4.6 Assume that(V,ω)is a simple non-degenerate CFT type vertex operator algebra and be generated strongly by the subspace V1⊕V2,where V1/=0 is an abelian Lie algebra as the weight-one subspace and V2is the weight-two subspace with dimV2=1.If Sc(V,ω)={0,ω′,ω′′,ω}and V=CV(< ω′＞)? CV(< ω′′＞),then(V,ω)is isomorphic to a simple twisted Heisenberg-Virasoro vertex operator algebra.

Proof Assume that< ω′＞ and< ω′′＞ have central charges c′,c′′as Virasoro vertex operator algebras,respectively.At first,since V=CV(< ω′＞)?CV(< ω′′＞),we note that ω′′= ω-ω′and CV(< ω′′＞)=CV(CV(< ω′＞)).By Lemma 4.3 2),we have V1=CV(< ω′＞)1⊕ CV(< ω′′＞)1and CV(< ω′＞)1is orthogonal to CV(< ω′′＞)1in V1.By Lemma 4.5,we know that either CV(< ω′＞)1=0 or CV(< ω′′＞)1=0.We can assume that CV(< ω′＞)1=0,then CV(< ω′′＞)1=V1.

Since V1is an abelian Lie algebra,then V1generates a simple Heisenberg VOA MV1(c′)in V and CV(< ω′′＞)=MV1(c′),where c′is the central charge of MV1(c′).According to the condition Sc(V,ω)={0,ω′,ω′′,ω}and the results of Ref.[15],we know that dimV1=1.Note that V is simple,then CV(< ω′′＞)and CV(< ω′＞)are both simple.On the other hand,since CV(< ω′＞)1=0 and dimV2=1,then CV(< ω′＞)=< ω′′＞,where< ω′′＞ is the simple Virasoro VOA with the central charge c′′.Finally,according to Theorem 3.8,we obtain that V is isomorphic to the twisted Heisenberg vertex operator algebraorfor some c∈ C as two cases in Corollary 3.12.

The twisted Heisenberg-Virasoro vertex operator algebra has two nontrivial semi-conformal vectors and it is also a tensor product of two vertex operator algebras.Such information will lead us to study the classification of VOAs with two nontrivial semi-conformal vectors in further work.

Remark 4.7 According to our present study,we know that some basic simple CFT type vertex operator algebras have no nontrivial semi-conformal vectors as follows

·M(1)(Ref.8),which is the Heisenberg vertex operator algebra with the rank 1 generated by =Ch;

· L(?,0),which is the simple Virasoro vertex operator algebra with the central charge?/=0(Ref.[29]);

· K(sl2,?),which is the parafermion vertex operator algebra with the level ?/=1(Refs.[12-13]).

It is interesting problem for us that the classification of vertex operator algebras without nontrivial semi-conformal vectors.Moreover,based on Theorem 4.6,we conjecture that for a vertex operator algebra(V,ω)with two nontrivial semi-conformal vectors,it should contain a conformal vertex operator subalgebra which is a tensor product of two vertex operator algebras without nontrivial semi-conformal vectors up to isomorphism.In fact,we expect to classify vertex operator algebras with two nontrivial semi-conformal vectors by tensor decompositions of vertex operator algebras.