Non-Abelian Quantum Hall States and their Quasiparticles: from the Pattern of Zeros to Vertex Algebra
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Non-Abelian Quantum Hall States and their Quasiparticles:from the Pattern of Zeros to Vertex Algebra
Yuan-Ming Lu, Xiao-Gang Wen,
2, 3
Zhenghan Wang, and Ziqiang Wang Department of Physics, Boston College, Chestnut Hill, MA 02467, USA Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2J 2Y5, Canada Microsoft Station Q, CNSI Bldg. Rm 2237, University of California, Santa Barbara, CA 93106 (Dated: Oct. 2009)In the pattern-of-zeros approach to quantum Hall states, a set of data { n ; m ; S a | a =1 , ..., n ; n, m, S a ∈ N } (called the pattern of zeros) is introduced to characterize a quantum Hallwave function. In this paper we find sufficient conditions on the pattern of zeros so that thedata correspond to a valid wave function. Some times, a set of data { n ; m ; S a } corresponds toa unique quantum Hall state, while other times, a set of data corresponds to several differentquantum Hall states. So in the latter cases, the patterns of zeros alone does not completely char-acterize the quantum Hall states. In this paper, We find that the following expanded set of data { n ; m ; S a ; c | a = 1 , ..., n ; n, m, S a ∈ N ; c ∈ R } provides a more complete characterization of quantumHall states. Each expanded set of data completely characterizes a unique quantum Hall state, atleast for the examples discussed in this paper. The result is obtained by combining the pattern ofzeros and Z n simple-current vertex algebra which describes a large class of Abelian and non-Abelianquantum Hall states Φ sc Z n . The more complete characterization in terms of { n ; m ; S a ; c } allows us toobtain more topological properties of those states, which include the central charge c of edge states,the scaling dimensions and the statistics of quasiparticle excitations. ContentsI. Introduction
II. Pattern-of-zeros approach to generic FQHstates a variables: the Pattern of Zeros 5C. Consistent conditions for the Pattern of Zeros 51. Translational invariance 52. Symmetry condition 53. Concave conditions 64. n -cluster condition 65. Summary 6D. Label the pattern of zeros by h sc a III. Constructing FQH wave functions from Z n vertex algebras Z n vertex algebra 7B. Relation between ˜ h sc a and h sc a h sc a and C a,b from theassociativity of vertex algebra 9 IV. Examples of generic FQH states described bythe Z n vertex algebra n = 1 case 10B. n = 2 case 10C. n = 3 case 11D. n = 4 case 11E. Including conditions (49) and (50) 11F. Summary 11 V. Z n simple-current vertex algebra Z n simple-current vertex algebra 12B. Consistent conditions from useful GJI’s 131. { A, B, C } = { ψ a , ψ b , ψ c } , a + b, b + c, a + c = 0 mod n { A, B, C } = { ψ a , ψ b , ψ − b } , a ± b = 0 mod n { A, B, C } = { ψ a , ψ a , ψ − a } , a = n/ n { A, B, C } = { ψ n/ , ψ n/ , ψ n/ } , n = even 14 VI. Representing quasiparticles in Z n simple-current vertex algebra { k sc γ ; a ; Q γ } { A, B, C } = { ψ a , ψ b , σ γ + c } , a + b = 0 mod n { A, B, C } = { ψ a , ψ − a , σ γ + c } VII. Examples of FQH states described by Z n simple-current vertex algebras Z n simple-current vertexalgebra 18B. The Z n parafermion vertex algebra: Z n parafermion states with { M k = 0; p = 1; m = 2 } Z parafermion state 22D. Quasiparticles in the Z parafermion state 23 E. Z n | Z n series: { M k = 0; p = 2 } Z | Z state 24G. Quasiparticles in the Z | Z state 25H. The Z | Z | Z state 27I. Gaffnian: a non unitary Z example 27J. The Z | Z state 28K. C n | C n series with { m ; h sc1 , · · · , h sc n − } = { n ; 2 , · · · , } VIII. Summary Acknowledgments A. Other ways to label the pattern of zeros { a j } { M k ; p ; m } { M k ; p ; m } labeling scheme 32b. Consistent conditions on { M k ; p ; m } B. Consistent conditions on the commutationfactor µ AB C. Determine the quasiparticle commutationfactor µ γ,a from the quasiparticle pattern ofzeros { k sc γ ; a } D. Generalized Jacobi Identity
E. Associativity of Z n vertex algebra and newconditions on h sc a and C ab A = ψ a , B = ψ b , C = ψ c ) 38b. Summary of new consistent conditions fromGJI 38 F. Subleading terms in OPE of a Z n simple-current vertex algebra and moreconsistent conditions { A, B, C } = { ψ a , ψ b , ψ c } , a + b , b + c , a + c , a + b + c = 0 mod n { A, B, C } = { ψ a , ψ b , ψ − a − b } , a, b, a + b = 0 mod n { A, B, C } = { ψ a , ψ b , ψ − b } , a ± b = 0 mod n { A, B, C } = { ψ a , ψ a , ψ − a } , a = n/ n { A, B, C } = { ψ n/ , ψ n/ , ψ n/ } , n =even 41f. { A, B, C } = { ψ a , ψ b , σ γ + c } , a + b = 0 mod n { A, B, C } = { ψ a , ψ − a , σ γ + c } References I. INTRODUCTION
Materials can have many different forms, which is par-tially due to the very rich ways in which atoms and elec- trons can organize. The different organizations corre-spond to different phases of matter (or states of matter).It is very important for physicists to understand these dif-ferent states of matter and the phase transitions betweenthem. At zero-temperature, the phases are described bythe ground state wave functions, which are complex wavefunctions Φ( r , r , · · · , r N ) with N → ∞ variables. Somathematically, to describe zero-temperature phases, weneed to characterize and classify the ground state wavefunctions with ∞ variables, which is a very challengingmathematical problem.For a long time we believe that all states of matterand all phase transitions between them are character-ized by their broken symmetries and the associated or-der parameters . A general theory for phases and phasetransitions is developed based on this symmetry breakingpicture. So within the paradigm of symmetry breaking,a many-body wave function is characterized by its sym-metry properties. Landau’s symmetry breaking theoryis a very successful theory and has dominated the the-ory of phases and phase transitions until the discovery offractional quantum Hall (FQH) effect .FQH states cannot be described by symmetry breakingsince different FQH states have exactly the same sym-metry. So different FQH states must contain a new kindof order. The new order is called topological order and the associated phase called topological phase, be-cause their characteristic universal properties (such asthe ground states degeneracy on a torus ) are invari-ant under any small perturbations of the system. Un-like symmetry-breaking phases described by local orderparameters, a topological phase is characterized by a pat-tern of long-range quantum entanglement . In Ref. 10,the non-Abelian Berry phases for the degenerate groundstates are introduced to systematically characterize andclassify topological orders in FQH states (as well as othertopologically ordered states). In this paper, we furtherdevelop another systematic characterization of the topo-logical orders in FQH states based on the pattern of zerosapproach. In the strong magnetic field limit, a FQH wave functionwith filling factor ν < { z i = x i + i y i } (except for a common factor that depends on geome-try: say, a Gaussian factor exp (cid:16) P i | z i | (cid:17) for a planargeometry). After factoring out an anti-symmetric fac-tor of Q i Within the pattern-of-zero approach, two questionsnaturally arise: (1) Does any pattern of zeros, ie an arbi-trary integer sequence { n ; m ; S a } corresponds to a sym-metric polynomial Φ( z , · · · , z N )? Are there any “illegal”patterns of zeros that do not correspond to any symmet-ric polynomial? (2) Given a “legal” pattern of zeros,can we construct a corresponding FQH many-body wavefunction? Is the FQH many-body wave function uniquelydetermined by the pattern of zeros?For question (1), it turns out that the pattern of ze-ros must satisfy some consistent conditions in orderto describe an existing symmetric polynomial. In otherwords, some sequences { n ; m ; S a } don’t correspond toany symmetric polynomials. However, Ref. 11,12 onlyobtain some necessary conditions on the pattern of zeros { n ; m ; S a } . We still do not have a set of sufficient condi-tions on pattern of zeros that guarantee a pattern of zerosto correspond to an existing symmetric polynomial.For the question (2), right now, we do not have an effi-cient way to obtain corresponding FQH many-body wavefunction from a “legal” pattern of zeros. Further more,while some patterns of zeros can uniquely determine theFQH wave function, it is known that some other pat-terns of zeros cannot uniquely determine the FQH wavefunction: ie in those cases, two different FQH wave func-tions can have the same pattern of zeros. This meansthat, some patterns of zeros do not provide complete in-formation to fully characterize FQH states. In this caseit is important to expand the data of pattern of zeros toobtain a more complete characterization of FQH states.We see that the above two questions are actuallyclosely related. In this paper, we will try to address those questions. Motivated by the conformal field theory(CFT) construction of FQH wave functions, we will try to use the patterns of zeros to define andconstruct vertex algebras (which are CFTs). Since thecorrelation function of the electron operator in the con-structed vertex algebra gives us the FQH wave function,once the vertex algebra is obtained from a pattern ofzeros, we effectively find the corresponding FQH wavefunction for the pattern of zeros. In this way, we estab-lish the connection between the pattern of zeros and theFQH wave function through the vertex algebra.In order for the correlation of electron operators in thevertex algebra to produce a single-valued electron wavefunction with respect to electron variables { z , · · · , z N } ,electron operators need to satisfy a so-called “simple-current” property (see eqn. (34) and eqn. (60)). Alsothe vertex algebra need to satisfy the generalized Ja-cobi identity (GJI) which guarantees the associativity ofthe corresponding vertex algebra. We find that only acertain set of patterns of zeros can give rise to simple-current vertex algebras that satisfy the GJI. So the GJIsin simple-current vertex algebras give us a set of sufficientconditions on a pattern of zeros so that this pattern ofzeros does correspond to an existing symmetric polyno-mial.In this paper, we first try to use the pattern of zeros { n ; m ; S a } to define a Z n vertex algebra. From some ofthe GJI of the Z n vertex algebra, we obtain more neces-sary conditions on the pattern of zeros { n ; m ; S a } thanthose obtained in Ref. 11,12 (see section III). It is notclear if those conditions are actually sufficient or not.Then, we try to use the pattern of zeros { n ; m ; S a } to define a Z n simple-current vertex algebra. From thecomplete GJI of the Z n simple-current vertex algebra,we obtain sufficient conditions on the pattern of zeros { n ; m ; S a } (see section V). B. How to expand the pattern-of-zeros data tocompletely characterize the topological order If a pattern of zeros { n ; m ; S a } can uniquely describethe topological order in a quantum Hall ground state,then from such a quantitative description, we shouldbe able to calculate the topological properties fromthe data { n ; m ; S a } . Indeed, this can be done. Firstdifferent types of quasiparticles can also be quantita-tively described and labeled by a set of sequences { S γ ; a } that can be determined from the pattern-of-zeros data { n ; m ; S a } . Those quantitative characterizations of thequantum Hall ground state and quasiparticles allow usto calculate the number of different quasiparticle types,quasiparticle charges, fusion algebra between the quasi-particles, and topological ground state degeneracy on aRiemann surface of any genus. However, from the pattern-of-zeros data, { n ; m ; S a } and { S γ ; a } , we still do not know how to calculate thequasiparticle statistics and scaling dimensions, as well asthe central charge c of the edge states. This difficultyis related to the fact that some patterns of zeros do notuniquely characterize a FQH state. Thus one cannot ex-pect to calculate the topological properties of FQH statefrom the pattern-of-zeros data alone in those cases.In this paper, we will try to solve this problem. Wefirst introduce a more complete characterization for FQHstates in terms of a expanded data set: { n ; m ; S a ; c } .Then, we use the data set { n ; m ; S a ; c } to define a socalled Z n simple-current vertex algebra. The Z n simple-current vertex algebra contain a subalgebra, Virasoro al-gebra, generated by the energy-momentum tensor T and c is the central charge of the Virasoro algebra. It containsonly n primary fields ψ a , a = 0 , , ..., n − Z n fusion rule ψ a ψ b ∼ ψ ( a + b ) mod n , ψ n = ψ . Those ψ a are called simple currents. The ex-tra data c is the one of the structure constants of the Z n simple-current vertex algebra. One may want to in-clude all the structure constants { C ab } in the data setto have a complete characterization. But for the ex-amples discussed in this paper, we find that data set { n ; m ; S a ; c } already provides a complete characteriza-tion. So in this paper, we will use { n ; m ; S a ; c } to char-acterize FQH states. If later we find that { n ; m ; S a ; c } isnot sufficient, we can always add additional data, suchas C ab . Every Z n simple-current vertex algebra uniquelydefine a FQH state, and the data { n ; m ; S a ; c } that de-fines a Z n simple-current vertex algebra also completelycharacterizes a FQH state.We would like to remark that although the data { n ; m ; S a ; c } and the corresponding Z n simple-currentvertex algebras describe a large class of FQH states, theydo not describe all FQH states. For example let Φ A i be the FQH wave function described by a Z n i simple-current vertex algebra A i , i = 1 , 2. Then, in general,the FQH state described by the product wave functionΦ = Φ A Φ A cannot be described by a simple-currentvertex algebra. Such a product state is described by theproduct vertex algebra A ⊗ A , which is in general nolonger a simple-current vertex algebra. So a more generalFQH state should have a formΦ = Y i Φ A i . (1)The study in Ref. 11,12,13 reveal that many FQH statesdescribed by pattern of zeros have the following formΨ( { z i } ) = Y a Φ Z ( ka ) na ( { z i } ) (2)where Φ Z ( ka ) na ( { z i } ) is the wave function described by Z ( k a ) n a parafermion vertex algebra. The Z n simple-current vertex algebra mentioned above is a natural gen-eralization of the Z ( k a ) n a parafermion vertex algebra, andeqn. (1) naturally generalizes eqn. (2). (Note that thereare many Z n simple-current vertex algebras even for afixed n , so there are many different Z n simple-currentstates.) For the subclass of FQH states described by Z n simple-current vertex algebra (which includes Virasoro algebraas an essential part), the quasiparticle statistics and scal-ing dimensions, as well as the central charge c of theedge states can be calculated from the data { n ; m ; S a ; c } .Certainly, we can also calculate the number of differentquasiparticle types, quasiparticle charges, fusion algebrabetween the quasiparticles, and topological ground statedegeneracy on a Riemann surface of any genus.Obviously, not every collection { n ; m ; S a ; c } corre-sponds to a Z n simple-current vertex algebra and aFQH state. GJIs of the Z n simple-current vertex alge-bra generate the consistent conditions on the data set { n ; m ; S a ; c } . Only those data sets { n ; m ; S a ; c } that sat-isfy the GJIs can describe a Z n simple-current vertexalgebra and FQH states.For some patterns of zeros { n ; m ; S a } , we find thatthey uniquely define the vertex algebras by completelydetermining the structure constants c and C ab . So thosepatterns of zeros completely specify the correspondingFQH wave functions. While for other patterns of ze-ros, we find that they cannot uniquely define the ver-tex algebras. For those patterns of zeros, many differ-ent sets of structure constants can satisfy the GJIs forthe same set of pattern of zeros. This corresponds tothe situation where there are many different FQH wavefunctions that share the same pattern of zeros. In thiscase, the pattern of zeros does not completely charac-terize FQH wave functions. We need additional datato completely characterize quantum Hall wave functions.Here we choose to add the structure constant c of theVirasoro algebra (which is the central charge) and use { n ; m ; S a ; c | a = 1 , ..., n ; n, m, S a ∈ N ; c ∈ R } to charac-terize FQH states. For all the examples that we con-sidered in this paper, the data { n ; m ; S a ; c } completelydetermine the simple-current vertex algebra. C. The organization of the paper This paper is organized as follows. In section II, we re-view and extend the pattern-of-zeros approach to quan-tum Hall states. In section III we use the pattern of zerosto define Z n vertex algebra, and then use associativityconditions ( ie the GJIs) of the vertex algebra to obtainextra conditions on the pattern of zeros that describegeneric FQH states. In section IV we list some numeri-cal solutions of the pattern of zeros for the generic FQHstates that also satisfy those extra consistent conditionsfound in section III. In section V we define and constructthe Z n simple-current vertex algebra from the pattern ofzeros. We list the consistent conditions obtained fromGJIs of Z n simple-current vertex algebra. The detailedderivations of those consistent conditions are discussed inappendix D, E and F. The consistent conditions on thepatterns of zeros that describe a Z n simple-current vertexalgebra are more restrictive than those for a generic Z n vertex algebra. Some of the solutions of the Z n simple-current pattern of zeros are listed in section VII. In sec-tion VI, we discuss how to represent quasiparticles inthe Z n simple-current vertex algebra, and to calculatethe topological properties of quasiparticles from the Z n simple-current pattern of zeros. In section VII, we ap-ply the vertex-algebra approach developed here to studysome simple (but non-trivial) examples of FQH states,which include Z n parafermion states (the Read-Rezayistates ), the Z n simple-current FQH states of Z n | Z n type, a Z simple-current FQH state of Z | Z type, etc. II. PATTERN-OF-ZEROS APPROACH TOGENERIC FQH STATES In this section, we will review how to use the patternof zeros to characterize and classify different FQH statesthat have one component. A discussion on two-component FQH states can be find in Ref. 26. A. FQH wave functions and symmetricpolynomials Generally speaking, to classify generic complex wavefunctions Φ( r , · · · , r N ) is not even a well-defined prob-lem. Fortunately, under a strong magnetic field, elec-trons are spin-polarized in the lowest Landau level (LLL)when the electron filling fraction ν e is less than 1. Thewave function of a single electron in LLL (we set mag-netic length l B = p ~ /eB to be unity hereafter) isΨ m ( z ) = z m e −| z | / in a planar geometry. m is the an-gular momentum of this single particle state. Thus themany-body wave function of spin-polarized electrons inthe LLL should beΨ e ( z , · · · , z N ) = ˜Φ e ( z , · · · , z N )e − P Ni =1 | zi | (3)where ˜Φ e ( { z i } ) is an anti-symmetric holomorphic polyno-mial of electron coordinates { z i = x i + i y i } . The electronfilling fraction ν e is defined as: ν e = lim N →∞ NN φ = lim N →∞ N N p (4)where N φ is the total number of flux quanta piercingthrough the sample, and N p is the total degree of poly-nomial ˜Φ e ( { z i } ). For FQH states ν e < 1, we can extracta Jastraw factor Q i In this section, we will review and summarize the con-sistent conditions on { S a } derived in Ref. 11,12. 1. Translational invariance A translational invariant wave functionΦ( z , · · · , z N ) = Φ( z − z, · · · , z N − z ) satisfiesΦ(0 , z , · · · , z N ) = 0. As a result we have S = 0. 2. Symmetry condition After we fuse a variables together to form an a -particlecluster ( a -cluster), it is natural to ask: what happenswhen we fuse an a -cluster and another b -cluster together?Let D a,b be the order of zeros obtained by fusing an a -cluster and another b -cluster together. It satisfies D a,b = D b,a ≥ 0. Since the final state is the same as fusing a + b variables together, we find an one-to-one relationbetween the two sets of data D a,b and S a D a,b = S a + b − S a − S b S a = a − X i =1 D i, (9)Since Φ( { z i } ) is a symmetric polynomial, it describesa state of bosonic particles seated at coordinates { z i } .Thus the a -cluster seated at z ( a ) i can also be regarded asa bosonic particle. The derived polynomial (see P S a ineqn. (7) as an example) should be symmetric with respectto interchange of two identical bosons seated at z ( a )1 and z ( a )2 . When we fuse such two identical bosonic clusters z ( a )1 and z ( a )2 together, we havelim z ( a )1 → z ( b )2 P ( z ( a )1 , z ( a )2 , · · · )= ( z ( a )1 − z ( a )2 ) D a,a ˜ P ( z ( a )1 + z ( a )2 , · · · )+ O (( z ( a )1 − z ( a )2 ) D a,a +1 ) (10)This leads to the symmetry condition D a,a = even ⇔ S a = even . (11) 3. Concave conditions The 1st concave condition is the non-negativity of D a,b D a,b ≥ ⇔ S a + b ≥ S a + S b (12)It comes naturally from the fusion of two clusters.When we fuse three clusters together, we find the totalorder of “off-particle” zeros to be∆ ( a, b, c ) = D a,b + c − D a,b − D a,c ≥ ( a, b, c ) = (14) S a + b + c + S a + S b + S c − S a + b − S a + c − S b + c ≥ n -cluster condition The above conditions eqn. (11), eqn. (12), andeqn. (14) have many solutions { S a } . Many of those solu-tions has a “periodic” structure that the whole sequence { S a } can be determined from first a few terms: S a + kn = S a + kS n + mn k ( k − kma, (15) where m ≡ D n, . (16)We will call such a pattern of zeros the one that satis-fies an n -cluster condition. We see that, for an n -clustersequence, only the first n terms, S , ..., S n +1 , are inde-pendent, and the whole sequence is determined by thefirst n terms.To understand the physical meaning of the n -clustercondition, we note that eqn. (15) is equivalent to thefollowing condition∆ ( kn, b, c ) = 0 for any k. (17)This means that a symmetric polynomial that satisfiesthe n -cluster condition has the following defining prop-erty: as a function of the n -cluster coordinate z ( n ) , thederived polynomial P ( z ( n ) , z ( a )1 , · · · ) has no off-particlezeros.Under the n -cluster condition, we see that D n,n = nm = even (18)We also note that the filling fraction is given by ν = nm (19)since S a → mn a as a → ∞ .We like to mention that the cluster condition plays avery important role in the Jack polynomial approach toFQH. However, in the pattern of zeros approach,the n -cluster condition only play a role of grouping andtabulating solutions of the consistent conditions. The so-lutions with larger n correspond to more complex wavefunctions which usually correspond to less stable FQHstates. Later, we will discuss the relation between thepattern of zeros and CFT. We find that the solutions thatdo not satisfy the n -cluster condition ( ie with n = ∞ )correspond to irrational CFT, which may always corre-spond to gapless FQH states. The Jack polynomial ap-proach and the pattern-of-zeros approach have some closerelations. The Jack polynomials are special cases of thepolynomials characterized by pattern of zeros. 5. Summary To summarize, we see that the pattern of zeros foran n -cluster polynomial is described by a set of positiveintegers { n ; m ; S , ..., S n } . Introducing S = 0 and S a + kn = S a + kS n + mn k ( k − kma (20)which define S n +1 , S n +2 , ..., we find that the data { n ; m ; S , ..., S n } must satisfy D a,b = S a + b − S a − S b ≥ D a,a = even (21)∆ ( a, b, c ) (22)= S a + b + c + S a + S b + S c − S a + b − S a + c − S b + c ≥ a, b, c = 1 , , · · · .The conditions (21) and (22) are necessary conditionsfor a pattern of zeros to represent a symmetric polyno-mial. Although eqn. (21) and eqn. (22) are very simple,they are quite restrictive and are quite close to be suffi-cient conditions. In fact if we add an additional condition∆ ( a, b, c ) = even (23)the three conditions (21), (22), and (23) may even be-come sufficient conditions for a pattern of zeros to rep-resent a symmetric polynomial. However, this con-dition is too strong to include many valid symmetricpolynomials such as Gaffnian, a nontrivial Z state dis-cussed in detail in section VII. We will obtain some ad-ditional conditions in section III C, which combined with(21) and (22) form a set of necessary and (potentially)sufficient conditions for a valid pattern of zeros. D. Label the pattern of zeros by h sc a In this section, we will introduce a new labeling schemeof the pattern of zeros. We can label the pattern of zerosin terms of h sc a = S a − aS n n + am − a m n . (24)This labeling scheme is intimately connected to the ver-tex algebra approach that we will discuss later.The n -cluster condition (20) of S a implies that h sc a isperiodic h sc0 = 0 , h sc a = h sc a + n (25)The two sets of data { n ; m ; S , ..., S n } and { n ; m ; h sc1 , ..., h sc n − } has a one-to-one correspondence,since S a = h sc a − ah sc1 + a ( a − m n . (26)We can translate the conditions on { m ; S a } to theequivalent conditions on { m ; h sc a } . First, we have2 nS a = 2 nh sc a − nah sc1 + a ( a − m = 0 mod 2 nnS a = nh sc2 a − nah sc1 + a (2 a − m = 0 mod 2 nm > , mn = even (27) nS n = 0 mod 2 n in eqn. (27) leads to 2 nh sc1 + m =0 mod 2, from which we see that 2 nh sc1 is an integer.From 2 nh sc a − a (2 nh sc1 ) + a ( a − m = even integer, wesee that 2 nh sc a are always integers. Also 2 nh sc2 a are alwayseven integers, and 2 nh sc2 a +1 are either all even or all odd.Since h sc n = 0, thus when n = odd, 2 nh sc a are all even.Only when n = even, can 2 nh sc2 a +1 either be all even or all odd. When m =even, 2 nh sc2 a +1 are all even. When m =odd, 2 nh sc2 a +1 are all odd.The two concave conditions become h sc a + b − h sc a − h sc b + abmn = D ab = integer ≥ h sc a + b + c − h sc a + b − h sc b + c − h sc a + c + h sc a + h sc b + h sc c = ∆ ( a, b, c ) = integer ≥ { n ; m ; h sc1 , ..., h sc n − } can be obtained bysolving eqn. (25), eqn. (27), eqn. (28), and eqn. (29).Choosing 1 ≤ a, b < a + b ≤ n in eqn. (29), we have0 ≤ ∆ ( a, b, n − a − b )= ( h sc n − a − b − h sc a + b ) − ( h sc n − a − h sc a ) − ( h sc n − b − h sc b )= − ∆ ( n − a, n − b, a + b ) ≤ reflection condition on { h sc a } : h sc a − h sc n − a = a ( h sc1 − h sc n − ) = 0 (30)From (30) we see that partially solving conditions (29)reduces the number of independent variables characteriz-ing a pattern of zeros from n − { S , · · · , S n } to [ n ] in { h sc1 , · · · , h sc[ n ] } . However, being a sequence of fractionsrather than integers, { h sc a } labeling scheme imposessome difficulty in numerically solving conditions (25),(27), (28), and (29). In Appendix (A 1) and (A 2) we willfurther use consistent conditions (29) to introduce twoschemes labeling the pattern of zeros with a sequence ofnon-negative integers or half-integers. They turn out tobe quite efficient for numerical studies, since consistentconditions (25), (27), (28), and (29) can be reducedto a much smaller set after introducing a new labelingscheme { M k ; p ; m } as in Appendix A 2. In particu-lar, this { M k ; p ; m } labeling scheme is the same one asadopted in the literature of parafermion vertex algebra. III. CONSTRUCTING FQH WAVE FUNCTIONSFROM Z n VERTEX ALGEBRAS If we use { n ; m ; h sc a } to characterize n -cluster symmet-ric polynomial Φ( { z i } ), the conditions (27), (28), and(29) are required by the single-valueness of the symmet-ric polynomial. Or more precisely, eqns. (27), (28), and(29) come from a simple requirement that the zeros inΦ( { z i } ) all have integer orders. However, the conditions(27), (28), and (29) are incomplete in the sense that somepatterns of zeros { n ; m ; h sc a } can satisfy those conditionsbut still do not correspond to any valid polynomial. A. FQH wave function as a correlation function in Z n vertex algebra To find more consistent conditions, in the rest of thispaper, we will introduce a new requirement for the sym-metric polynomial. We require that the symmetric poly-nomial can be expressed as a correlation function in avertex algebra. More specifically, we have Φ( { z i } ) = lim z ∞ →∞ z h N ∞ h V ( z ∞ ) N Y i =1 V e ( z i ) i (31)where V e ( z ) is an electron operator and V ( ∞ ) representsa positive background to guarantee the charge neutralcondition. This new requirement, or more precisely, theassociativity of the vertex algebra, leads to new condi-tions on h sc a .The electron operator has the following form V e ( z ) = ψ ( z ) : e i φ ( z ) / √ ν : (32)where : e i φ ( z ) / √ ν : (:: stands for normal ordering, whichis implicitly understood hereafter) is a vertex operator ina Gaussian model. It has a scaling dimension of ν andthe following operator product expansion (OPE) e i aφ ( z ) e i bφ ( w ) = ( z − w ) ab e i( a + b ) φ ( w ) + O (cid:16) ( z − w ) ab +1 (cid:17) (33)The operator ψ is a primary field of Virasoro algebraobeys an quasi-Abelian fusion rule ψ a ψ b ∼ ψ a + b + ..., ψ a ≡ ( ψ ) a . (34)where ... represent other primary fields of Virasoro al-gebra whose scaling dimensions are higher than that of ψ a + b by some integer values. We believe that the integraldifference of the scaling dimensions is necessary to pro-duce a single-valued correlation function(see eqn. (31)).Let ˜ h sc a be the scaling dimension of the simple current ψ a . Therefore the a -cluster operator V a ≡ ( V e ) a = ψ a ( z )e i aφ ( z ) / √ ν (35)has a scaling dimension of h a = ˜ h sc a + a ν (36)The vertex algebra is defined through the followingOPE of the a -cluster operators V a ( z ) V b ( w ) = C Va,b V a,b ( w )( z − w ) h a + h b − h a + b + O (cid:16) ( z − w ) h a + b − h a − h b +1 (cid:17) . (37)where C Va,b are the structure constants. However, theabove OPE is not quite enough. To fully define the ver-tex algebra, we also need to define the relation between V a ( z ) V b ( w ) and V b ( w ) V a ( z ).The correlation functions is calculated throughthe expectation value of radial-ordered operator product. The radial-ordered operator product isdefined through( z − w ) α VaVb R [ V a ( z ) V b ( w )]= ( ( z − w ) α VaVb V a ( z ) V b ( w ) , | z | > | w | µ ab ( w − z ) α VaVb V b ( w ) V a ( z ) , | z | < | w | (38)where α V a V b = h a + h b − h a + b . (39)Note that the extra complex factor µ ab is introduced inthe above definition of radial order. In the case of stan-dard conformal algebras, where α V a V b ∈ Z , we choose µ ab = − e i πα VaVb if both V a and V b are fermionic and µ ab = e i πα VaVb if at least one of them is bosonic. Butin general, the commutation factor can be different from ± V a = e i aφ and V b = e i bφ . The scalingdimension of V a and V b are h a = a and h b = b . α V a V b = h a + h b − h a + b . We see that α V a V b ∈ Z if a, b ∈ Z and such a Gaussian model is an example ofstandard conformal algebras. If both a and b are odd,then h a and h b are half integers and V a and V b arefermionic operators. In this case α V a V b = − ab = odd.So under the standard choice µ ab = − e i πα VaVb , we have µ ab = 1. If one of a and b is even, α V a V b = − ab = evenand one of V a and V b is bosonic operators. Under thestandard choice µ ab = e i α VaVb , we have again µ ab = 1.Even when a and b are not integers, in the Gaussianmodel, the radial order of V a = e i aφ and V b = e i bφ isstill defined with a choice µ ab = 1. This is a part of thedefinition of the Gaussian model. In this paper, we willchoose a more general definition of radial order where µ ab are assumed to be generic complex phases | µ ab | = 1.The vertex algebra generated by ψ have a form ψ a ( z ) ψ b ( w ) = C a,b ψ a + b ( w )( z − w ) ˜ h sc a +˜ h sc b − ˜ h sc a + b + O (cid:16) ( z − w ) ˜ h sc a + b − ˜ h sc a − ˜ h sc b +1 (cid:17) . (40)where C a,b = 0 . (41)When combined with the U (1) Gaussian model, theabove vertex algebra can produce the wave function fora FQH state (see eqn. (31)).We will also limit ourselves to the vertex algebra thatsatisfies the n -cluster condition: ψ n = 1 (42)where 1 stands for the identity operator defined in Ap-pendix B. Those vertex algebras are in some sense “fi-nite” and correspond to rational conformal field theory.We will call such vertex algebra Z n vertex algebra. Wesee that in general, a FQH state can be described by thedirect product of a U (1) Gaussian model and a Z n vertexalgebra. Some exmaples of Z n vertex algebra are studiedin Ref. 36,37.Note that the Z n vertex algebras are different from the Z n simple-current vertex algebras that will be defined insection V. The Z n simple-current vertex algebras arespecial cases of the Z n vertex algebras. In this and thenext sections, we will consider Z n vertex algebras. Wewill further limit ourselves to Z n simple-current vertexalgebras in section V and later.As a result˜ h sc a = ˜ h sc a + n , ˜ h sc n = 0 µ ab = µ a + n,b = µ a,b + n , µ n,a = µ a,n = 1 C a,b = C a + n,b = C a,b + n , C n,a = C a,n = 1 C a,b = µ a,b C b,a (43)By choosing proper normalizations for the operators ψ a ,we can have C a, − a = (cid:26) , a mod n ≤ n/ µ a, − a , a mod n > n/ C a,b = 1 , if a or b = 0 mod n (44)To summarize, we see that the Z n vertex algebras(whose correlation functions give rise to electron wavefunctions) are characterized by the following set of data { n ; m ; ˜ h sc a ; C a,b , ... | a, b = 1 , ..., n } , where m = n/ν . Herethe ... represent other structure constants in the sub-leading terms. The commutation factors µ ab are notincluded in the above data because they can be ex-pressed in terms of ˜ h sc a and are not independent (seeeqn. (E8)). Since the Z n vertex algebra encodes themany-body wave function of electrons, we can say thatthe data { n ; m ; ˜ h sc a ; C a,b , ... | a, b = 1 , ..., n } also char-acterize the electron wave function. We can studyall the properties of electron wave functions by study-ing the data { n ; m ; ˜ h sc a ; C a,b , ... | a, b = 1 , ..., n } . In thepattern-of-zero approach, we use data { n ; m ; h sc a } tocharacterize the wave functions. We will see that the { n ; m ; ˜ h sc a ; C a,b , ... | a, b = 1 , ..., n } characterization is morecomplete, which allows us to obtain some new results. B. Relation between ˜ h sc a and h sc a What is the relation between the two characteriza-tions: { n ; m ; h sc a | a = 1 , ..., n } and { n ; m ; ˜ h sc a ; C ab | a, b =1 , ..., n } ? The single-valueness of the correlation functionΦ( { z i } ) requires that the zeros in Φ( { z i } ) all have inte-ger orders. In this section, we derive conditions on thescaling dimension ˜ h sc a , just from this integral-zero con-dition. This allows us to find a simple relation between { n ; m ; h sc a | a = 1 , ..., n } and { n ; m ; ˜ h sc a ; C ab | a, b = 1 , ..., n } . From the definition of D ab and the OPE (37), we seethat D a,b ≡ S a + b − S a − S b = h a + b − h a − h b = ˜ h sc a + b − ˜ h sc a − ˜ h sc b + abν = D b,a (45)We see that D ,n = nν . So nν is an positive integer whichis called m .From eqn. (45), we can show that S a = a − X i =1 D i, = h a − ah = ˜ h sc a − a ˜ h sc1 + a ( a − ν (46)and ˜ h sc a = S a − aS n n + am − a m n . (47)Therefore, the h sc a introduced before is nothing butthe scaling dimensions ˜ h sc a of the simple currents ψ a (see eqn. (24)). In the following, we will use h sc a to describe the scaling dimensions of ψ a . Thus thedata { n ; m ; ˜ h sc a ; C a,b | a, b = 1 , ..., n } can be rewritten as { n ; m ; h sc a ; C a,b | a, b = 1 , ..., n } . Those h sc a satisfy eqns.(27), (28), and (29).As emphasized in Ref. 11,12, the conditions (27), (28),and (29), although necessary, are not sufficient. In thefollowing, we will try to find more conditions from thevertex algebra. C. Conditions on h sc a and C a,b from theassociativity of vertex algebra The multi-point correlation of a Z n vertex algebra canbe obtained by fusing operators together, thus reducingthe original problem to calculating a correlation of fewerpoints. It is the associativity of this vertex algebra thatguarantees any different ways of fusing operators wouldyield the same correlation in the end, so that the elec-tron wave function would be single-valued. The asso-ciativity of a Z n vertex algebra requires h sc a and C a,b tosatisfy many consistent conditions. Those are the extraconsistent conditions we are looking for. The consistentconditions come from two sources. The first source is theconsistent conditions on the commutation factors µ a,b asdiscussed in appendix B. When applied to our vertexalgebra (40), we find that some consistent conditions on µ a,b allow us to express µ a,b in terms of h sc a . Then otherconsistent conditions on µ a,b will become consistent con-ditions on h sc a (see appendix E 1). The second source isGJI for the vertex algebra (40) as discussed in appendixE 2. We like to stress that the discussions so far are verygeneral. The consistent conditions that we have obtainedfor generic Z n vertex algebra are necessary conditions forany FQH states.A detailed derivation of those conditions on h sc a and C a,b is given in appendix E. Here we just summarize the0new and old conditions in a compact form. The consis-tent conditions can be divided in two classes. The firsttype of consistent conditions act only on the pattern ofzeros { n ; m ; h sc a } (see eqns. (27), (28), (29), (E9), (E10),(E12), (E14), and (E31)): nh sc2 a − nah sc1 + a (2 a − m = 0 mod 2 n,m > , mn = even ,h sc a + b − h sc a − h sc b + abmn ∈ N ,h sc a + b + c − h sc a + b − h sc b + c − h sc a + c + h sc a + h sc b + h sc c ∈ N ,nα , = even ,a α , − α a,a = even ∀ a = 1 , , · · · n − , ∆ ( n , n , n h sc n = 1 , if n = even , (48)where h sc a = h sc a + n and α a,b = h sc a + h sc b − h sc a + b .The second type of consistent conditions act on thestructure constants (see eqns. (E21), (E22), (E27), and(E28)): For any a, b, cC a,b C a + b,c = C b,c C a,b + c = µ a,b C a,c C b,a + c if ∆ ( a, b, c ) = 0 ,C a,b C a + b,c = C b,c C a,b + c + µ a,b C a,c C b,a + c if ∆ ( a, b, c ) = 1 , (49)where µ a,b is a function of the pattern of zeros { h sc a } : µ ij = ( − ijα , − α i,j = ± . For any a = n/ C a, − a = C a,a C a, − a = 1 if ∆ ( a, a, − a ) = 0 , C a, − a = C a,a C a, − a if ∆ ( a, a, − a ) = 1 . (50)Here C a,b satisfies the normalization condition (44).There may be additional conditions when ∆ ( a, b, c ) =0 , 1. But we do not know how to derive those conditionssystematically at this time. IV. EXAMPLES OF GENERIC FQH STATESDESCRIBED BY THE Z n VERTEX ALGEBRA To obtain the examples of generic FQH states, wehave numerical solved the conditions (48). (We don’trequire eqn. (E16) to be satisfied, in order to includesome valid interesting solutions, like Gaffnian which vio-lates eqn. (E16).) In this section, we list some of thosesolutions in terms of { n ; m ; h sc a | a =1 ,...,n − } . First we notethat, for two n -cluster symmetric polynomial Φ and Φ described by { n ; m ; h sc1 ,a } and { n ; m ; h sc2 ,a } , the productΦ = Φ Φ is also an n -cluster symmetric polynomial. Φis described by the pattern of zeros { n ; m ; h sc a } = { n ; m + m ; h sc1 ,a + h sc2 ,a } . (51) Most of the solutions can be decomposed according toeqn. (51). We will call the solutions that cannot be de-composed primitive solutions. We will only list thoseprimitive solutions. We only searched solutions with afilling fraction ν ≥ / 4. We can see that most solu-tions shown also satisfy condition (E16), which meansthey obey OPE (68) and correspond to special Z n ver-tex algebras. However, some solutions such as a 4-clusterstate called Gaffnian, explicitly violates condition (E16)and their OPE’s take the more general form (E18) and(E19). They are described by generic Z n vertex algebras. A. n = 1 case There is only one n = 1 primitive solution: n = 1 : c = 0 { m ; h sc1 ..h sc n − } = { }{ p ; M ..M n − } = { }{ n ..n m − } = { } . (52)It is ν = 1 / h sc a = 0, indi-cating that the simple-current part of vertex algebra istrivial and has a zero central charge c = 0. The vertexalgebra contains only the U (1) Gaussian part. B. n = 2 case We note that the n = 1 primitive solution also appearsas a n = 2 primitive solution. We find only one new n = 2 primitive solution: n = 2 : c = 1 / Z parafermion state) { m ; h sc1 ..h sc n − } = { 2; 12 }{ p ; M ..M n − } = { 1; 0 }{ n ..n m − } = { } . (53)It is the ν = 1 Pfaffian state Φ Z . The simple currentpart of the vertex algebra is a Z parafermion CFT. Ifwe only use the conditions (27), (28), and (29) obtainedin Ref. 11,12, then n = 2 : { m ; h sc1 ..h sc n − } = { 2; 14 }{ p ; M ..M n − } = { 12 ; 0 }{ n ..n m − } = { } . (54)will be a solution. Such a solution does not correspondto any symmetric polynomial, indicating that the con-ditions (27), (28), and (29) are incomplete. An extracondition (E12) from commutation factors remove suchan incorrect solution.1 C. n = 3 case Apart from the n = 1 primitive solution, we find onlyone new n = 3 primitive solution n = 3 : c = 4 / Z parafermion state) { m ; h sc1 ..h sc n − } = { 2; 23 23 }{ p ; M ..M n − } = { 2; 0 0 }{ n ..n m − } = { } . (55)It is the Z parafermion state Φ Z . D. n = 4 case Apart from the n = 1 primitive solutions, we find onlytwo new n = 4 primitive solutions using conditions (27),(28), (29), (E10), (E12), and (E14): n = 4 : c = 1 ( Z parafermion state) { m ; h sc1 ..h sc n − } = { 2; 34 1 34 }{ p ; M ..M n − } = { 2; 0 0 0 }{ n ..n m − } = { } . (56)which is the Z parafermion state Φ Z , and n = 4 : { m ; h sc1 ..h sc n − } = { 2; 14 0 14 }{ p ; M ..M n − } = { 1; 12 1 12 }{ n ..n m − } = { } . (57)We like to point out that a non primitive solution { m ; h sc1 , .., h sc n − } = 2 × { , , } = { , , } is the Z parafermion state (the Pfaffian state). Consistent condi-tions from a study of useful GJI’s show that it has cen-tral charge c = 1 / Z Pfaffian state) and µ ,a = 1 , ∂ψ = 0, indicating that ψ = 1 is the iden-tity operator here. In other words, this Z simple-currentvertex algebra is generated by a Z simple current.Another non primitive solution { m ; h sc1 , .., h sc n − } =3 × { , , } = { , , } is the Gaffnian state .Gaffnian vertex algebra is a Z simple-current vertexalgebra with µ , = µ , = µ , = − ∂ψ = 0.In comparison with Z Pfaffian, this Z Gaffnian vertexalgebra cannot be generated by any Z simple current.This example will be analyzed in detail in section VII. E. Including conditions (49) and (50) In the above, we only considered the conditions (48).Those patterns of zeros that satisfy eqn. (48) may not satisfy the conditions (49) and (50), ie one may not beable to find C a,b that satisfy eqns. (49) and (50). How-ever, we do not know how to check the conditions (49)and (50) systematically. We have to check them on acase by case basis.For the Z and Z parafermion states, we find thateqns. (49) and (50) reduce to trivial identities after usingeqn. (44). So the non-trivial C , and C , for the Z parafermion vertex algebra cannot be determined fromeqns. (49) and (50), which means that the conditions(49) and (50) can be satisfied by any choices of C a,b thatare consistent with eqn. (44).For the state with pattern of zeros { n ; m ; h sc a } = { 4; 2; } , we find that by choosing ( a, b, c ) = (1 , , , , 3) in (49), we can obtain the following equations C , C , = C , C , = − , C , C , − . (58)Clearly no { C a,b } can satisfy the above two equations.Thus the n = 4 pattern of zeros { m ; h sc a } = { } do not correspond to any valid symmetric polynomial.It’s interesting to note that the n = 4 pattern of zeros { m ; h sc a } = 2 × { } = { } correspond tothe Z parafermion state and the n = 4 pattern of zeros { m ; h sc a } = 3 × { } = { } correspond to theGaffnian state, both being valid symmetric polynomials.For the state with pattern of zeros { n ; m ; h sc a } = { 4; 4; 1 1 1 } , we find that by choosing ( a, b, c ) = (1 , , , , , , 0) in (49), we can obtain the followingequations C , C , = C , C , + C , C , C , = C , = C , C , = C , = − C , (59)which can be reduced to − C , = 2 C , . We see that theonly solution is C , = C , = C , = 0, which is notallowed by eqn. (41). Thus the n = 4 pattern of zeros { m ; h sc a } = { 4; 1 1 1 } do not correspond to any validsymmetric polynomial. F. Summary In Ref. 11,12, we have seen that the conditions (27),(28), and (29) are not enough since they allow the fol-lowing pattern of zeros { n ; m ; h sc a } = { 2; 1; } . Such apattern of zeros does not correspond to any valid poly-nomial. The conditions (48) obtained in this paper ruleout the above invalid solution. So the conditions (48)is more complete than the conditions (27), (28), and(29). However, the conditions (48) is still incomplete,since they allow the invalid patterns of zeros such as { n ; m ; h sc a } = { 4; 2; } and { 4; 4; 1 1 1 } . Both ofthem can be ruled out by the conditions (49) and (50).The conditions (48), (49), and (50) are the consistentconditions that we can find from some of GJI, based on2the most general form of OPE (40). So those conditionsare necessary, but may not be sufficient.The correspondence between the patterns of zeros { n ; m ; h sc a } and FQH states is not one-to-one. Therecan be many polynomials that have the same patternof zeros. This is not surprising since the pattern of zerosonly fixes the highest-order zeros in electron wave func-tions (symmetric polynomials), while different patternsof lower-order zeros could lead to different polynomialsin principle. In other words, the leading-order OPE (40)alone might not suffice to uniquely determine the cor-relation function of the vertex algebra. The examplesstudied in this section support such a belief. Explicitcalculations for some examples suggest that the patternof zeros together with the central charge c and simple cur-rent condition would uniquely determine the FQH state.This is a reason why we introduce Z n simple-current ver-tex algebra in the next section. V. Z n SIMPLE-CURRENT VERTEX ALGEBRA In the last section, we discuss “legal” patterns of zerosthat satisfy the consistent conditions (48), (49), and (50)and describe existing FQH states. If we believe that a“legal” pattern of zeros { n ; m ; h sc a } , or more precisely thedata { n ; m ; h sc a ; c } , can completely describe a FQH state,then we should be able to calculate all the topologicalproperties of the FQH states. But so far, from the pat-tern of zeros { n ; m ; h sc a } , we can only calculate the num-ber of different quasiparticle types, quasiparticle charges,and the fusion algebra between the quasiparticles. Even with the more complete data { n ; m ; h sc a ; c } , we stilldo not know, at this time, how to calculate the quasipar-ticle statistics and scaling dimensions.One idea to calculate more topological properties fromthe data { n ; m ; h sc a ; c } is to use the data to define andconstruct the corresponding Z n vertex algebra, and thenuse the Z n vertex algebra to calculate the quasiparticlescaling dimensions and the central charge c . However, sofar we do not know how to use the data { n ; m ; h sc a ; c } tocompletely construct a Z n vertex algebra in a systematicmanner.Starting from this section, we will concentrate on asubset of “legal” patterns of zeros that correspond toa subset of Z n vertex algebra. Such a subset is called Z n simple-current vertex algebras. The FQH states de-scribed by those Z n simple-current vertex algebras arecalled Z n simple-current states. We will show thatin many cases the quasiparticle scaling dimensions andthe central charge c can be calculated from the data { n ; m ; h sc a ; c } for those Z n simple-current states. A. OPE’s of Z n simple-current vertex algebra The Z n simple-current vertex algebra is definedthrough an Abelian fusion rule with cyclic Z n symme- try for primary fields { ψ a } of Virasoro algebra ψ a ψ b ∼ ψ a + b , ψ a ≡ ( ψ ) a . (60)Compared to eqn. (34), here we require that ψ a and ψ b fuse into a single primary field of Virasoro algebra ψ a + b . Such operators are called simple currents. The Z n simple-current vertex algebra is defined by the followingOPE of ψ a : ψ a ( z ) ψ b ( w ) = C a,b ψ a + b ( w )( z − w ) α a,b + O (cid:16) ( z − w ) − α a,b (cid:17) (61) ψ a ( z ) ψ − a ( w ) = 1 + h sc a c ( z − w ) T ( w )( z − w ) h sc a + O (cid:16) ( z − w ) − h sc a (cid:17) (62)where we define α a,b ≡ h sc a + h sc b − h sc a + b (63) ψ − a ≡ ψ n − a and ψ a = ψ n + a is understood due to the Z n symmetry. In the context the subscript a of Z n simplecurrents is always defined as a mod n .We like to point out here that the form of the OPE(62) is a special property of the Z n simple-current ver-tex algebra. For a more general Z n vertex algebra thatdescribes a generic FQH state, the correspond OPE hasa more general form ψ a ( z ) ψ − a ( w ) = 1 + h sc a c ( z − w ) T ( w )( z − w ) h sc a (64)+ T ′ a ( z − w ) h sc a − + O (cid:16) ( z − w ) − h sc a (cid:17) where T ′ a are dimension-2 primary fields of Virasoro al-gebra ( { T ′ a , a = 1 , · · · , [ n ] } may not be linearly inde-pendent though). Also, for the Z n simple-current vertexalgebra, the subleading terms in (61) are also determined.For more details, see appendix F. { C a,b } are the structure constants of this vertex alge-bra. We also have conformal symmetry T ( z ) ψ a ( w ) = h sc a ( z − w ) ψ a ( w ) + 1 z − w ∂ψ a ( w ) + O (1)(65)and Virasoro algebra T ( z ) T ( w ) = c/ z − w ) + 2 T ( w )( z − w ) + ∂T ( w ) z − w + O (1)(66)where T ( z ) represents the energy-momentum tensor,which has a scaling dimension of 2. c stands for the cen-tral charge as usual, which is also a structure constant.3Using the notation of generalized vertex algebra (seeAppendix B), we have[ ψ i ψ j ] α i,j = C i,j ψ i + j , i + j = 0 mod n (67)[ ψ i ψ − i ] α i, − i = 1 , [ ψ i ψ − i ] α i, − i − = 0 , [ ψ i ψ − i ] α i, − i − = 2 h sc i c T (68)[ T ψ i ] = h sc i ψ i = [ ψ i T ] , [ T ψ i ] = ∂ψ i (69)[ ψ i T ] = ( h sc i − ∂ψ i [ T T ] = c , [ T T ] = 0 , [ T T ] = 2 T, [ T T ] = ∂T (70)with α T,ψ i = 2 , α T,T = 4. We call it a special Z n simple-current vertex algebra if it satisfies OPE’s (67)-(70). Forexample, the Z n parafermion states correspond to aseries of special Z n simple-current vertex algebras.The commutation factor µ AB equals unity if either A or B is the energy-momentum tensor T : µ T,ψ i = µ ψ i ,T = µ T,T = 1. Similarly we have µ A, = µ ,A = 1 forthe identity operator 1 and any operator A . However, µ i,j ≡ µ ψ i ,ψ j given in eqn. (E8) can be ± µ i, − i = 1 , ∀ i ,which is not necessary. For example, the Z Gaffniandoes not satisfy µ i, − i = 1 , ∀ i . So, we will adopt the moregeneral OPE (E18) and (E19) instead of eqn. (68) to in-clude examples like Gaffnian which do give a FQH wavefunction. OPE (68) is for a special Z n simple-currentvertex algebra that satisfies µ i, − i = 1 , ∀ i . For a moregeneral Z n simple-current vertex algebra, they become[ ψ i ψ − i ] α i, − i = C i, − i , [ ψ i ψ − i ] α i, − i − = 0 , [ ψ i ψ − i ] α i, − i − = 2 C i, − i h sc i c T (71) C i, − i = (cid:26) , i ≤ n/ nµ i, − i , i > n/ n so that we always have C a,b = µ a,b C b,a for any subscripts a and b in such an associative vertex algebra.The OPE’s (67), (71), (69), (70), (116) and (117) de-fine the generalized Z n simple-current vertex algebra , orsimply Z n simple-current vertex algebra. The Gaffnianstate corresponds to a generalized Z simple-current ver-tex algebra with µ a, − a = 1. When µ a, − a = 1, we have aspecial Z n simple-current vertex algebra.What kind of pattern of zeros { n ; m ; h sc a } , or moreprecisely what kind of data { n ; m ; h sc a ; c, C ab } , can pro-duce a Z n simple-current vertex algebra? Since the Z n simple-current vertex algebras are special cases of Z n vertex algebras, the data { n ; m ; h sc a ; c, C ab } must satisfythe conditions (48), (49), and (50). However, the data { n ; m ; h sc a ; c, C ab } for Z n simple-current vertex algebrasshould satisfy more conditions. Those conditions can be obtained from the GJI of Z n simple-current vertex alge-bras. In Appendix E 2, we derived all those extra consis-tent conditions for a generic Z n vertex algebra, from theuseful GJI’s based on OPE (40). Now based on OPE’ssummarized in this section, we can similarly derive a setof extra consistent conditions for a Z n simple-current ver-tex algebra. These conditions are summarized in sectionV B. For those valid data that satisfy all the consistentconditions, the full properties of simple-current vertexalgebra can be obtained. This in turn allows us to calcu-late the physical topological properties of the FQH statesassociated with those valid patterns of zeros.We like to point out that many examples of Z n simple-current vertex algebra have been studied in detail.They include the simplest Z n simple-current vertex alge-bra – the Z n parafermion algebra. More generalexmaples that have been studied are the higher genera-tions of Z n parafermion algebra and gradedparafermion algebra. In those exapmles, the Z n simple-current algebras are studied by embedding the al-gebras into some known CFT, such as coset models ofKac-Moody current algebras and/or Coulomb gas mod-els. However, in this paper, we will not assume such kindof embeding. We will try to calculate the properties of Z n simple-current vertex algebra directly from the data { n ; m ; h sc a , c, ... } without assuming any embedding. B. Consistent conditions from useful GJI’s In Appendix E 2 a, we show how to obtain the con-sistent conditions on the data { n ; m ; h sc a ; c, C ab } charac-terizing a generic Z n vertex algebra from a set of usefulGJI’s as described in Appendix D, requiring that OPE(E1) is obeyed. Here for a Z n simple-current vertex alge-bra, requiring that OPE’s (67), (71), (69) and (70) (116)and (117) are obeyed, we can derive a larger set of con-sistent conditions on the data { n ; m ; h sc a ; c, C ab } . For theexamples studied in this paper, we find that C ab can beuniquely determined from { n ; m ; h sc a ; c } using those con-sistent conditions. Thus, for those states, C ab are notindependent and can be dropped.Since ∆ ( A, B, C ) = ∆ ( A, C, B ) = ∆ ( C, A, B ) foran associative vertex algebra, we can combine the con-sistent conditions obtained from GJI’s of all possible 6permutations of 3 operators ( A, B, C ) together. In thissection we summarize the consistent conditions obtainedfrom useful GJI’s (just like in Appendix E 2 a) and listthem in a compact manner. These extra consistent con-ditions, together with conditions (48) should form a com-plete set of consistent conditions, which allows us to ob-tain a valid pattern of zeros and construct the associatedsimple-current vertex algebra and FQH wave function.4 { A, B, C } = { ψ a , ψ b , ψ c } , a + b, b + c, a + c = 0 mod n For ∆ ( a, b, c ) = 0, we have the following consistentconditions: C a,b C a + b,c = C b,c C a,b + c = µ a,b C a,c C b,a + c (72)Notice that the consistent conditions obtained from use-ful GJI’s of ( ψ a , ψ b , ψ c ) and of ( ψ b , ψ a , ψ c ) only differ bya factor of µ a,b since C a,b = µ a,b C b,a . Thus they are notindependent conditions. Similarly it’s easy to show thatother permutations yield consistent conditions linearlydependent with the above condition, using the fact that µ a,b µ a,c = µ a,b + c here since ∆ ( a, b, c ) = 0.For ∆ ( a, b, c ) = 1, we have the following consistentconditions: C a,b C a + b,c = C b,c C a,b + c + µ a,b C a,c C b,a + c (73)For ∆ ( a, b, c ) ≥ { A, B, C } = { ψ a , ψ b , ψ − b } , a ± b = 0 mod n For ∆ ( a, b, − b ) = 0 we have the following independentconsistent conditions h sc a h sc b = α a, ± b = 0 , h sc b ∂ψ a ≡ C a,b C a + b, − b = µ a,b C a, − b C b,a − b = C b, − b (74)since we know eqn. (B7) and µ a, = 1.For ∆ ( a, b, − b ) = 1 the independent consistent condi-tions are c = 4 h sc a h sc b α a,b (1 − α a,b ) ,C a,b C a + b. − b = (1 − α a,b ) C b, − b µ a,b C a, − b C b,a − b = − α a,b C b, − b (75)For ∆ ( a, b, − b ) = 2 the independent consistent condi-tions are µ a,b C a, − b C b,a − b = [ α a,b ( α a,b − h sc a h sc b c ] C b, − b C a,b C a + b, − b = [ ( α a,b − α a,b − h sc a h sc b c ] C b, − b (76)For ∆ ( a, b, − b ) = 3 the independent consistent condi-tions are C a,b C a + b, − b − µ a,b C a, − b C b,a − b = [ ( α a,b − α a,b − h sc a h sc b c ] C b, − b (77)For ∆ ( a, b, − b ) ≥ { A, B, C } = { ψ a , ψ a , ψ − a } , a = n/ mod n For ∆ ( a, a, − a ) = α a,a + 2 h sc a = 0 the consistent con-ditions are summarized as: h sc a = α a,a = 0 , ∂ψ a ≡ C a,a C a, − a = C a, − a = C − a,a = µ a, − a = 1 (78)For ∆ ( a, a, − a ) = α a,a + 2 h sc a = 1 the correspondingconsistent conditions are: α a,a = − , h sc a = 1 , c = − , µ a, − a = − C a,a C a, − a = 2 C a, − a , C a, − a = − C − a,a (79)For ∆ ( a, a, − a ) = 2 the independent consistent con-ditions are c = 2 h sc a − h sc a ,C a,a C a, − a = 2 h sc a , C a, − a = C − a,a = µ a, − a = 1 (80)since we have C a, − a = C − a, a here.For ∆ ( a, a, − a ) = α a,a + 2 h sc a = 3 the extra consistentconditions are µ a, − a = − c = − h sc a ) (2 h sc a − h sc a − , C − a,a = − C a, − a C a,a C a, − a = 4( h sc a − C a, − a (81)For ∆ ( a, a, − a ) = 4 the independent consistent con-ditions are C a, − a = C − a,a = µ a, − a = 1 C a,a C a, − a = h sc a (2 h sc a − 3) + 2( h sc a ) c (82)For ∆ ( a, a, − a ) = α a,a +2 h sc a = 5 there is only 1 usefulGJI and the consistent conditions is: µ a, − a = − , C a, − a = − C − a,a C a,a C a, − a = 2[( h sc a − h sc a − 3) + 2( h sc a ) c ] C a, − a For ∆ ( a, a, − a ) = α a,a + 2 h sc a ≥ { A, B, C } = { ψ n/ , ψ n/ , ψ n/ } , n = even Just like shown in Appendix E 2 a,we require that∆ ( n/ , n/ , n/ = 1 , , . (83)otherwise the useful GJI’s would yield a contradiction ψ n/ = 0.For ∆ ( n/ , n/ , n/ 2) = 0 the extra consistent condi-tions are h sc n/ = 0 , ∂ψ n/ ≡ ( n/ , n/ , n/ 2) = 2 the extra consistent condi-tions are c = h sc n/ = 1 / ( n/ , n/ , n/ 2) = 4 the extra consistent condi-tions are c = h sc n/ = 1 (86)For ∆ ( n/ , n/ , n/ 2) = 6 there are no extra consistentconditions.For ∆ ( n/ , n/ , n/ 2) = 7 the extra consistent condi-tions are c = 49 , h sc n/ = 7 / ( n/ , n/ , n/ ≥ VI. REPRESENTING QUASIPARTICLES IN Z n SIMPLE-CURRENT VERTEX ALGEBRA Since the Z n simple-current vertex algebras completelydetermine the FQH states and their topological orders,we should be able to calculate all the topological prop-erties from the vertex algebras. In this section, we willdiscuss how to represent quasiparticles and how to cal-culate quasiparticle properties from the vertex algebras. A. The pattern of zeros for quasiparticles and itsconsistent conditions First, let us review the pattern of zeros description forquasiparticles in FQH states. 1. Definition and consistent conditions The pattern of zeros for the ground state wave functioncan be easily generalized to describe the wave functionswith quasiparticle excitations. If a symmetric polynomialΦ( { z i } ) has a quasiparticle at z = 0, Φ( { z i } ) will havea different pattern of zeros { S γ ; a } as z = λξ , · · · , z a = λξ a approach 0:lim λ → + Φ( { z i } ) = λ S γ ; a P γ ( ξ , · · · , ξ a ; z a +1 , · · · )+ O ( λ S γ ; a +1 ) . (88)Thus we can use the sequence of non-negative integers { S γ ; a } to quantitatively characterize quasiparticles.It was shown that there are similar consistent con-ditions on the quasiparticle pattern of zeros { S γ ; a } :First concave condition D γ + a,b ≡ S γ ; a + b − S γ ; a − S b ≥ ( γ + a ; b, c ) ≡ S γ ; a + b + c + S γ ; a + S b + S c − S γ ; a + b − S γ ; a + c − S b + c ≥ n -cluster condition S γ ; a + kn = S γ ; a + k ( S γ ; n + ma ) + mn k ( k − { S γ ; a } is a quantitative way to label all types of thequasiparticles in the FQH state described by { S a } . Thequestion is that is { S γ ; a } an one-to-one label of thequasiparticles? Can two different quasiparticles sharethe same pattern of zeros? The answer is yes and no.For certain FQH states (such as all the generalized andcomposite parafermion FQH states), { S γ ; a } is an one-to-one label of all the quasiparticles. While for other FQHstates, such as Z | Z and Z | Z in section VII, { S γ ; a } isnot an one-to-one label and two different quasiparticlescan have the same pattern of zeros.If we assume { S γ ; a } to be an one-to-one label of all thequasiparticles, then by solving the above consistent con-ditions, we can obtain the number of quasiparticle types,which happens to equal the ground state degeneracy ofthe FQH state on a torus. We can also calculate otherphysical properties of quasiparticles from { S γ ; a } . For ex-ample, the quasiparticle charge Q γ can be obtained fromthe pattern of zeros as Q γ = S γ ; n − S n m (92)(The above formula is valid even when { S γ ; a } is not anone-to-one label.) 2. Label quasiparticle pattern of zeros by { k sc γ ; a ; Q γ } Another way to label the quasiparticle pattern of zeroscan be obtained by introducing the { k sc γ ; a } vector (whichis denoted by ˜ l sc γ ; a in Ref. 13): k sc γ ; a ≡ S γ ; a − S γ ; a − + h sc1 − m ( Q γ + a − n (93)Conversely we have S γ ; a = a X i =1 k sc γ ; i + a ( mQ γ n − h sc1 ) + ma ( a − n (94)The n -cluster condition (91) of { S γ ; a } sequence resultsin the periodic property of { k sc γ ; a } k sc γ ; n + a = k sc γ ; a (95)Therefore we can use the set of data { k sc γ ;1 , ..., k sc γ ; n ; Q γ } .to describe quasiparticles.6Let a = n in eqn. (94) and use eqn. (92) we can seethat n X i =1 k sc γ ; i = 0 (96)The two concave conditions (89) and (90) for this set ofdata now becomes D γ + a,b = b X i =1 k sc γ ; a + i − h sc b + b mQ γ n + mabn ∈ N (97)∆ ( γ + a, b, c ) = c X i =1 ( k sc γ ; a + b + i − k sc γ ; a + i )+ h sc b + h sc c − h sc b + c ∈ N (98)A set of { k sc γ ;1 , ..., k sc γ ; n ; Q γ } satisfying the above two con-ditions and S γ ; a ≥ { h sc a } .We note that γ + 1 corresponds to a bound state be-tween a γ -quasiparticle and a hole (the absence of anelectron). The ( γ + 1)-quasiparticle is labeled by { k sc γ +1;1 , · · · , k sc γ +1; n ; Q γ +1 } = { k sc γ ;2 , · · · , k sc γ ; n , k sc γ ;1 ; Q γ +1 } . Since two quasiparticles that differ by an electron areregarded as equivalent, we can use the above equivalencerelation to pick an equivalent label that has the minimalcharge and satisfies S γ ; a ≥ 0. For each equivalence class,there exists only one such label. In this paper, we willuse such a label to label inequivalent quasiparticles. B. Quasiparticle wave functions and quasiparticleoperators Just like the ground state wave function (31), the wavefunction with a quasiparticle can also be written as a R-ordered correlation function between electron operatorsand quasiparticle operators in the vertex algebraΦ γ ( w ; { z i } ) = lim z ∞ →∞ z h N ∞ h V ( z ∞ ) h Y i V e ( z i ) i V γ ( w ) i , (99)where w is the location of the quasiparticle and V γ ( w )is the quasiparticle operator. By definition, a quasipar-ticle operator can be any operator that is mutually localrespect to the electron operators V e ( z ).In our simple-current × U (1) vertex algebra, the quasi-particle operator V γ has the following form V γ ( z ) = σ γ ( z ) : e i φ ( z ) Q γ / √ ν : (100)where σ γ is a disorder operator that generates a repre-sentation of the simple current part of the vertex algebra. When a electrons and one quasiparticle are fused togetherwe have V γ + a ( z ) ∝ V a V γ = σ γ + a ( z ) : e i φ ( z )( Q γ + a ) / √ ν : σ γ + a ∝ ψ a σ. (101)The OPE between the quasiparticle operator and theelectron operator can be written as V e ( z ) V γ + a ( w ) ∝ ( z − w ) l γ ; a +1 V γ + a +1 ( w ) + · · · (102)The mutual locality between the quasiparticle operatorand the electron operator requires l γ ; a to be integers. Inorder for the quasiparticle wave function Φ γ ( w ; { z i } ) tocontain no poles, we also require that l γ ; a ≥ l γ ; a , a = 1 , , ... , provides a quan-titative way to label the quasiparticles (and quasiparticleoperators). We have introduced another quantitative la-bel of the quasiparticles in terms of S γ ; a , a = 1 , , ... Thetwo labeling schemes are related by S γ ; a = a X i =1 l γ ; i , l γ ; a = S γ ; a − S γ ; a − . (103)We can also convert the orbital sequence l γ ; a into an oc-cupation sequence n γ ; l . If we view l γ ; a as the index of theorbital occupied by the a -th particle, then n γ ; l is simplythe number of particles occupying the l th orbital.Let us denote the scaling dimension of disorder opera-tors σ γ as h sc γ . Can we calculate those scaling dimensionsfrom the data l γ ; a that characterize the quasiparticle?From the OPE of the quasiparticle operators, we findthe following relations l γ ; a +1 = h sc γ + a +1 − h sc γ + a − h sc1 + m ( Q γ + a ) n (104)and S γ ; a = a X i =1 l γ ; i (105)= h sc γ + a − h sc γ + a ( mQ γ n − h sc1 ) + ma ( a − n , Making use of eqn. (93) we immediately obtain therelations between { k sc γ ; a } and { h sc γ + b } k sc γ ; a = h sc γ + a − h sc γ + a − (106)which implies that h sc γ + a = h sc γ + a X i =1 k sc γ ; i . (107)Moreover, eqn. (95) and eqn. (96) lead to the periodiccondition on h sc γ + a h sc γ + a + n = h sc γ + a (108)7which is implied by the fusion rule ψ n σ = σ since ψ n = 1.We know that we can use { k sc γ ;1 , ..., k sc γ ; n ; Q γ } that satis-fies the two concave conditions (97) and (98) to describe(or label) a quasiparticle operator V γ (or a quasiparticle γ ). The above result (107) only allows us to determinethe scaling dimensions { h sc γ + a } of the associated disor-der operators up to a constant. That is if we know thescaling dimension h sc γ of a disorder operator σ γ , then thescaling dimensions of a family of disorder operators σ γ + a can be determined. However, the scaling dimension h sc γ cannot be determined from the considerations discussedhere. Can we do a better job by fully using the struc-ture of the vertex algebra? In section VI D and VII wewill show how to extract the scaling dimension h sc γ + a fromuseful GJI’s defined in Appendix D. C. A more complete characterization ofquasiparticles Through a study of Z n vertex algebra, we have re-alized that the pattern-of-zero data { n ; m ; h sc a } does notfully describe a symmetric polynomial ( ie a quantum Hallwave function). We need to at least expand { n ; m ; h sc a } to { n ; m ; h sc a ; c } to characterize a quantum Hall wavefunction. Similarly, the data { k sc γ ; a ; Q γ } does not fullydescribe a quasiparticle either, ie some times, differ-ent quasiparticles can have the same pattern of zeros { k sc γ ; a ; Q γ } .To see how to extend { k sc γ ; a ; Q γ } , we note that a genericOPE between σ γ + b and ψ a has a form ψ a ( z ) σ γ + b ( w ) = C a,γ + b ( z − w ) α a,γ + b σ γ + a + b ( w ) + · · · (109)where α a,γ + b = h sc a + h sc γ + b − h sc γ + a + b . (110)We also need to introduce the commutation factor µ a,γ + b :( z − w ) α a,γ + b ψ a ( z ) σ γ + b ( w )= µ a,γ + b ( w − z ) α a,γ + b σ γ + b ( w ) ψ a ( z ) , (111)to describe the commutation relation between σ γ + b and ψ a . We see that in vertex algebra, we need additionaldata, C a,γ + b , C γ + b,a , µ a,γ + b , and µ γ + b,a , to describethe quasiparticle γ . (In appendix C, we give a discus-sion about the relation between the quasiparticle com-mutation factor µ a,γ and quasiparticle pattern of zeros { k sc γ ; a ; Q γ } .)However, if we put the quasiparticle at w = 0 (seeeqn. (99)), then we do not need to use commutationfactor µ a,γ when we calculate the R-ordered correlationfunction (99). Thus, the electron wave function with aquasiparticle do not depend on the commutation factor µ a,γ . Similarly, the R-ordered correlation function onlydepend on C a,γ + b . Therefore, we only need to add C a,γ + b to describe the quasiparticle γ more completely. Therefore, within the simple-current vertex algebra, wecan use the following more complete data { k sc γ ; a ; Q γ ; C a,γ + b } (112)to describe a quasiparticle. By considering the fullstructure of the vertex algebra (see next section VI D),we can obtain many self-consistent conditions on thedata { k sc γ ; a ; Q γ ; C a,γ + b } . In particular, we can calcu-late the scaling dimension h sc γ of σ γ from the data { k sc γ ; a ; Q γ ; C a,γ + b } .Once we find the scaling dimension h sc γ of a disorderoperator σ γ , the scaling dimension h γ of the associatedquasiparticle operator V γ can be determined from h γ = h sc γ + h ga γ = h sc γ + mQ γ n . (113)where h ga γ is the scaling dimension of the U (1) parte i φQ γ / √ ν of the quasiparticle operator. h γ is the intrinsicspin of the quasiparticle which is closely related to thestatistics of the quasiparticle. (Note that in 2+1D theintrinsic spin is not quantized as half integer.) D. Consistent conditions for quasiparticles fromuseful GJI’s 1. Complete vertex algebra with quasiparticle operators To find more consistent conditions on the quasiparticledata { k sc γ ; a ; Q γ ; C a,γ + b } , we need to write down the com-plete OPE between the disorder operators σ γ + b and thesimple currents ψ a ( z ) σ γ + b ( w ) = C a,γ + b σ γ + a + b ( w )( z − w ) α a,γ + b + O (( z − w ) − α a,γ + b ) T ( z ) σ γ + a ( w ) = h sc γ + a ( z − w ) σ γ + a ( w ) + 1 z − w ∂σ γ + a ( w ) + O (1)(114)where we define α a,γ + b ≡ h sc a − ( h sc γ + a + b − h sc γ + b ) = h sc a − a X i =1 k sc γ ; b + i , (115)In other words we have[ ψ a σ γ + b ] α a,γ + b = C a,γ + b σ γ + a + b (116)[ σ γ + b ψ a ] α a,γ + b = µ γ + b,a C a,γ + b σ γ + a + b [ T σ γ + a ] = h sc γ + a σ γ + a = [ σ γ + a T ] , (117)[ T σ γ + a ] = ∂σ γ + a , [ σ γ + a T ] = ( h sc γ + a − ∂σ γ + a with α T,σ γ + a = 2. We set C a,γ = 1 as the definition ofdisorder operators σ γ + a , a = 0 mod n , which possess Z n h sc γ + a of disorder opera-tors, as will be shown in examples.The consistent conditions on the quasiparticle data, { k sc γ ; a ; Q γ ; C a,γ + b } or { h sc γ + a ; Q γ ; C a,γ + b } , can also be ob-tained from useful GJI’s with respect to the OPE’s (116)and (117), just as we did in Appendix E 2 a for simplecurrents of a generic Z n vertex algebra. In the following,we’ll list the obtained consistent conditions from GJI’s. 2. Consistent conditions: { A, B, C } = { ψ a , ψ b , σ γ + c } , a + b = 0 mod n Apply the GJI to the quasiparticle algebra (116) and(117), we can obtain many new consistent conditions.For ∆ ( a, b, γ + c ) = 0 the independent consistent con-ditions are µ a,b C a,γ + c C b,γ + a + c = C a,b C a + b,γ + c = C b,γ + c C a,γ + b + c (118)For ∆ ( a, b, γ + c ) = 1 the only independent consistentcondition is: µ a,b C a,γ + c C b,γ + a + c (119)= C a,b C a + b,γ + c − C b,γ + c C a,γ + b + c For ∆ ( a, b, γ + c ) ≥ 3. Consistent conditions: { A, B, C } = { ψ a , ψ − a , σ γ + c } For ∆ ( a, − a, γ + b ) = 0 the independent consistentconditions are h sc a ∂σ γ + b ≡ , α ± a,γ + b = h sc a h sc γ + b = 0 ,C a,γ + b C − a,γ + a + b = C − a,γ + b C a,γ + b − a = C a, − a . (120)since µ γ + b, = 1.For ∆ ( a, − a, γ + b ) = 1 the independent consistentconditions are h sc a c h sc γ + b = α a,γ + b (1 − α a,γ + b )4 ,C − a,γ + b C a,γ + b − a = C a, − a α a,γ + b ,C a,γ + b C − a,γ + a + b = C − a,a α − a,γ + b . (121)Notice here the quasiparticle scaling dimension h sc γ is de-termined through useful GJI’s.For ∆ ( a, − a, γ + b ) = 2 the independent consistentconditions are C − a,γ + b C a,γ + b − a =[ 2 h sc a h sc γ + b c + α a,γ + b ( α a,γ + b − C a, − a ,C a,γ + b C − a,γ + a + b = (122)[ 2 h sc a h sc γ + b c + ( α a,γ + b − α a,γ + b − C − a,a . For ∆ ( a, − a, γ + b ) = 3 the independent consistentcondition is µ a, − a C a,γ + b C − a,γ + a + b + C − a,γ + b C a,γ + b − a = [ 2 h sc a h sc γ + b c + ( α a,γ + b − α a,γ + b − C a, − a . (123)For ∆ ( a, − a, γ + b ) ≥ VII. EXAMPLES OF FQH STATES DESCRIBEDBY Z n SIMPLE-CURRENT VERTEX ALGEBRAS In this section, we will examine some examples of FQHstates that can be described by Z n simple-current vertexalgebra. A. Pattern of zeros for Z n simple-current vertexalgebra When we consider FQH states described by Z n simple-current vertex algebra, the patterns of zero for those FQHstates satisfy many additional conditions on top of theconditions (48), (49), and (50) for generic FQH states. Insection V B, we list those additional consistent conditionsobtained from GJI. Many conditions do not contain thestructure constants C ab , and those conditions become theextra conditions on the pattern of zeros. We have numer-ically solved all those conditions on the pattern of zeros.In this section, we list some of the numerical solutions.We like to point out that the patterns of zeros forFQH states described by simple-current vertex algebrado not have the additive property. This is because giventwo FQH wave functions described by simple-current ver-tex algebra, their product in general cannot be describedany simple-current vertex algebra. The direct product oftwo simple-current vertex algebra, in general, contains atleast one dimension-2 primary field of Virasoro algebrathat violates the Abelian fusion algebra. Thus the di-rect product of two simple-current vertex algebra is nota simple-current vertex algebra in general.Among many solutions of the consistent conditionsare the Z n parafermion algebras, which are the simplestsimple-current vertex algebra. The Z n parafermion al-gebras give rise to Z n parafermion wave functions Φ Z n .As an example of no additive property, the pattern ofzeros for the product wave function Φ Z ⊗ Z ≡ Φ Z Φ Z does not satisfy the consistent conditions for the simple-current vertex algebra, indicating that the direct productof Z and Z parafermion vertex algebras is not a simple-current vertex algebra. In the following, we only list somesolutions that are not Z n parafermion algebras.9 Z simple-current vertex algebra: n = 2 : c = 1 ( Z | Z state) { m ; h sc1 ..h sc n − } = { 4; 1 }{ p ; M ..M n − } = { 2; 0 }{ n ..n m − } = { } . (124) n = 2 : ( Z | Z | Z state) { m ; h sc1 ..h sc n − } = { 6; 32 }{ p ; M ..M n − } = { 3; 0 }{ n ..n m − } = { } . (125) Z simple-current vertex algebra: n = 3 : ( Z | Z state) { m ; h sc1 ..h sc n − } = { 4; 43 43 }{ p ; M ..M n − } = { 2; 0 0 }{ n ..n m − } = { } . (126) Z simple-current vertex algebra: n = 4 : c = 1 ( C state) { m ; h sc1 ..h sc n − } = { 4; 1 1 1 }{ p ; M ..M n − } = { 2; 12 1 12 }{ n ..n m − } = { } . (127) n = 4 : c = 1 ( Z | Z state) { m ; h sc1 ..h sc n − } = { 6; 54 1 54 }{ p ; M ..M n − } = { 3; 1 2 1 }{ n ..n m − } = { } . (128) n = 4 : (Gaffnian state) { m ; h sc1 ..h sc n − } = { 6; 34 0 34 }{ p ; M ..M n − } = { 3; 32 3 32 }{ n ..n m − } = { } . (129) n = 4 : ( C | C state) { m ; h sc1 ..h sc n − } = { 8; 2 2 2 }{ p ; M ..M n − } = { 4; 1 2 1 }{ n ..n m − } = { } . (130) Z simple-current vertex algebra: n = 6 : { m ; h sc1 ..h sc n − } = { 6; 32 2 52 2 32 }{ p ; M ..M n − } = { 3; 1 2 2 2 1 }{ n ..n m − } = { } . (131) n = 6 : c = 1 { m ; h sc1 ..h sc n − } = { 8; 43 43 1 43 43 }{ p ; M ..M n − } = { 3; 2 4 5 4 2 }{ n ..n m − } = { } . (132)We like to stress that the above pattern of zeros areonly checked to satisfy the consistent conditions that donot contain structure constants C a,b . It remains to beshown that there exist C a,b for those patterns of zerosthat satisfy all the consistent conditions for structureconstants (from GJI’s). When we check those additionalconditions for C a,b , we find that the C pattern of zero { n ; m ; h sc a } = { 4; 4; 1 1 1 } does not correspond to anysymmetric polynomial as discussed in section IV E.We will discuss some other patterns of zeros in de-tail later. We will show how the central charge c , thestructure constants C a,b and the quasiparticle scaling di-mension h sc γ + a of the corresponding vertex algebra can bedetermined from the pattern of zeros { n, m ; h sc a } , throughthe consistent conditions in section V B, VI D and in Ap-pendix F. Those consistent conditions are generated byuseful GJI’s: (D3) or (D6) with eqn. (D11).To calculate the central charge and the quasiparticlescaling dimensions { c ; C a,b ; h sc γ + a } , in the first step wewill try to determine them from conditions in sectionV B, ie we don’t specify subleading order term (F1) inOPE. If these conditions don’t give enough information,then we will resort to more conditions in Appendix F,which is based on the subleading OPE term (F1).We note that some pattern of zeros can directly fixthe central charge, and we list the central charge forthose patterns of zeros as in above. The Z n parafermionpatterns of zeros are examples in this class. While forother patterns of zeros, the central charges depend on thestructure constants C a,b . We will calculate those centralcharges below. There are even patterns of zeros that donot completely determine the simple-current vertex alge-bra. We need to include additional information C ab todetermine the corresponding simple-current vertex alge-bra. The Z | Z , Z | Z states etc are examples in thisclass of pattern of zeros.We also give names for some patterns of zeros. Forexample, the C n | C n pattern of zeros { m ; h sc1 ..h sc n − } = { n ; 2 2 ... } is the sum of two C n pattern of zeros { m ; h sc1 ..h sc n − } = { n ; 1 1 ... } . Also, the Z | Z pat-tern of zeros is described by { m ; h sc1 ..h sc n − } = { } which is a sum of { m ; h sc1 ..h sc2 n − } = { } for the Z parafermion state and { m ; h sc1 ..h sc n − } = { } forthe Z parafermion state. (Note that the Z parafermionstate is also described by { m ; h sc1 ..h sc n − } = { } . )However, the wave function of such a Z | Z state is dif-ferent from the product of a Z parafermion wave func-tion and a Z parafermion wave function. The prod-uct wave function called the Z ⊗ Z state, is describedby a Z vertex algebra given by the direct product ofthe Z parafermion algebra and Z parafermion alge-bra. Such a Z vertex algebra is different from any0 Z simple-current vertex algebras, featured by an extradimension-2 primary field. However, both the Z | Z and Z ⊗ Z states have the same pattern of zeros. Thisis an example showing that the same pattern of zeros { m ; h sc1 ..h sc n − } = { } can correspond to more thanone FQH wave functions. B. The Z n parafermion vertex algebra: Z n parafermion states with { M k = 0; p = 1; m = 2 } In this simplest case we have p = 1 , M k = 0. Forexample the Z parafermion state is described by thefollowing pattern of zeros: n = 3 : Z state { m ; h sc1 ..h sc n − } = { 2; 23 23 }{ p ; M ..M n − } = { 1; 0 0 }{ n ..n m − } = { } . (133)In general, we have (we don’t specify p = 1 until nec-essary, trying to obtain some general conclusions on Z n | ... | Z n series): h sc a = p a ( n − a ) n ,α a,b = 2 pabn − p ( a + b − n ) θ ( a + b − n ) (134)As a result we have µ a,b = 1 , C a,b = C b,a (135)Besides, d a,b defined in eqn. (F4) has a simple form inthis case: d a,b ≡ 12 (1 + h sc a − h sc b h sc a + b ) = d n − a,n − b = aa + b if a + b < n (136)In eqn. (A18) we have ∆ M [ a, b, c ] = 0 and ∆ ( a, b, c ) canonly be multiples of 2 p .At first let’s take a look at { A, B, C } = { ψ a , ψ b , ψ c } , a + b, b + c, a + c = 0 mod n . Onlywhen ∆ ( a, b, c ) = 0 there are extra consistent condi-tions in section V B 1, ie a + b + c ≤ n or a + b + c ≥ n we have C a,b C a + b,c = C b,c C a,b + c = C a,c C b,a + c (137)Particularly when a + b + c = 0 mod n we have C a,b = C a,c = C b,c (138)The other consistent condition is satisfied by eqn. (136).For A = B = C = ψ n/ , n = even we know that∆ ( n/ , n/ , n/ 2) = 4 h sc n/ = pn . Only when np/ ≤ ie when n ≤ p = 1, n ≤ p = 2. The above conclusions hold for any p ∈ N . Now let’senforce p = 1 for this special series.For { A, B, C } = { ψ a , ψ b , ψ − b } , a ± b = 0 mod n weknow ∆ ( a, b, − b ) ≥ p . Only when ∆ ( a, b, − b ) = 2there are extra consistent conditions in section V B 2, ie when a = 1 < b, n − b < n , b = 1 < a < n − ≤ n, n − b < a = n − { A, B, C } = { ψ a , ψ a , ψ − a } , a = n/ n , sim-ilarly only when ∆ ( a, a, − a ) = 2 , ie when a = 1 , a = n − , n − (1 , , − 1) = ∆ ( − , − , 1) = 2 and h sc1 =1 − /n, α , = 2 /n , from section V B 3 we have c = 2( n − n + 2 C , C ,n − = C n − ,n − C n − , = 2( n − n With central charge c in hand, from ∆ (2 , , − 2) =∆ ( − , − , 2) = 4 we have C , C ,n − = C n − ,n − C n − , = 6( n − n − n ( n − 1) (139)Similarly from ∆ (1 , b, − b ) = ∆ ( a, , − 1) =∆ ( − , − b, b ) = ∆ ( − a, − , 1) = 2 we have C b,n − C n − b,b − = b ( n + 1 − b ) nC a,n − C ,a − = a ( n + 1 − a ) n (140)These are all the extra consistent conditions. Usingeqn. (137) and eqn. (138) repeatedly we find out thatthe independent conditions besides eqn. (137)-eqn. (138)and C a,b = C b,a are just C ,a C n − ,n − a = ( a + 1)( n − a ) n (141)Other structure constants can be expressed as C a,b = C a,b − C ,a + b − C ,b − = · · · = Q b − i =0 C ,a + i Q b − j =1 C ,j = Q a + b − i =1 C ,i Q a − i =1 C ,i Q b − i =1 C ,i (142)if a + b ≤ n ; C a,b = C a,b +1 C n − ,a + b +1 C n − ,b +1 = · · · = Q n − b − i =0 C n − ,a − i Q n − b − j =1 C n − ,n − j = Q n − a + n − b − i =1 C n − ,n − i Q n − a − i =1 C n − ,n − i Q n − b − i =1 C n − ,n − i (143)if a + b > n . Notice that the above two equations arecompatible with eqn. (137)! Using eqn. (141) we imme-1diately have C a,b C n − a,n − b = Q b − i =0 C ,a + i C n − ,n − a − i Q b − j =1 C ,j C n − ,n − j = Γ( a + b + 1)Γ( n − a + 1)Γ( n − b + 1)Γ( n + 1)Γ( a + 1)Γ( b + 1)Γ( n − a − b + 1) , (144) ∀ ≤ a, b < a + b ≤ n (145)To summarize, the consistent conditions in section V Bdetermine the central charge and fix the structure con-stants to the following form: C a,b = Q b − i =0 λ a + i Q b − j =1 λ j × (146) s Γ( a + b + 1)Γ( n − a + 1)Γ( n − b + 1)Γ( n + 1)Γ( a + 1)Γ( b + 1)Γ( n − a − b + 1)if 1 ≤ a, b < a + b ≤ n . Free parameters { λ a | a =1 , · · · , n − } are nonzero complex numbers, defined by C ,a = λ a C n − ,n − a . Moreover, the condition (138) re-quires that C a,n − a = 1, so from eqn. (142) we have thefollowing “reflection” condition on { λ a } λ n − = λ n − λ = · · · = λ n − − k λ k = · · · = 1 (147)We point out that the above conclusions are all ob-tained from conditions in section V B, ie we haven’t in-troduced the subleading order OPE (F1) and new con-ditions in Appendix F yet. Now we apply conditions inAppendix F to see whether the normalization constants { λ a | a = 1 , · · · , [ n − ] } can be determined or not.According to Appendix F 2 a, choosing those∆ ( a, b, c ) = 2 p = 2 with a + b, b + c, a + c =0 mod n, a + b + c = n + 1 leads to the following newconstraints: λ a − λ n − a = 1 (148)which means that λ a = ± 1. Other useful GJI’s like∆ ( a, b, c ) = 4 p = 4 with a + b, b + c, a + c = 0 mod n doesn’t result in any new constraints. So finally wecan conclude that considering the subleading order OPE(F1), the structure of such a Z n simple-current vertex algebra is determined self-consistently as: c = 2( n − n + 2 ; λ a = ± 1; (149) C a,b = Q b − i =0 λ a + i Q b − j =1 λ j × s Γ( a + b + 1)Γ( n − a + 1)Γ( n − b + 1)Γ( n + 1)Γ( a + 1)Γ( b + 1)Γ( n − a − b + 1) , (150) C n − a,n − b = Q b − j =1 λ j Q b − i =0 λ a + i × s Γ( a + b + 1)Γ( n − a + 1)Γ( n − b + 1)Γ( n + 1)Γ( a + 1)Γ( b + 1)Γ( n − a − b + 1) , (151) ∀ a + b ≤ n ; λ a − = λ n − a , λ = λ n − = 1 , λ a + n = λ a . (152)It is interesting to see that the Z n -parafermion patternof zeros does not completely fix the structure constants C a,b . Do those different structure constants C a,b corre-sponding to different choices of λ a give rise to differentFQH wave functions, even through they all have the samepattern of zeros? In general, the different structure con-stants (even with the same pattern of zeros) will give riseto different FQH wave functions. But in this particularcase, all the above different structure constants for differ-ent choices of λ a = ± ψ a = χ a ψ a , χ a = ± , χ a + n = χ a . (153)Those modified simple current operators will generatethe same FQH state. But the structure constants of ˜ ψ a is changed ˜ C a,b = C a,b χ a χ b χ a + b (154)So such kind of change in structure constants C a,b → ˜ C a,b = C a,b χ a χ b χ a + b (155)does not generate new FQH wave function. Therefore C a,b and ˜ C a,b = C a,b χ a χ b χ a + b describe the same FQH state.We will call the transformation (155) an equivalencetransformation.Note that the factors can be rewritten as Q b − i =0 λ a + i Q b − j =1 λ j = Q a + b − i =0 λ i Q a − i =0 λ i Q b − i =0 λ i (156) Q b − j =1 λ j Q b − i =0 λ a + i = Q n − b − i =0 λ n − a + i Q n − b − j =1 λ j = Q n − a + n − b − i =0 λ i Q n − a − i =0 λ i Q n − b − i =0 λ i , I I na n γ ;0 ..m − nk sc γ ;1 ..n Q h sc + h ga na − na − na + TABLE I: The pattern of zeros and the charges Q for thequasiparticles in the Z parafermion state. n γ ;1 ...n γ,m − isthe occupation sequence characterizing the quasiparticle γ (defined below eqn. (103)). The quasiparticles are labeled bythe index I . The scaling dimensions of the quasiparticle op-erators are sums of the contributions from the simple-currentvertex algebra and the Gaussian model: h γ = h sc + h ga . where we have used eqn. (148). So if we choose χ a = Q a − i =0 λ i , the equivalence transformation (155) will re-move the λ a dependent factors in the structure con-stants. This completes our proof. We see that the Z n parafermion patterns of zeros, { M k = 0; p = 1; m = 2 } ,completely determine the structure constants C ab and thecentral charge c .It is interesting to note that we can use the equivalencetransformation (155) to make C a,b = 1 for all a + b ≤ n ,as one can see from (142). We can also use the transfor-mation (155) to make C a,b = 1 for all a + b ≥ n , as onecan see from (143). But we cannot make all C a,b = 1. C. Quasiparticles in the Z parafermion state In this section, we will study the quasiparticles in the Z parafermion state. For the Z parafermion state, wehave a simple current with scaling dimension h sc1 = 1 / c = 1 / 2. According to Ref. 13, thepatterns of zeros { k sc γ ; a ; Q γ } for the quasiparticles are ob-tained by solving the conditions (97) and (98). The resultis listed in table I. There are three types of quasiparti-cles. In fact, these three quasiparticles belong to two dif-ferent families. The quasiparticles in the same family canchange into each other by combining an Abelian quasi-particle. The two quasiparticles in the first family differfrom each other merely by a magnetic translation ( ie by an insertion of an Abelian magnetic flux tube),while the 3rd quasiparticle differs from the first two intheir non-Abelian content. For a family of quasiparticlesdiffer by magnetic translations, we only need to studyone of them to obtain all the information of simple cur-rent part (the difference between different quasiparticlesin such a family comes solely from a U (1) factor).First let us study the 3rd quasiparticle { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , } . Using the follow-ing relations derived from eqn. (107) (which will be usedfrequently in the quasiparticle calculation) α a,γ + b = h sc a − a X k =1 k sc γ ; b + k ∆ ( a, b, γ + c ) = α a,γ + c + α b,γ + c − α a + b,γ + c , (157) I I na n γ ;0 ..m − nk sc γ ;1 ..n Q h sc + h ga na − na − na − + na − + TABLE II: The pattern of zeros and the charges Q for thequasiparticles in the Z parafermion state. The quasiparticlesare labeled by the index I . The scaling dimensions of thequasiparticle operators are sums of the contributions fromthe simple-current vertex algebra and the Gaussian model: h γ = h sc + h ga . we find ∆ (1 , , γ ) = ∆ (1 , , γ + 1) = 1, α ,γ = α ,γ +1 =1 / 2. So from section VI D 3, we see that h sc γ = h sc γ +1 = 1 / C ,γ +1 = 1 / Z or Isingvertex algebra.Next we study the 1st quasiparticle { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , − ; 0 } from its family.With ∆ (1 , , γ ) = 0 and ∆ (1 , , γ + 1) = 2 we havefrom section VI D 3 that h sc γ = 0 , h sc γ +1 = 1 / C ,γ +1 = 1 , ∂σ γ ≡ σ γ is proportional to the identity oper-ator. This means that this quasiparticle is simply thetrivial vacuum modulo electrons.We like to stress that for the above two quasipar-ticles, the structure constants C a,γ + b are uniquely de-termined by the quasiparticles pattern of zeros { k sc γ ; a } .So the quasiparticles in the Z parafermion theory areuniquely described by the quasiparticles pattern of zeros { k sc γ ; a ; Q γ } . Each index I in the table I label a uniquequasiparticle pattern of zeros, and so the index I also la-bel a unique quasiparticle for the Z parafermion state.We can also obtain the fusion algebra using the methodin Ref. 13. We find that0 × , × , × , × , × , × , (160)where we have used the index I to label different quasi-particles (see table I). We have regarded two quasipar-ticles to be equivalent if they differ by some electrons.The index I really label the above equivalent classes ofquasiparticles. We may also define a different equivalentclass of quasiparticles by regarding two quasiparticles tobe equivalent if they differ by some electrons or by someAbelian magnetic flux tubes. Such classes of quasiparti-cles are characterized by { k sc γ ;1 , · · · , k sc γ ; n } up to a cyclicpermutation. We introduce an index I na to label thosenon-Abelian classes of quasiparticles. From the relationbetween the two sets of indices I and I na as shown in3table I, we can reduce the fusion algebra (160) to a sim-pler fusion algebra between the non-Abelian classes ofquasiparticles0 na × na = 0 na , na × na = 1 na , na × na = 0 na + 0 na . (161) D. Quasiparticles in the Z parafermion state The Z simple-current vertex algebra is characterizedby h sc1 = h sc2 = 23 , c = 45 ,C , = C , = 2 √ , C , = C , = 1 (162)where we have fixed the normalization factors to be λ a =1. There are two families of quasiparticles obtained from(97) and (98) (see table II). The 1st family is representedby quasiparticle { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , , − ; 0 } .With ∆ (1 , , γ ) = ∆ (2 , , γ ) = ∆ (1 , , γ + 1) =∆ (2 , , γ + 2) = 0, we find that (see section VI D 2 orAppendix F 2 f) C ,γ +1 = C ,γ +2 = C , = C , C ,γ +1 = C ,γ +2 (163)Then with ∆ (1 , , γ ) = 0 and ∆ (1 , , γ + 1) =∆ (1 , , γ + 2) = 2 we have from section VI D 3 that h sc γ = 0 , h sc γ +1 = h sc γ +2 = 23 , ∂σ γ ≡ C ,γ +1 = C ,γ +2 = 1 , C ,γ +1 C ,γ +2 = 43 (164)Therefore this quasiparticle is characterized by h sc γ = 0 , h sc γ +1 = h sc γ +2 = 23 , ∂σ γ = 0 C ,γ +1 = C ,γ +2 = 1 , C ,γ +1 = C ,γ +2 = 2 √ ∂σ γ = 0 and h sc γ = 0 imply that the quasiparticle oper-ator σ γ is a constant operator with scaling dimension 0.Such an operator is the trivial identity operator.The 2nd family is represented by a quasiparticle with { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , − , } . With ∆ (1 , , γ ) =∆ (2 , , γ + 2) = 0 and ∆ (1 , , γ + 1) = ∆ (1 , , γ + 2) =∆ (2 , , γ ) = ∆ (2 , , γ +1) = 1, we find that (see sectionVI D 2 or Appendix F 2 f) C ,γ +1 = C , , C ,γ +2 = C , / C ,γ +2 = C , C , / , C ,γ +1 = C , C , / (1 , , γ ) = ∆ (1 , , γ + 2) = 1 and∆ (1 , , γ + 1) = 2, we find that (see section VI D 3) h sc γ = h sc γ +2 = 115 , h sc γ +1 = 25 ,C ,γ +1 = C ,γ +1 C ,γ +2 = 23 , C ,γ +2 = 13 (167) This nontrivial quasiparticle is characterized by h sc γ = h sc γ +2 = 115 , h sc γ +1 = 25 ,C ,γ +1 = 23 , C ,γ +2 = 13 ,C ,γ +1 = C , = 2 √ , C ,γ +2 = C , / √ C a,γ + b are uniquely determined by the quasi-particles pattern of zeros { k sc γ ; a } . So the quasiparticles inthe Z parafermion theory are uniquely described by thequasiparticles pattern of zeros { k sc γ ; a ; Q γ } . Each index I in the table II label a unique quasiparticle pattern of ze-ros, and so the index I also label a unique quasiparticlefor the Z parafermion state.The full fusion algebra between the quasiparticles is × , × , × , × , × , × , × , × , × , × . (169)The fusion algebra between the non-Abelian classes ofquasiparticles is0 na × na = 0 na , na × na = 1 na , na × na = 0 na + 1 na . (170) E. Z n | Z n series: { M k = 0; p = 2 } The Z n | Z n vertex algebra is called the “second gen-eration” of Z n parafermion algebra and is studied inRef. 40,41,42,43. In this case we have h sc a = 2 a ( n − a ) n (171)As a result we still have eqn. (135), eqn. (136), eqn. (137)and eqn. (138), therefore eqn. (142) and eqn. (143) stillhold.Apparently in this case the extra conditions in sectionV B are not enough to determine the full structure ofthis vertex algebra, since now ∆ ( a, b, c ) are multiples of2 p = 4! So we introduce the subleading order OPE (F1)and resort to new conditions in Appendix F.Since now we have ∆ (1 , b, − b ) = ∆ ( − , b, − b ) =∆ (1 , a, − 1) = 2 p = 4, from Appendix F 2 c we have: C ,a C n − ,n − a = ( a + 1)( n − a ) n ( n − × h a ( n − a − n − c + ( n − a )( n − a − i (172)Representing the central charge c in terms of a continuousvariable g in the following way c = 4( n − g ( n + g − n + 2 g − n + 2 g ) (173)4yields an expression of structure constants in termsof g and normalization constants { λ a = λ n − − a ; a =1 , · · · , [ n − ] } , which is totally similar with Z n parafermion states: C ,a = λ a s ( a + 1)( n − a ) n ( a + g )( n + g − a − g ( n + g − 1) (174) C a,b = Q a + b − i =1 C ,i Q b − j =1 C ,j Q a − k =1 C ,k = Q b − i =0 λ a + i Q b − j =1 λ j × s Γ( a + b + 1)Γ( n − a + 1)Γ( n − b + 1)Γ( n + 1)Γ( a + 1)Γ( b + 1)Γ( n − a − b + 1) × s Γ( a + b + g )Γ( n − a + g )Γ( n − b + g )Γ( g )Γ( n + g )Γ( a + g )Γ( b + g )Γ( n − a − b + g ) , (175) C n − a,n − b = ( Q b − j =1 λ j Q b − i =0 λ a + i ) C a,b , a + b ≤ n (176)where reflection condition (147) should also be satisfiedfor the normalization constants λ a . It’s easy to verifythat ∆ ( a, b, c ) = 2 p = 4 doesn’t result in any newconstraints on free parameters { g ; λ a | a = 1 , · · · , [ n − ] } .Therefore the above are all conditions on this Z n | Z n se-ries of vertex algebra.Using the equivalence transformation (155), we canchange the normalization constants to λ a = 1. So onlydifferent g in the structure constants give rise to differ-ent FQH states. All those different FQH states have thatsame pattern of zeros, and we need an additional param-eter g to completely characterize the FQH state. Forthe simple ideal Hamiltonian introduced in Ref. 11,12,all those different FQH states have a zero energy. InRef. 19, additional terms are introduced in the Hamilto-nian so that only the Z | Z state with a particular g canbe the zero energy states. F. Quasiparticles in the Z | Z state The Z | Z state is described by the following patternof zeros: n = 2 : c = 1 ( Z | Z state) { m ; h sc1 ..h sc n − } = { 4; 1 }{ n ..n m − } = { } . (177)Here we have n = 2 , p = 2 , M = 0 and thus h sc1 = 1.Since ∆ (1 , , 1) = 4 we have according to section V B 4: c = 1 (178)There is no free parameter in such a Z simple currentvertex algebra.Now let’s turn to the quasiparticles of this state. Thereare three families of different quasiparticles, and we willdiscuss them one by one. I I na n γ ;0 ..m − nk sc γ ;1 ..n Q h sc + h ga na − na − na − na − na − + na − + na − + na − + na η + na η + 1TABLE III: The pattern of zeros and the charges Q for thequasiparticles in the Z | Z parafermion state. The quasipar-ticles are labeled by the index I . The scaling dimensions ofthe quasiparticle operators are given by h γ = h sc + h ga , where η = C ,γ +1 / I =8, 9 each actually corresponds to a class of quasiparticles pa-rameterized by a continuous parameter η . The 1st family has { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , − 1; 0 } asits representative. With ∆ (1 , , γ ) = 0 and ∆ (1 , , γ +1) = 4 we have h sc γ = 0 , ∂σ γ ≡ , C ,γ +1 = 1 (179)indicating this quasiparticle is trivial. We also have h sc γ +1 = h sc γ + k sc γ ;1 = 1 from eqn. (107).The 2nd family is represented by { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , − ; } . With ∆ (1 , , γ ) = 0and ∆ (1 , , γ + 1) = 3, we find that (see section VI D 3) h sc γ = 1 / , h sc γ +1 = 9 / , C ,γ +1 = 1 / Q γ =1 / / { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , } . With ∆ (1 , , γ ) = ∆ (1 , , γ + 1) = 2, we findthat (see section VI D 3) h sc γ = h sc γ +1 = C ,γ +1 / . (181)Remember that the quasiparticles are described bythe data { k sc γ ; a ; Q γ ; C a,γ + b } . For the first two family ofthe quasiparticles, the quasiparticle structure constants C a,γ + b are uniquely determined by the quasiparticle pat-tern of zeros { k sc γ ; a ; Q γ } . In this case, a pattern of zeroscorrespond to a single type of quasiparticle. For the 3rdfamily, the pattern of zeros does not fix C a,γ + b . There-fore the quasiparticles in the third family are labeledby the pattern of zeros { k sc γ ; a ; Q γ } and a free parameter η = C ,γ +1 / 2. So there are infinite types of quasiparti-cles in the 3rd family. The energy gap for such kind ofquasiparticles must vanish at least in the η → ψ a σ γ + b ] α a,γ + b − = C a,γ + b d a,γ + b ∂σ γ + a + b , like we did in Appendix F cannot5fix this free parameter here. There are indeed infinitetypes of quasiparticles in the Z | Z simple-current FQHstate. This suggests that the Z | Z simple-current FQHstate is gapless for the ideal Hamiltonian introduced inRef. 11.Using the method in Ref. 13, we obtain the full fusionalgebra between the quasiparticles (expressed in terms ofthe index I in table III):0 × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × I = 8 or I = 9 does not correspond to asingle quasiparticle. They each actually corresponds toa class of quasiparticles parameterized by a continuousparameter η . We can use (8 , η ) and (9 , η ) to uniquelylabel those quasiparticles. Thus, for example, the fusionrule 8 × , η ) × (9 , η ′ ) = 1 + 3 + (8 , η ′′ ), for some η , η ′ , and η ′′ .The fusion algebra between the non-Abelian classes ofquasiparticles is0 na × na = 0 na na × na = 1 na (183)0 na × na = 2 na na × na = 0 na + 2 na na × na = 1 na + 1 na na × na = 0 na + 0 na + 2 na where the relation between I and I na is given in table III. G. Quasiparticles in the Z | Z state The Z | Z state is described by the following patternof zeros: n = 3 : ( Z | Z state) { m ; h sc1 ..h sc n − } = { 4; 43 43 }{ n ..n m − } = { } . (184) Here we have n = 3 , p = 2 , M = M = 0 and thus h sc1 = h sc2 = . As shown in section VII E we have C , C , = 49 ( 8 c − 1) (185)as the only extra consistent condition of this vertex alge-bra from useful GJI’s. We can use two free parameters { c, λ } to express the structure constants: C , = λ, C , = 49 λ ( 8 c − . (186)However, using the equivalence transformation (seeeqn. (155)) ψ → χψ , ψ → χ − ψ , λ → λ/χ , (187)we can set λ = 1. So the infinite Z | Z simple-currentvertex algebras are parameterized by only a single realnumber c .There are 5 classes of non-Abelian quasiparticles asshown in TABLE IV. We shall study these 5 classes oneby one in this section. The 1st class is the trivial one,represented by the data { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , , − 43 ; 0 } . (188)With ∆ (1 , , γ ) = ∆ (1 , , γ + 1) = ∆ (2 , , γ ) =∆ (2 , , γ + 2) = 0 we have for the structure constants: C ,γ +1 = C , , C ,γ +2 = C , , C ,γ +2 = C ,γ +1 . (189)Then with ∆ (1 , , γ ) = 0 we have C ,γ +2 = C ,γ +1 = 1 , h sc γ = 0 , ∂σ γ = 0 . (190)which dictates that this is a trivial quasiparticle, propor-tional to the identity operator.The 2nd class is represented by the data { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , − , − 23 ; 14 } . (191)With ∆ (1 , , γ ) = ∆ (2 , , γ + 2) = 0 and ∆ (1 , , γ +1) = ∆ (2 , , γ ) = 1 we have for the structure constants: C ,γ +1 = C , , C ,γ +2 = C , / , C ,γ +1 = 2 C ,γ +2 . (192)Then with ∆ (1 , , γ ) = 1 and ∆ (1 , , γ +2) = 3 we have C ,γ +2 = 13 , C ,γ +1 = 23 ,h sc γ = c , h sc γ +2 = c C ,γ +1 C ,γ +2 c 24 + 23 . (193)after using the structure constants (186). The above re-sults from GJI’s are consistent with (107).6The 3rd class is represented by the data { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , , − 1; 12 } . (194)With ∆ (1 , , γ ) = ∆ (2 , , γ + 2) = 1 and ∆ (1 , , γ +1) = ∆ (2 , , γ ) = 0 we have for the structure constants: C ,γ +1 = C , / , C ,γ +2 = C , , C ,γ +2 = 2 C ,γ +1 . (195)Then with ∆ (1 , , γ ) = 1 and ∆ (1 , , γ +1) = 3 we have C ,γ +2 = 23 , C ,γ +1 = 13 ,h sc γ = c , h sc γ +1 = c 24 + 23 . (196)where we have used (186) in calculating h sc γ +1 as well.The 4th class is represented by the data { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , − , 0; 12 } . (197)With ∆ (1 , , γ ) = ∆ (2 , , γ + 2) = 0 we have for thestructure constants: C ,γ +1 = C , , C ,γ +2 C ,γ +1 = C , C ,γ +2 . (198)Then with ∆ (1 , , γ ) = ∆ (1 , , γ + 2) = 2 we have thefollowing consistent conditions: C ,γ +1 = C ,γ +2 + 13 = C ,γ +1 C ,γ +2 h sc γ = h sc γ +2 = − c 12 + 3 c C ,γ +1 (199)Solving the above nonlinear equations gives us the struc-ture constants and quasiparticle scaling dimensions: C ,γ +1 = C , , C ,γ +2 = C ,γ +1 − ,C ,γ +1 = − 29 + 169 c ± p ( c − c − c ,C ,γ +2 = C , ( C ,γ +1 − 13 ) /C ,γ +1 ,h sc γ = h sc γ +2 = 23 − c ± p ( c − c − ≡ η ± (200) ± corresponds to two different branches of solutions.Here c ≤ c ≥ h sc γ to be a real number.For a Z | Z simple-current algebra described by a fixed c and a quasiparticle pattern of zeros indexed by I =12,13, 14, or 15, there are two sets of quasiparticle structureconstants that satisfy all the consistent conditions forthe GJI. This implies that the index I =12, 13, 14, 15in the table IV each actually corresponds to two types ofquasiparticles parameterized by the two sets of structureconstants. Those quasiparticles are uniquely labeled by( I, +) and ( I, − ), I =12, 13, 14, 15. When c = 2 or I I na n γ ;0 ..m − nk sc γ ;1 ..n Q h sc + h ga na − na − na − na − na − − c + na − − c + na − − c + na − − c + ) + na − c + na − c + 10 2 na − c + 11 2 na − c + ) + 12 3 na − η ± + 13 3 na − η ± + 14 3 na − η ± + 15 3 na − η ± + 16 4 na − η + 17 4 na − η + 18 4 na − η + ) + 19 4 na − η + ) + TABLE IV: The pattern of zeros and the charges Q for thequasiparticles in the Z | Z parafermion state parameterizedby { c, λ } . Note that the quasiparticle quantum numbers donot depend on the second parameter λ . The quasiparticlesare labeled by the index I . The scaling dimensions of thequasiparticle operators are sums of the contributions fromthe simple-current vertex algebra and the Gaussian model: h γ = h sc + h ga , where η ± is given by eqn. (200). Note theindex I =16, 17, 18, 19 each actually corresponds to a classof quasiparticles parameterized by a continuous parameter η .Similarly the index I =12, 13, 14, 15 each corresponds to twotypes of quasiparticles parameterized by ± . c = 8, then there is only one type of quasiparticle foreach I =12, 13, 14, 15.The 5th class is represented by the data { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , , − 13 ; 34 } . (201)With ∆ (1 , , γ ) = ∆ (1 , , γ + 1) = ∆ (2 , , γ ) =∆ (2 , , γ + 2) = 1 we have for the structure constants: C ,γ +1 = C , / , C ,γ +2 = C , / , C ,γ +2 = C ,γ +1 . (202)Then with ∆ (1 , , γ ) = 2 and ∆ (1 , , γ + 1) =∆ (1 , , γ + 2) = 3 we have h sc γ = 3 c C ,γ +2 ≡ η,h sc γ +1 = h sc γ +2 = 3 c C ,γ +2 + C ,γ +1 C ,γ +2 ) + c 24= 3 c C ,γ +2 + 13 . (203)7where we have used (186). Just like the Z | Z states,there are infinite sets of quasiparticles structure constantsthe satisfy the consistent conditions. Those sets of struc-ture constants is parameterized by a single real number η = c C ,γ +2 = c C ,γ +1 . This implies that the index I =16, 17, 18, 19 in table IV each corresponds to a classof quasiparticles parameterized by a continuous parame-ter η . Those quasiparticles are uniquely labeled by ( I, η ), I =16, 17, 18, 19. We see that there are infinite typesof quasiparticles in the Z | Z state, suggesting that the Z | Z state is gapless for the ideal Hamiltonian intro-duced in Ref. 11,19. H. The Z | Z | Z state This Z simple-current state is described by the pat-tern of zeros: n = 2 : ( Z | Z | Z state) { m ; h sc1 ..h sc n − } = { 6; 3 / }{ p ; M ..M n − } = { 3; 0 }{ n ..n m − } = { } . (204)Since there are no structure constants for a Z vertexalgebra after choosing the proper normalization, the onlyfree parameter in this simple-current vertex algebra isthe central charge c . However, since ∆ (1 , , 1) = 6 inthis case, consistent conditions from GJI’s cannot fix thecentral charge according to section V B 4.Explicit calculations of simple currents correlationfunctions suggest that the electron wave functionsuniquely depends on the central charge c . We like tostress that the Z | Z | Z state provides an interesting ex-ample that the vertex algebra is not determined by thestructure constants C ab of the leading terms, but by astructure constant of a subleading term.In table V, we list 21 distinct quasiparticle patterns ofzeros which give rise to at least 21 different quasiparticles.Those quasiparticles group into 4 classes of non-Abelianquasiparticles. I. Gaffnian: a non unitary Z example A Z solution { m ; h sc1 , · · · , h sc n − } = { , , } iscalled Gaffnian in literature . It has the following com-mutation factors: µ , = µ , = µ , = − Z simple-current vertex al-gebra.With ∆ (2 , , 2) = 0 we know from section V B 4 that ∂ψ ≡ I I na n γ ;0 ..m − nk sc γ ;1 ..n Q na − na − na − na − na − na − na − na − na − na − na − na − na − na − na − na − na − na − na na na Q for thequasiparticles in the Z | Z | Z state. The quasiparticles arelabeled by the index I . Since ∆ (1 , , 2) = ∆ (2 , , 3) = 0 we know from sectionV B 1 that C , = C , = − C , C , = C , = − C , C , = − , C , = 1 (207)With ∆ (1 , , 3) = ∆ (1 , , 3) = 3 we know from sectionV B 2 that c = − h sc1 ) (2 h sc1 − h sc1 − 2) = − C , C , = C , C , = − (1 , , 3) = ∆ (2 , , 3) =∆ (1 , , 2) = 0 in section V B 2 and ∆ (1 , , 1) =∆ (3 , , 3) = 3 in section V B 1 don’t produce any newconditions. Further calculations show that even intro-ducing the subleading order OPE (F1) and applying newconditions in Appendix F wouldn’t not supply any extraconditions. In summary we have c = − , ∂ψ ≡ C , = C , = − C , = λ = 0 C , = C , = − C , = − λ − C , = 1 , C , = − I I na n γ ;0 ..m − nk sc γ ;1 ..n Q h sc + h ga na − − na − − na − − na − − − + na − − − + na − − + TABLE VI: The pattern of zeros and the charges Q forthe quasiparticles in the Gaffnian state. The quasiparticlesare labeled by the index I . The scaling dimensions of thequasiparticle operators are sums of the contributions fromthe simple-current vertex algebra and the Gaussian model: h γ = h sc + h ga . for this Z simple-current vertex algebra, which corre-sponds to the Gaffnian wave function. Using the equiva-lence transformation (see eqn. (155))( ψ , ψ , ψ ) → ( χψ , ψ , χ − ψ ) , λ → λχ − (210)we can set λ = 1. So there is only a single Gaffnian wavefunction.Gaffnian state { m ; h sc1 , · · · , h sc n − } = { , , } hastwo families of different quasiparticles according to con-ditions (97) and (98) (see table VI). The 1st familyhas the following representative: { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , − , , − ; 0 } . With ∆ (1 , , γ ) = ∆ (1 , , γ + 2) =∆ (1 , , γ + a ) = ∆ (2 , , γ + a ) = ∆ (3 , , γ ) =∆ (3 , , γ + 2) = 0 we obtain all the structure constantsfrom section VI D 2 or Appendix F 2 f C ,γ +1 = C ,γ +2 = − C ,γ +1 = λ,C ,γ +3 = C ,γ +2 = − C ,γ +1 = 1 ,C ,γ +3 = C ,γ +3 = − C ,γ +2 = − λ − . (211)With ∆ (1 , , γ ) = 1 = ∆ (1 , , γ +2) , ∆ (1 , , γ +1) =∆ (1 , , γ + 3) = 2 and ∆ (2 , , γ + a ) = 0 we have thequasiparticle scaling dimensions from section VI D 3 h sc γ = h sc γ +2 = 0 , h sc γ +1 = h sc γ +3 = 34 ,∂σ γ = ∂σ γ +2 ≡ . (212)Since C a,b = C a,γ + b and ∂σ γ = 0 here, we know thisquasiparticle σ γ must be proportional to the identity op-erator 1.The 2nd family has the following representa-tive: { k sc γ ;1 , · · · , k sc γ ; n ; Q γ } = { , − , , − ; } . With∆ (1 , , γ ) = ∆ (1 , , γ + 2) = ∆ (3 , , γ ) = ∆ (3 , , γ +2) = 1 and ∆ (1 , , γ + a ) = ∆ (2 , , γ + a ) = 0 we ob-tain all the structure constants from section VI D 2 orAppendix F 2 f C ,γ +1 = λ/ , C ,γ +3 = − λ − / ,C ,γ +3 = − C ,γ +1 = 1 / , C ,γ +2 = 1 ,C ,γ +2 = − C ,γ +1 = λ,C ,γ +2 = − C ,γ +3 = λ − . (213) With ∆ (1 , , γ ) = ∆ (1 , , γ + 2) = ∆ (2 , , γ + a ) = 0and ∆ (1 , , γ + 1) = ∆ (1 , , γ + 3) = 3 we have thequasiparticle scaling dimensions from section VI D 3 h sc γ = h sc γ +2 = − , h sc γ +1 = h sc γ +3 = 15 (214)and the structure constants are consistent with all theuseful GJI’s. Apparently this quasiparticle is a nontrivialone.Using the method in Ref. 13, we obtain the full fusionalgebra between the quasiparticles (expressed in terms ofthe index I in table VI):0 × × × × × × × × × × × × × × × × × × × × × na × na = 0 na na × na = 1 na na × na = 0 na + 1 na . (216) J. The Z | Z state This solution { m ; h sc1 , · · · , h sc n − } = { , , } isa direct product of a n = 4 Pfaffian state { m ; h sc1 , · · · , h sc n − } = { , , } and a Z parafermionstate { m ; h sc1 , · · · , h sc n − } = { , , } .In this case we have p = 3 , M = M = 1 , M = 2 (217)It’s easy to verify that µ i,j = 1 and thus C i,j = C j,i .From section V B 4 we see that ∆ (2 , , 2) = 4 deter-mines the central charge c = 1 (218)Then with ∆ (1 , , 2) = ∆ (2 , , 3) = 0 we know fromsection V B 1 that C , = C , = C , C , = C , = C , (219)∆ (1 , , 3) = ∆ (1 , , 3) = 4 in section V B 3 and∆ (1 , , 3) = ∆ (1 , , 2) = ∆ (2 , , 3) = 2 in sectionV B 2 both lead to the following conclusions: C , C , = C , C , = 52 c = − 58 + 258 c = 52 (220)Note that ∆ (1 , , 1) = ∆ (2 , , 2) = 2 doesn’t bring usany new constraints. Further studies after introducing9 I I na n γ ;0 ..m − Q I I na n γ ;0 ..m − Q na na na na na na na na na na 10 1 na 11 1 na 12 2 na 13 2 na na 15 2 na 16 2 na na 18 3 na 19 3 na na 21 4 na 22 4 na 23 4 na 24 4 na 25 4 na 26 4 na 27 5 na 28 5 na na 30 5 na 31 5 na na 33 6 na 34 6 na 35 6 na 36 6 na 37 6 na 38 6 na 39 7 na 40 7 na 41 7 na Q forthe quasiparticles in the Z | Z state. The quasiparticles arelabeled by the index I . subleading order OPE (F1) show that there are no newconstraints on the structure constants, so we concludethat: c = 1 , C , = C , = 1 C , = C , = C , = λ = 0 C , = C , = C , = 52 λ (221)characterizes this Z simple-current vertex algebra. Us-ing the equivalence transformation (see eqn. (155))( ψ , ψ , ψ ) → ( χψ , ψ , χ − ψ ) , λ → λχ − (222)we can set λ = 1. So there is only a single Z | Z simple-current vertex algebra which correspond to a single FQHwave function.In table VII, we list 42 distinct quasiparticle patternsof zeros which give rise to at least 42 different quasi-particles. Those quasiparticles group into 8 classes ofnon-Abelian quasiparticles. K. C n | C n series with { m ; h sc , · · · , h sc n − } = { n ; 2 , · · · , } This corresponds to a series of FQH states with fill-ing fraction ν = 1 / ν = 1 / I I na n γ ;0 ..m − Q I I na n γ ;0 ..m − Q na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na Q for thequasiparticles in the C | C state (which also the Z | Z | Z state). The quasiparticles are labeled by the index I . for fermionic electrons). A C | C example is given ineqn. (130).First, from eqn. (E8) we know that µ a,b = 1 for such a C n | C n simple-current vertex algebra, since all the simplecurrent scaling dimensions are even integers and so areall α a,b , ∀ a, b . As a result we have C a,b = C b,a , ∀ a, b ∈ Z (223)It’s straightforward to check that if we don’t have thesubleading term (F1) in OPE, this solution only has thefollowing extra consistent conditions shown in sectionV B 1 with ∆ ( a, b, − a − b ) = 0 , a, b, a + b = 0 mod n : C a,b = C a, − a − b = C b, − a − b (224)which for sure can be satisfied for all a, b ∈ Z .Now we introduce the subleading OPE term (F1) andthe new consistent conditions in Appendix F to seewhether they are satisfied for this vertex algebra. Note0that here we have α a,b = (cid:26) , a + b = 0 mod n , a + b = 0 mod n (225)for any a, b = 0 mod n , and also d a,b = 1 / , a + b =0 mod n from eqn. (F4).Taking any integers a, b, c = 0 mod n , for this such a Z n simple current vertex algebra we have:∆ ( a, b, c ) = 2 for a + b, b + c, a + c, a + b + c = 0 mod n ,so according to Appendix F 2 a we have: C a,b C a + b,c = C b,c C a,b + c = C a,c C b,a + c (226)Then all consistent conditions are satisfied without re-quiring that ∂ψ a = 0.∆ ( a, b, c = − a − b ) = 0 for a + b = 0 mod n , soaccording to Appendix F 2 b we have: C a,b = C a, − a − b = C b, − a − b (227)∆ ( a, b, c = − b ) = 4 for a ± b = 0 mod n , so accordingto Appendix F 2 c we have: C a,b C a + b, − b = C a, − b C b,a − b = 8 c (228)∆ ( a, b = a, c = − a ) = 6 for 2 a = 0 mod n , so accord-ing to Appendix F 2 d we have: d a,a = 1 / ( a = n/ , b = n/ , c = n/ 2) = 8 for n = even, soaccording to section V B 4 there are no extra consistentconditions.In summary, this series of solutions { m ; h sc1 , · · · , h sc n − } = { n ; 2 , · · · , } corresponds toa Z n simple-current vertex algebra satisfying thefollowing consistent conditions: C a,b = C b,a , ∀ a, b ∈ Z ; (230) C a,b C a + b,c = C a,c C b,a + c = C b,c C a,b + c , if a + b, b + c, a + c = 0 mod n ; (231) C a,b C a + b, − b = C a, − b C b,a − b = 8 c , if a ± b = 0 mod n ; (232)By solving the above conditions in the similarly wayas with the Z n and the Z n | Z n series, we find that C a,b = Q a + b − i =1 λ i Q a − i =1 λ i Q b − j =1 λ j r c , a + b < n (233) C n − a,n − b = Q a − i =1 λ i Q b − j =1 λ j Q a + b − i =1 λ i r c , a + b < n (234) C a,n − a = 1 . (235) where nonzero complex parameters { λ i | i = 1 , , · · · , n − } satisfy the following constraint: λ a − = λ n − a , ≤ a ≤ n − , λ = 1 . (236)If we choose χ a = Q a − i =0 λ i , the equivalence transforma-tion (155) will remove the λ a dependent factors in thestructure constants. We find that the C n | C n series ischaracterized by the following data: C a,b = r c , a + b < n or a + b > n (237) C a,n − a = 1 . (238)Therefore this theory has one free parameters c if n > C | C state (or Z | Z | Z state). Thosequasiparticles group into 10 classes of non-Abelian quasi-particles. VIII. SUMMARY The pattern-of-zeros is a powerful way to character-ize FQH states. However, the pattern-of-zeros ap-proach is not quite complete. It is known that some pat-terns of zeros do not uniquely describe the FQH states.As a result, we cannot obtain all the topological prop-erties of FQH states from the data of pattern of zeros { n ; m ; S a } .In this paper, we combine the pattern-of-zero approachwith the vertex algebra approach. We find that wecan generalize the data of pattern of zeros { n ; m ; S a } to { n ; m ; S a ; c } to completely describe a FQH state, at leastfor the many examples discussed in this paper. Manyconsistent conditions on the new set of data { n ; m ; S a ; c } are obtained from the GJI of the simple-current vertexalgebra. Those consistent conditions are sufficient: ie ifthe data { n ; m ; S a ; c } satisfy those conditions, then thedata will define a Z n simple-current vertex algebra anda FQH wave function. Using the new characterizationscheme and the Z n simple-current vertex algebra, wecan calculate quasiparticle scaling dimensions, fractionalstatistics, the central charge of the edge states, as well asmany other properties, from the data { n ; m ; S a ; c } .For example, for the Z parafermion state character-ized by pattern of zeros { n ; m ; h sc1 ..h sc n − } = { 2; 2; } , wefind the well known scaling dimensions (the non-Abelianpart) 0, , and for the three kind of quasiparticles.For the Z | Z state characterized by pattern of zeros { n ; m ; h sc1 ..h sc n − } = { 2; 4; 1 } , we find the scaling dimen-sions and the charges for all its quasiparticles (see tableIII). We find that the FQH state described by the Z | Z simple-current vertex algebra contains infinite types ofquasiparticles and two classes of them are parameterizedby a real parameter. This indicates that the Z | Z stateis gapless for the ideal Hamiltonian introduced in Ref. 11.1We also studied the Z | Z state described by the Z | Z simple-current vertex algebra, with the pattern of ze-ros { n ; m ; h sc1 ..h sc n − } = { 3; 4; 43 43 } . Such a state isalso studied in Ref. 19. We show that the Z | Z statecannot be completely characterized by the pattern-of-zeros data { n ; m ; h sc1 ..h sc n − } = { 3; 4; 43 43 } . We need toadd one more parameter c and use the expanded data { n ; m ; h sc1 ..h sc n − ; c } = { 3; 4; 43 43 ; c } to completely charac-terize the Z | Z state. We find the scaling dimensionsand the charges for all its quasiparticles (see table IV).Again there are infinite types of quasiparticles and fourclasses of them are parameterized by a real parameter.This again suggests that the Z | Z state is gapless forthe ideal Hamiltonian introduced in Ref. 11,19.The study in this paper is based on the Z n simple-current vertex algebra. But the Z n simple-current vertexalgebra makes some unnecessary assumptions. It is muchmore natural to study FQH state based on the more gen-eral Z n vertex algebra. This will be a direction of futureexploration. Acknowledgments YML is grateful to Boris Noyvert for many helpfuldiscussions on the algebraic approach to conformal fieldtheory. This research is supported by DOE Grant DE-FG02-99ER45747 (YML,ZQW), NSF Grant No. DMR- 0706078 (XGW), and by NSF Grant No. DMS-034772(ZHW). XGW is also supported by Perimeter Institutefor Theoretical Physics. Research at Perimeter Instituteis supported by the Government of Canada through In-dustry Canada and by the Province of Ontario throughthe Ministry of Research & Innovation. APPENDIX A: OTHER WAYS TO LABEL THEPATTERN OF ZEROS In section II, we have discussed two ways to label thepattern of zeros, one in terms of { n ; m ; S a } and the otherin terms of { n ; m ; h sc a } . In this section, we will introducetwo other more efficient ways to label pattern of zeros.The new ways of labeling automatically satisfy more selfconsistent conditions. 1. Label the pattern of zeros by a set ofnon-negative integers { a j } Since ∆ ( a, b, c ) in eqn. (29) is just a linear combina-tion of the h sc a , there are only n − n choices of ( i, j, k ) in eqn. (22).A convenient choice would be ( i, j, k ) = (1 , , a ) , a =1 , , · · · , n − 1. These equations are∆ (1 , , 1) = (2 h sc1 − h sc2 ) + h sc1 − h sc2 + h sc3 = a ∈ N · · · ∆ (1 , , j ) = (2 h sc1 − h sc2 ) + h sc j − h sc j +1 + h sc j +2 = a j ∈ N · · · ∆ (1 , , n − 3) = (2 h sc1 − h sc2 ) + h sc n − − h sc n − + h sc n − = a n − ∈ N ∆ (1 , , n − 2) = (2 h sc1 − h sc2 ) + h sc n − − h sc n − = a n − ∈ N ∆ (1 , , n − 1) = (2 h sc1 − h sc2 ) + h sc n − + h sc1 = a n − ∈ N Here we only used the h sc n = h sc0 = 0 condition. Addingup these equations together we immediately obtain thefollowing equation: n (2 h sc1 − h sc2 ) = n − X i =1 a i (A1)By defining another vector { A j } : A j = a j − (2 h sc1 − h sc2 ) = a j − n n − X i =1 a i (A2)we have a simple relation: X · h sc = A (A3) where h sc = ( h sc1 , · · · , h sc n − ) T and A = ( A , · · · , A n − ) T are column vectors and the matrix X = − · · · − · · ·· · ·· · · − · · · − · · · ( n − × ( n − (A4)It would be straightforward to check that the above ma-trix is not singular and its inverse equals n · X − i,j = { ij − jn j < iij − ( i − n j ≥ i (A5)2So we can express ( h sc1 , ..., h sc n − ) in terms of the( a , ..., a n − ): h sc a = n − X b =1 X − a,b A b = a ( n − a )2 n n − X j =1 a j + a − nn n − X j =1 ( j + 1) a j + n − X j = a ( j + 1 − a ) a j (A6)Since the date { n ; m ; h sc1 , ..., h sc n − } and { n ; m ; a , ..., a n − } have an one-to-one correspon-dence, we can also use { n ; m ; a , ..., a n − } to label thepattern of zeros. The { n ; m ; a , ..., a n − } labeling schemeis more efficient: once we choose non-negative integers { a , ..., a n − } , we generate h sc a that already satisfy apart of eqn. (29).From the reflection conditions (30) of { h sc a } , we canobtain similar reflection conditions for a j : a j = a n − − j (1 ≤ j ≤ n − a n − = 0 (A7)then the independent sequence of non-negative integersis actually { a j , ≤ j ≤ [ n ] − a n − } , which contain[ n ] integers. (We use [ x ] to denote the biggest integerno larger than x .) The { a j } label of the patterns ofzeros provides us an efficient way to numerically find thesolutions of eqn. (25), eqn. (27), eqn. (28), and eqn. (29). 2. Label patterns of zeros by { M k ; p ; m } a. The { M k ; p ; m } labeling scheme Using reflection conditions (A7) we can define: p ≡ n − X j =1 a j (A8)= a n − [ n ] − X j =1 a j + ( a [ n ] − , n = odd a [ n ] − / , n = evenand M k ≡ n − kn n − X j =1 ( j + 1) a j − n − X j = k ( j + 1 − k ) a j (A9)It’s easy to verify the following reflection condition for { M k } for k = 1 , , · · · , [ n M k = M n − k = k − X j =1 ( j + 1) a j + k [ n ] − X j = k − a j + k · ( a [ n ] − , n = odd a [ n ] − / , n = even (A10) and another important relation M = 2 M (A11)Then we can express { h sc a } in terms of this new set ofindependent variables { M k , k = 1 , , , · · · , [ n ]; p } (also[ n ]-dimensional) h sc a = p a ( n − a ) n − M a (A12)From definitions we see that both p and { M k , k =odd } can be half integers, while { M k , k = even } mustbe integers. When n = odd, { M k , k = odd } must beintegers too. In fact, the simplest parafermion vertexalgebra (which describes the Z n parafermion states in a FQH context) corresponds to the case in which p =1 , M k = 0.Certainly, not all possible choices of { M k ; p ; m } corre-spond to valid patterns of zeros. Only those that satisfythe conditions eqn. (27), eqn. (28), and eqn. (29) arevalid. But { M k ; p ; m } labeling scheme is an efficient wayto generate the valid patterns of zeros.Now we have two [ n ]-dimensional vectors describingthe { h sc a } : { a j , j = 1 , , · · · , [ n ] − a n − } and { M k , k =1 , , , · · · , [ n ]; p } . The latter is expressed in terms of theformer in eqn. (A8) and eqn. (A10). Conversely, we canexpress the former in terms of the latter in the followingway a j = − M j + 2 M j +1 − M j +2 j = 1 , , · · · , [ n − a [ n ] − = ( M [ n ] − M [ n ] − ) · (cid:26) , n = odd2 , n = even a n − = 2 p − M (A13)The { M k ; p ; m } label of the pattern of zeros has a closetie to simple parafermion CFT. b. Consistent conditions on { M k ; p ; m } Now we use this new labeling scheme in (A12) to seewhat are the constraints on { M k ; p ; m } , from all the con-sistent conditions (27), (28) and (29) on { h sc a ; m } .At first, with M = 2 M (27) leads to S = D , = m − pn = even (A14)therefore we have m = 2 p + 2 nk m , k m ∈ N . (A15)This determines the electron filling fraction ν e = (1 + m/n ) − = n p +(2 k m +1) n .3To guarantee mn = even with (A15) we have another condition on p : p ∈ N , if n = odd . (A16)Since S a = P a − i =2 D i, , (28) naturally guarantees S a ∈ N . Moreover we have D a + n,b = D a,b + mb , thus we only needto satisfy the following conditions for (28): D a,b = (cid:26) M a + M b − M a + b + 2 k m ab, ≤ a, b ≤ a + b ≤ nM a + M b − M a + b + 2 k m ab + 2 p ( a + b − n ) , < a, b ≤ n < a + b ∈ N (A17)Since ∆ ( i, j, k ) , ≤ i ≤ j ≤ k < n should be non-negative integers, M k and p satisfy some additional conditionsas shown in (29): ∆ ( i, j, k ) = h sc i + h sc j + h sc k + h sc i + j + k − h sc i + j − h sc i + k − h sc j + k = − ∆ M [ i, j, k ] i + j + k ≤ n − ∆ M [ i, j, k ] + 2 p ( i + j + k − n ) j + k ≤ n < i + j + k ≤ n − ∆ M [ i, j, k ] + 2 pi i + k ≤ n < j + k ≤ i + j + k ≤ n − ∆ M [ i, j, k ] + 2 p ( n − k ) i + j ≤ n < i + k ≤ j + k ≤ i + j + k ≤ n − ∆ M [ i, j, k ] + 2 p (2 n − i − j − k ) n < i + j ≤ i + k ≤ j + k ≤ i + j + k ≤ n − ∆ M [ i, j, k ] n < i + j ≤ i + k ≤ j + k ≤ n < i + j + k ∈ N (A18)where we defined∆ M [ i, j, k ] = M i + M j + M k + M ( i + j + k mod n ) − M ( i + j mod n ) − M ( i + k mod n ) − M ( j + k mod n ) (A19)By partially solving the consistent conditions (27), (28)and (29) on { h sc a ; m } , we obtain a finite set of conditions(A15)-(A18). They are the consistent conditions to besatisfied by the pattern of zeros { M k ; p ; m } , a sequenceof integers and half-integers. For instance, the simplest Z n parafermion states correspond to the pattern ofzeros { M k = 0; p = 1; m = 2 p = 2 } , by choosing non-negative integer k m = 0. APPENDIX B: CONSISTENT CONDITIONS ONTHE COMMUTATION FACTOR µ AB To introduce some useful notations, let us write theOPE between two generic operators A ( z ) and B ( w ) asthe follwoing : A ( z ) B ( w ) = 1( z − w ) α AB (cid:16) [ AB ] α AB ( w )+ ( z − w )[ AB ] α AB − ( w )+ ( z − w ) [ AB ] α AB − ( w ) + · · · (cid:17) (B1)where α AB = h A + h B − h [ AB ] αAB , (B2)and h A ia the scaling dimension of operator A . A and B satisfy the following commutation relation( z − w ) α AB A ( z ) B ( w ) = µ AB ( w − z ) α AB B ( w ) A ( z ) . (B3) Let us derive some conditions on µ AB from the asso-ciativity of the vertex algebra. By exchanging A and B twice we have µ AB µ BA = 1 (B4)which immediately leads to µ AA = 1 (B5) µ AA = − A fields would vanish otherwise.Let B ( z ) C ( w ) = D ( w )( z − w ) α BC + · · · (B6)To exchange A with B and then with C is equivalent toexchange A with D = [ BC ] α BC , so we have µ AB ( − α AB µ AC ( − α AC = µ AD ( − α AD ⇒ µ AD = µ AB µ AC r ABC (B7)in which r ABC ≡ ( − α AB + α AC − α AD = ± ( A, B, C ) ≡ α AB + α AC − α AD = h sc A + h sc B + h sc C + h sc[ AD ] αAD − h sc[ AB ] αAB − h sc[ AC ] αAC − h sc[ BC ] αBC ∈ N (B9)4In a vertex algebra the identity operator 1 (e.g. ψ in a Z n simple-current vertex algebra) is a zero-scaling-dimension operator with the following OPEs:[1 , A ] − j = A δ j, ; [ A, − j = 1 j ! ∂ j A (B10)and α ,A = α A, = 0 , µ ,A = µ A, = 1 for any operator A in the vertex algebra. ∂ j , j ≥ A (let’s suppose that A is not the identityoperator: A = 1), ie ∂A could be zero or not, dependingon the definition of this operator A . For example, simplecurrent ψ in a Z Gaffnian vertex algebra obeys ∂ψ =0. However, this simple current ψ is not the identityoperator 1 = ψ since it has nontrivial commutationsfactors µ , = µ , = − = 1. APPENDIX C: DETERMINE THEQUASIPARTICLE COMMUTATION FACTOR µ γ,a FROM THE QUASIPARTICLE PATTERN OFZEROS { k sc γ ; a } The quasiparticle commutation factors µ γ,a are notfully independent of the patter-of-zero data { k sc γ ; a ; Q γ } .In this section, we try to determine µ γ + b,a from k sc γ ; a .Note that C γ + b,a = µ γ + b,a C a,γ + b (C1)By choosing A = σ γ + a , B = ψ b , C = ψ c , D = ψ b + c we see from eqn. (B7) that: µ γ + a,b + c = µ γ + a,b µ γ + a,c e i π ( α γ + a,b + α γ + a,c − α γ + a,b + c ) (C2)where we have defined α γ + a,b ≡ h sc γ + a + h sc b − h sc γ + a + b (C3)Repeatedly using eqn. (C2) we immediately have µ γ + a,b = µ γ + a,b − µ γ + a, e i π ( α γ + a, + α γ + a,b − − α γ + a,b ) = µ γ + a,b − µ γ + a, e i π (2 α γ + a, + α γ +1 ,b − − α γ + a,b ) = · · · = µ bγ + a, e i π ( bα γ + a, − α γ + a,b ) (C4)By requiring that µ γ + a,n = 1 since ψ n is the identityoperator, we have µ γ + a, = e − i πα γ + a, +2 π i κγ + an (C5)where κ γ + a is an Z n integer. As a result we can obtainall commutation factors: µ γ + a,b = e − i πα γ + a,b +2 π i bκγ + an (C6) Due to the consistency condition (B4) we also have µ b,γ + a = e i πα γ + a,b − π i bκγ + an (C7)Now we implement eqn. (B7) again with A = ψ b , B = σ γ + a , C = ψ c , D = σ γ + a + c to see whether there areany new consistency conditions for quasiparticle scalingdimensions h sc γ + a µ b,γ + a µ b,c µ γ + a + c,b e i π ( α γ + a,b + α b,c − α b,γ + a + c ) = e π i( α γ + a,b + α b,c − α γ + a + c,b ) µ b,c e − i πα b,c +2 π i b ( κγ + a + c − κγ + a ) n = e − i πbcα , +2 π i b ( κγ + a + c − κγ + a ) n = e π i b κγ + a + c − pc − κγ + an (C8)where we used eqn. (E8), q ≡ nα , / ∈ Z , andthat ∆ ( γ + a, b, c ) = α γ + a,b + α b,c − α γ + a + c,b ∈ N .Eqn. (B7) with A = σ γ + a , B = ψ b , C = ψ c , D = ψ b + c does not produce any new conditions. We cansee that all the consistency conditions on { µ γ + a,b } , i.e.eqn. (B4) and eqn. (B7) can be guaranteed by choosingeqn. (C6), eqn. (C7) and the integer κ γ + a as κ γ + a = κ γ + qa mod n (C9)We find that µ γ + a,b and µ b,γ + a can almost be deter-mined from k sc γ ; a : µ γ + a,b = e − i πα γ + a,b +2 π i b ( κγ + aq ) n ,µ b,γ + a = e i πα γ + a,b − π i b ( κγ + aq ) n ,q = nα , / ∈ Z . (C10)However, we need to supply a Z n integer κ γ to fully fix µ γ + a,b and µ b,γ + a from k sc γ ; a . (Note that α γ + a,b = h sc b − P a + bi = a k sc γ ; i .) APPENDIX D: GENERALIZED JACOBIIDENTITY1. GJI’s of an associative vertex algebra In the above, we only considered the associativity ofthe vertex algebra through the commutation factor µ AB .Although some new conditions on pattern of zeros andsome relations between the quasiparticle scaling dimen-sions are obtained, the associativity of the algebra is notfully utilized. To fully use the associativity condition ofthe vertex algebra, we need to derive the generalized Ja-cobi Identity.5Choose f ( z, w ) in the following relation I | z | > | w | d z I d wf ( z, w ) − I | w | > | z | d w I d zf ( z, w ) = I d w I w d zf ( z, w ) (D1)to be the operator function f ( z, w ) = A ( z ) B ( w ) C (0)( z − w ) γ AB w γ BC z γ AC (D2)with α AB − γ AB , α AC − γ AC , α BC − γ BC ∈ Z , we obtain the generalized Jacobi Identity (GJI) α BC − γ BC − X j =0 ( − j (cid:18) γ AB j (cid:19) [ A [ BC ] γ BC + j +1 ] γ AB + γ AC +1 − j − µ AB ( − α AB − γ AB α AC − γ AC − X j =0 ( − j (cid:18) γ AB j (cid:19) [ B [ AC ] γ AC + j +1 ] γ AB + γ BC +1 − j = α AB − γ AB − X j =0 (cid:18) γ AC j (cid:19) [[ AB ] γ AB + j +1 C ] γ BC + γ AC +1 − j (D3)where (cid:0) nm (cid:1) is the binomial function.When we choose γ AB − α AB = γ AC − α AC = γ BC − α BC = 0, eqn. (D2) is a regular function with the asymptoticbehavior lim z → lim w → f ( z, w ) = lim z → z α AB + α AC A ( z ) D (0) = lim z → z α AB + α AC − α AD [ AD ] α AD (0) (D4)Since f ( z, w )) is an analytic function of both z and w , z α AB + α AC − α AD should still be an analytic function of z . Thus α AB + α AC − α AD should be a non-negative integer, allowing us to obtain the consistency condition (B9).For clarity we introduce three integers n AB , n AC , n BC as γ AB = α AB − − n AB γ AC = α AC − − n AC γ BC = α BC − − n BC (D5)and the GJI (D3) can be rewritten as( − n BC n BC X j =0 ( − j (cid:18) α AB − − n AB n BC − j (cid:19) [ A [ BC ] α BC − j ] α AB + α AC − − ( n AB + n AC + n BC )+ j + µ AB ( − n AB + n AC n AC X j =0 ( − j (cid:18) α AB − − n AB n AC − j (cid:19) [ B [ AC ] α AC − j ] α AB + α BC − − ( n AB + n AC + n BC )+ j = n AB X j =0 (cid:18) α AC − n AC − n AB − j (cid:19) [[ AB ] α AB − j C ] α BC + α AC − − ( n AB + n AC + n BC )+ j (D6)The GJI (D3) or (D6) is the associativity condition ofa vertex algebra. It generalizes the usual Jacobi identityof a Lie algebra to the case of an infinite-dimensionalLie algebra (the vertex algebra here), with the usual Liebracket (the commutator) defined by OPE in eqn. (B1)We say that the theory is associative up to a certain or-der if all the GJI’s are satisfied up to this order in OPE.Applying the GJI, more conditions on the patterns of ze-ros can be found. More importantly, those conditions arelikely to be the necessary and the sufficient conditions.For example, by choosing C in GJI (D6) to be theidentity operator 1 (note that we have ∆ ( A, B, 1) = 0), n AB = k ≥ , n BC = − , n AC = 0 and makinguse of OPE (B10), we immediately reach the followingrelation µ AB [ BA ] α AB − k = k X j =0 ( − j ( k − j )! ∂ k − j [ AB ] α AB − j (D7)This allows us to obtain the OPE of [ B, A ] to the sameorder with the OPE of [ A, B ] to a certain order in hand.For example, we have[ ψ i T ] = ∂ [ T ψ i ] − [ T ψ i ] = ( h sc i − ∂ψ i µ T,ψ i = 1 and α T,ψ i = 2. As a special case ofeqn. (D7) we have[ AA ] α AA − k − = 12 k X j =0 ( − j (2 k + 1 − j )! ∂ k +1 − j [ AA ] α AA − j (D8)This relation is actually an example, showing how we“derive” (or more precisely, obtain the consistent condi-tions of) higher order OPE’s from the known OPE’s upto a certain order based on GJI’s. 2. “Useful” GJI’s of a vertex algebra up to acertain order In practice we need to extract the consistent conditionsof a vertex algebra from a set of “useful” GJI’s concerningonly the OPE’s up to a certain order. The OPE’s up tothis order are determined already except for some struc-ture constants (usually complex numbers). Other GJI’sinvolving higher order OPE’s do not serve as constraintsto the vertex algebra up to this order, since we can alwaysintroduce new operators into this infinite-dimensional Liealgebra in higher order OPE’s. For example, a generic Z n vertex algebra is defined by OPE’s between currents { ψ a } up to leading order with [ ψ i , ψ j ] α i,j = C i,j ψ i + j .Let’s now consider the GJI (D6) of three operators( A, B, C ), with the corresponding vertex algebra definedup to ( N AB , N BC , N AC ) order, i.e. [ AB ] α AB − i is knownup to structure constants for all 0 ≤ i ≤ N AB in the OPE(B1) of operators A and B . For example, in a special Z n simple-current vertex algebra defined by OPE’s (67)-(70), with ( A = ψ i , B = ψ j ) we have N AB = 0 if i + j =0 mod n or N AB = 2 if i + j = 0 mod n . Let’s furtherassume the following relation α [ AB ] αAB − j ,C = α [ AB ] αAB ,C + j, if [ AB ] α AB − j = 0(D9)is satisfied by any operators ( A, B, C ) of this vertex alge-bra. It’s straightforward to verify that Z n simple-currentvertex algebra indeed obeys the above relation: e.g.( A = ψ i , B = ψ n − i , C = ψ j ) we have 1 = [ AB ] α AB , T =[ AB ] α AB − and α T,ψ j = 2 , α ,ψ j = 0. Defining the fol-lowing quantity: N ABC ( n AB , n BC , n AC ) ≡ min { N [ AB ] αAB − j ,C , ≤ j ≤ N AB ; N B, [ AC ] αAC − j , ≤ j ≤ N AC ; N A, [ BC ] αBC − j , ≤ j ≤ N BC } (D10)then we can obtain all the “useful” consistent conditionsof the vertex algebra from the GJI (D6), by choosing n AB ≤ N AB , n BC ≤ N BC , n AC ≤ N AC ;∆ ( A, B, C ) − ≤ n AB + n BC + n AC ≤ ∆ ( A, B, C ) − N ABC ( n AB , n BC , n AC ) (D11) Any other choice with larger ( n AB , n AC , n BC ) will in-volve higher order OPE’s. Generally speaking, the setof “useful” GJI’s satisfying eqn. (D11) will be translatedinto a set of nonlinear equations of structure constants.(here “structure constants” have a broader meaning thanusual, e.g. h sc i c in eqn. (68) and h sc i in eqn. (69) shouldalso be considered as structure constants.) Some of theseequations become consistent conditions of this vertex al-gebra, while others help define this vertex algebra e.g. bydetermining the structure constants { C a,b , C a,γ + b } andcentral charge c of a Z n simple-current vertex algebra, asis shown in section V and VII. APPENDIX E: ASSOCIATIVITY OF Z n VERTEXALGEBRA AND NEW CONDITIONS ON h sc a AND C ab In this section, we apply the consistency conditions ofcommutation factor discussed in Appendix B and GJIdiscussed in Appendix D to a Z n vertex algebra. Thisallows us to derive additional conditions on the scalingdimension h sc a from the associativity of Z n vertex algebra.As mentioned earlier, a generic FQH wave function canbe expressed as a correlation function of an associativevertex algebra obeying the following OPE: ψ a ( z ) ψ b ( w ) = C a,b ψ a + b ( w ) + O ( z − w )( z − w ) α a,b (E1)where we have α a,b ≡ h sc a + h sc b − h sc a + b mod n . This guar-antees the quasi-Abelian fusion rule ψ a ψ b ∼ ψ a + b (see34). Moreover, we choose the normalization of simplecurrents ψ a to be eqn. (44). 1. New conditions from the commutation factors If we use the radial order (38) to calculate correlationfunction, then the continuity of the correlation functionrequires that( z − w ) α VaVb V a ( z ) V b ( w ) = µ V a ,V b ( w − z ) α VaVb V b ( w ) V a ( z )(E2)Since the operators e i aφ ( z ) / √ ν in the Gaussian model sat-isfy ( z − w ) a ν + b ν − ( a + b )22 ν e i aφ ( z ) / √ ν e i bφ ( w ) / √ ν = ( w − z ) a ν + b ν − ( a + b )22 ν e i bφ ( w ) / √ ν e i aφ ( z ) / √ ν , (E3)the simple current operator satisfy the following commu-tation relation( z − w ) α ab ψ a ( z ) ψ b ( w ) = µ ab ( w − z ) α ab ψ b ( w ) ψ a ( z ) , (E4)where α ab = h sc a + h sc b − h sc a + b . (E5)7We stress that the relation (E4) is required by the conti-nuity of the correlation function of the electron operators.The commutation factors µ ij satisfy some consistencyrelations, which is discussed in the appendix B under amore general setting. The conditions (B4), (B5), and(B7) are the conditions on the commutation factors µ ij that were obtained from the associativity of the vertexalgebra. From the n -cluster condition ψ n = ψ n = 1, wealso have µ ij = 1 , if i = 0 mod n or j = 0 mod n. (E6)due to the definition of the identity operator ψ = 1shown in Appendix B.Those conditions, (B4), (B5), (B7), and (E6), can beexpressed as the extra condition on the scaling dimen-sions h sc a . We note that according to eqn. (B7) andeqn. (B8) µ ij = µ i, µ i,j − ( − α i, + α i,j − − α i,j = µ i, µ i,j − ( − α i, + α i,j − − α i,j = · · · = µ ji, ( − jα i, − α ij (E7)A similar manipulation leads to µ ,i = µ i , ( − iα , − α ,i ,and we can write the commutation factor in a symmetricway µ ij = ( − ijα , − α i,j = ± . (E8)We see that µ ij can be expressed in terms of h sc i . Eq.(E8) also implies that ijα , − α i,j = integer , (E9)which is actually guaranteed by eqn. (29). The condition(E6) becomes ijα , − α i,j = even , if i = 0 mod n or j = 0 mod n. (E10)Now we use { M k , k = 1 , , , · · · , [ n ]; p } to describe h sc a (see section A 2). So the consistent conditions on { h sc a } can be translated into the consistent conditions on { M k ; p } . We note that α i,j ≡ α ψ i ,ψ j = h sc i + h sc j − h sc i + j = p ijn − M i − M j + M i + j mod n − p ( i + j − n ) θ ( i + j − n ) (E11)for 1 ≤ i, j ≤ n − j = n in (E10) we have nα , = 2 p = even (E12)As a result p ∈ N . (E13) n = odd M k ∈ Z n = even M Z +1 ∈ Z M Z ∈ Z TABLE IX: The parity of M k with different parity of n fora special Z n simple-current vertex algebra with OPE (68),according to constraint (E17). The parafermion scaling di-mension { h sc a } is given by h sc a = pa ( n − a ) /n − M a , with p being a non-negative integer. Besides, µ i,i = 1 becomes another constraint i α , − α i,i = even ∀ i = 1 , , · · · n − Z n simple cur-rent vertex algebra and the definition of commutationfactor (E4) we immediately have µ a,b = 1 , if a + b = 0 mod n (E15)which becomes an extra constraint i ( n − i ) α , − α i,n − i = even ∀ i = 1 , , · · · n − M i − M i mod n = 2 M i = even (E17)This determines the parity of { M k } as summarized inTable IX. Notice that we always have M k ∈ Z for sucha special Z n vertex algebra.However, the constraint (E16) is too strong and is notnecessary (this is why we use ”special” as a descriptionhere). For example, a 4-cluster state called Gaffnian ex-plicitly violates it since we have µ , = − = 1 for aGaffnian vertex algebra. To remove the constraint (E16),we need to modify the normalization C a, − a = 1 in OPE(40) to:for a ≤ n/ nψ a ( z ) ψ − a ( w ) = 1( z − w ) h sc a + ... (E18)and for a > n/ nψ a ( z ) ψ − a ( w ) = µ a, − a ( z − w ) h sc a + ... (E19)If we adopt the more general OPE (E18) and (E19),the corresponding consistent conditions on { M k } fromeqn. (E11) and eqn. (E14) become2 M i − M i mod n = even (E20)This leads to some conclusions on the parity of { M k } assummarized in Table X. Generally we don’t have M k ∈ Z for such a generic Z n vertex algebra.To summarize, (E12) and (E14) are the extra condi-tions from commutation factors for a generic Z n vertexalgebra. They become (E13) and (E20) when translatedinto consistent conditions of { M k ; p ; m } , as a supplementto conditions (A15)-(A18) in Appendix A 2.8 n = odd M k ∈ Z n = 2 mod 4 M Z +1 ∈ Z M Z ∈ Z n = 0 mod 4 2 M i − M i mod n ∈ Z M Z ∈ Z TABLE X: The parity of M k with different parity of n for a Z n simple current vertex algebra with OPE (71), according toconstraint (E20). The parafermion scaling dimension { h sc a } isgiven by h sc a = pa ( n − a ) /n − M a , with p being a non-negativeinteger. 2. New conditions from GJI As shown in Appendix D, all GJI’s must be satisfiedfor the associativity of the vertex algebra. With the OPE(E1), we have N a,b ≡ N ψ a ,ψ b = 0 and the useful GJI’s arevery limited. We list the consistent conditions from use-ful GJI’s of this generic Z n vertex algebra below. Thenwe summarize the new consistent conditions on { h sc a } and { c } . a. A list of useful GJI’s: ( A = ψ a , B = ψ b , C = ψ c ) Using the notations in Appendix D, here we have N AB = N BC = N AC = 0.If ∆ ( a, b, c ) = 0, all the 3 useful GJI’s satisfyingeqn. (D11) are n AB = − , n BC = 0 , n AC = 0 ⇒ ( C b,c C a,b + c − µ a,b C a,c C b,a + c ) ψ a + b + c = 0 n AB = 0 , n BC = − , n AC = 0 ⇒ ( µ a,b C a,c C b,a + c − C a,b C a + b,c ) ψ a + b + c = 0 n AB = 0 , n BC = 0 , n AC = − ⇒ ( C b,c C a,b + c − C a,b C a + b,c ) ψ a + b + c = 0If ∆ ( a, b, c ) = 1, the only 1 useful GJI satisfyingeqn. (D11) is n AB = 0 , n BC = 0 , n AC = 0 ⇒ ( C a,b C a + b,c − C b,c C a,b + c − µ a,b C a,c C b,a + c ) ψ a + b + c = 0For ∆ ( a, b, c ) ≥ C i,j . b. Summary of new consistent conditions from GJI As shown in Appendix E 2 a, for ∆ ( a, b, c ) ≡ ∆ ( ψ a , ψ b , ψ c ) = 0 the extra consistent conditions are C a,b C a + b,c = C b,c C a,b + c = µ a,b C a,c C b,a + c (E21) For ∆ ( a, b, c ) = 1 the corresponding consistent condi-tion is C a,b C a + b,c = C b,c C a,b + c + µ a,b C a,c C b,a + c (E22)For ∆ ( a, b, c ) ≥ a, b, c ) are. Now let’s further specify ( a, b, c ) anduse the normalization (44) of structure constants to ob-tain new conditions.If a + b, a + c, b + c = 0 mod n :For ∆ ( a, b, c ) = 0 we have C a,b C a + b,c = C b,c C a,b + c = µ a,b C a,c C b,a + c (E23)For ∆ ( a, b, c ) = 1 we have C a,b C a + b,c = C b,c C a,b + c + µ a,b C a,c C b,a + c (E24)If a ± b = 0 mod n :For ∆ ( a, b, − b ) = 0 we have C a,b C a + b, − b = C b, − b = µ a,b C a, − b C b,a − b (E25)For ∆ ( a, b, − b ) = 1 we have C a,b C a + b, − b = C b, − b + µ a,b C a, − b C b,a − b (E26)If a = n/ n :For ∆ ( a, a, − a ) = 0 we have µ a, − a = C a, − a = C a,a C a, − a = 1 (E27)For ∆ ( a, a, − a ) = 1 we have µ a, − a = µ − a,a = − , (E28) C a,a C a, − a = 2 C a, − a . (E29)If n = even, we require ∆ ( n , n , n ) = 1 since otherwise ψ n/ ≡ { C a,b } , while othersserve as the new conditions on the pattern of zeros { h sc a } or { S a } . As a summary, the extra consistent conditionsfor the pattern of zeros from GJI’s are µ a, − a = 1 if ∆ ( a, a, − a ) = 0 ,µ a, − a = − , if ∆ ( a, a, − a ) = 1 , ∆ ( n , n , n h sc n = 1 , n = even . (E31)where we need (E8) to relate commutation factor µ a,b with the pattern of zeros. Note that the first twoconditions in the above can be rewritten as a α , + α a, − a =even if ∆ ( a, a, − a ) = 0 and a α , + α a, − a =oddif ∆ ( a, a, − a ) = 1. Since a α , − α a,a =even and9∆ ( a, a, − a ) = α a,a + α a, − a , the two conditions are al-ways satisfied.Obviously these extra conditions, based on the mostgeneric OPE (E1) of a Z n vertex algebra, are not enoughto determine the structure constants { C a,b } of this vertexalgebra. In order to have more consistent conditions andto determine the structure constants, we need to spec-ify higher order terms in the OPE (E1). This is donethrough defining Z n simple current vertex algebra in sec-tion V, essentially by introducing the energy momentumtensor T and Virasoro algebra into the vertex algebra.The corresponding extra consistent conditions are sum-marized in section V B and VI D. We can obtain evenmore extra conditions from GJI’s when we fix the sub-leading term of OPE’s between simple currents, as shownin Appendix F. APPENDIX F: SUBLEADING TERMS IN OPEOF A Z n SIMPLE-CURRENT VERTEXALGEBRA AND MORE CONSISTENTCONDITIONS1. “Deriving” subleading terms in OPE from GJI’s In this section we show how to “derive” the subleadingterm in OPE (67) of a Z n simple-current vertex algebraas an example.First we notice that the subleading term [ ψ a ψ b ] α a,b − should have a scaling dimension of h sc a + b + 1, thus wepropose the following conclusion:[ ψ a ψ b ] α a,b − = C a,b d a,b ∂ψ a + b (F1)Then we can use GJI’s to determine the expression of d a,b in terms of scaling dimensions { h sc i } .First we choose ( A = T, B = ψ a , C = 1) and( n AB = − n BC = 1 , n AC = − 1) in GJI (D6). Since∆ ( T, ψ a , 1) = 0 and [ ψ a , = ∂ψ a we have the follow-ing consistent conditions from this GJI:[ T ∂ψ a ] = 2[ T ψ a ] = 2 h sc a ψ a (F2)Then we choose ( A = T, B = ψ a , C = ψ b ) , a + b =0 mod n and ( n AB = 0 , n BC = 1 , n AC = 0) in GJI(D6). It’s easy to verify that ∆ ( T, ψ a , ψ b ) = 2 since wehave α T,ψ i = 2 , ∀ i . Plugging in eqn. (F1) and this GJIyields a consistent condition on d a,b :2 d a,b h sc a + b = ( h sc a + b + h sc a − h sc b ) (F3)Thus we conclude that as long as h sc a + b = 0 (this shouldhold in most cases except for “strange” examples likeGaffnian) we have d a,b = 12 (1 + h sc a − h sc b h sc a + b ) (F4) 2. More consistent conditions due to subleadingterms in OPE Now with the subleading terms we have more usefulGJI’s and therefore more consistent conditions on thedata { n ; m ; h sc a ; c } characterizing a Z n simple current ver-tex algebra. In this section we shall show the extra con-sistent conditions accompanied with the introduction ofthe subleading term (F1), (F4) in OPE of [ ψ a , ψ b ].It turns out that there are many more useful GJI’s con-sidering the subleading order OPE (F1) with eqn. (F4).In many cases the new consistent conditions are ex-tremely complicated, so we will only show the completeconsistent conditions in several cases (which will be uti-lized in section VII for some examples). a. { A, B, C } = { ψ a , ψ b , ψ c } , a + b , b + c , a + c , a + b + c = 0 mod n Right now we have N AB = N BC = N AC = 1 > ( a, b, c ) = 0 the complete consistent conditionsare summarized as: C a,b C a + b,c = C b,c C a,b + c = µ a,b C a,c C b,a + c α a,b + c d b,c = α b,a + c d a,c = α a,b ( d a,b + c + d b,a + c − d a + b,c ) ∂ψ a + b + c ≡ d a,b + c − d a,c (1 − d b,a + c )] ∂ψ a + b + c ≡ d a,b + c − d a,c d a + c,b ) ∂ψ a + b + c ≡ a, b, c ).For ∆ ( a, b, c ) = 1 some of the new consistent condi-tions are: C b,c C a,b + c C a,b C a + b,c = (1 − α a,b + d b,c α a,b + c ) − µ a,b C a,c C b,a + c C a,b C a + b,c = α a,b − d b,c α a,b + c α a,b − d b,c α a,b + c − α a,b + d a,c α b,a + c )( α a,b − d b,c α a,b + c ) = 1(1 − α a,b + d b,c α a,b + c )( α a,c − d a,b α a + b,c ) = 1 (F6)we didn’t show those lengthy consistent conditions withthe form of ( · · · ) ∂ψ a + b + c = 0 here.For ∆ ( a, b, c ) = 2 the new consistent condition are:( α a,b − d b,c α a,b + c ) C b,c C a,b + c (F7)= µ a,b C a,c C b,a + c ( α a,b − α b,a + c d a,c ) , ( α a,b − d b,c α a,b + c − C b,c C a,b + c (F8)= − C a,b C a + b,c ( α a,c − α a + b,c d a,b ) , ( α a,b − d a,c α b,a + c − C a,b C a + b,c (F9)= µ a,b C a,c C b,a + c ( α a,b − α b,a + c d a,c − , ≡ ∂ψ a + b + c { C a,b C a + b,c d a + b,c (F10)+ C b,c C a,b + c [ d a,b + c ( α a,b + d b,c − − d b,c α a,b + c ) − d b,c ]+ µ a,b C a,c C b,a + c [ d b,a + c ( α a,b + d a,c − − d a,c α b,a + c ) − d a,c ] } , ≡ ∂ψ a + b + c { C b,c C a,b + c d a,b + c (F11)+ C a,b C a + b,c d a + b,c ( − α a,c − d a,b + 1 + d a,b α a + b,c )+ µ a,b C a,c C b,a + c [ d b,a + c ( α a,b + d a,c − − d a,c α b,a + c ) − d a,c ] } , ≡ ∂ψ a + b + c { µ a,b C a,c C b,a + c d b,a + c (F12)+ C b,c C a,b + c [ d a,b + c ( α a,b + d b,c − − d b,c α a,b + c ) − d b,c ]+ C a,b C a + b,c d a + b,c ( α a,c + d a,b − − d a,b α a + b,c ) } . For ∆ ( a, b, c ) = 4 the new consistent conditions are: n AB = 1 , n BC = 1 , n AC = 1 ⇒ C a,b C a + b,c (2 − α a,c + d a,b α a + b,c )+ C b,c C a,b + c (2 − α a,b + d b,c α a,b + c ) − µ a,b C a,c C b,a + c (2 − α a,b + d a,c α b,a + c ) = 0 (F13)For ∆ ( a, b, c ) ≥ b. { A, B, C } = { ψ a , ψ b , ψ − a − b } , a, b, a + b = 0 mod n For ∆ ( a, b, − a − b ) = 0 the new consistent conditionare: C a,b C a + b, − a − b = C b, − a − b C a, − a = µ a,b C a, − a − b C b, − b α a,b = α a, − a d b, − a − b = α b, − b d a, − a − b α a, − a − b = α a, − a d − a − b,b = α − a − b,a + b d a,b d b, − a − b h sc a = d a, − a − b h sc b = α a,b / d − a − b,b h sc a = d a,b h sc a + b = α a, − a − b / d a,b h sc a = α a, − a − b ( h sc a + 1 − h sc a + b ) / d − a − b,b h sc a + b = α − a − b,a ( h sc a + b + 1 − h sc a ) / d a, − a − b h sc a = α a,b ( h sc a + 1 − h sc b ) / a ↔ b exchange.For ∆ ( a, b, − a − b ) = 4 there is still only 1 useful GJIfor ( A, B, C ) in a certain order and the new consistentcondition are: n AB = 1 , n BC = 1 , n AC = 1 ⇒ C a,b C a + b, − a − b (2 − α a, − a − b + d a,b α a + b, − a − b )+ C b, − a − b C a, − a (2 − α a,b + d b, − a − b α a, − a ) − µ a,b C a, − a − b C b, − b (2 − α a,b + d a, − a − b α b, − b ) = 0 (F15)For ∆ ( a, b, − a − b ) ≥ c. { A, B, C } = { ψ a , ψ b , ψ − b } , a ± b = 0 mod n Now we have N ψ a ,ψ ± b = 1 , N ψ b ,ψ − b = 2.For ∆ ( a, b, − b ) = 0 the consistent conditions are: h sc b h sc a = α a, ± b = 0 , h sc b ∂ψ a ≡ C a,b C a + b, − b = µ a,b C a, − b C b,a − b = C b, − b ( d a ± b, ∓ b − ∂ψ a = ( d a ± b, ∓ b − ∂ψ a = d ± b,a ∂ψ a = d ± b,a ∓ b ∂ψ a ≡ ( a, b, − b ) = 2 the consistent conditions are: µ a,b C a, − b C b,a − b C b, − b = α a,b α a,b − d a, − b α b,a − b = α a,b ( α a,b − h sc a h sc b c (F17) C a,b C a + b, − b C b, − b = 2 − α a,b α a, − b − d a,b α a + b, − b = ( α a,b − α a,b − h sc a h sc b c (F18) C a,b C a + b, − b µ a,b C a, − b C b,a − b = α a, − b − − d a,b α a + b, − b α a,b − − d a, − b α b,a − b (F19) { d ± b,a ∓ b [ 2 h sc a h sc b c + α a, ± b ( α a, ± b − − d a ± b, ∓ b )[ 2 h sc a h sc b c + ( α a, ± b − α a, ± b − − h sc b c } ∂ψ a ≡ { [ 2 h sc a h sc b c + α a, ± b ( α a, ± b − d a, − b ( d b,a − b − 1) + 1]+ α a,b d b,a − b − h sc b c } ∂ψ a ≡ { [ ( α a,b − α a,b − h sc a h sc b c ](1 − d a,b d a + b, − b )+ ( α a,b − d a + b, − b − h sc b c } ∂ψ a ≡ { [ ( α a,b − α a,b − h sc a h sc b c ] d a + b, − b ( d a,b − α a,b ( α a,b − h sc a h sc b c ] d b,a − b + ( α a,b − − d a + b, − b ) } ∂ψ a ≡ { [ ( α a,b − α a,b − h sc a h sc b c ] d a + b, − b + [ α a,b ( α a,b − h sc a h sc b c ]( d a, − b d b,a − b − d a, − b − d b,a − b )+ α a,b ( d b,a − b + 1) − } ∂ψ a ≡ { [ ( α a,b − α a,b − h sc a h sc b c ](2 − d a,b ) d a + b, − b + [ ( α a,b − α a,b h sc a h sc b c ]( d a, − b d b,a − b − d a, − b − d b,a − b )+ 1 + α a,b d b,a − b + ( α a,b − d a + b, − b } ∂ψ a ≡ c can be determined consis-tently from the first two conditions. Notice that aftera b ↔ − b exchange the above conditions should also besatisfied.For ∆ ( a, b, − b ) = 4 there are 4 useful GJI’s for( A, B, C ) in a certain order now and the consistent con-ditions are: C a,b C a + b, − b = µ a,b C a, − b C b,a − b ( α b,a − b d a, − b − α a,b + 1)+ C b, − b [ ( α a,b − α a,b − h sc a h sc b c ] (F26) µ a,b C a, − b C b,a − b = C a,b C a + b, − b ( d a,b α a + b, − b − α a, − b + 1)+ C b, − b [ ( α a,b − α a,b − h sc a h sc b c ] (F27) { C a,b C a + b, − b d a + b, − b (2 − α a, − b − d a,b + d a,b α a + b, − b )+ C b, − b [ ( α a,b − α a,b − h sc a − h sc b c ]+ µ a,b C a, − b C b, − a − b [ d b,a − b ( α a,b + d a, − b − − d a, − b α b,a − b ) − d a, − b ] } ∂ψ a ≡ b ↔ − b .For ∆ ( a, b, − b ) ≥ d. { A, B, C } = { ψ a , ψ a , ψ − a } , a = n/ mod n Now we have N ψ a ,ψ a = 1 , N ψ a ,ψ − a = 2.For ∆ ( a, a, − a ) = α a,a + 2 h sc a = 0 the consistent con-ditions are the same as in section V B: h sc a = h sc2 a = α a,a = 0 , ∂ψ a ≡ C a,a C a, − a = C a, − a = C − a,a = µ a, − a = 1 (F29)For ∆ ( a, a, − a ) = α a,a + 2 h sc a = 1 the consistent con-ditions are: h sc2 a = 3 , h sc a = 1 , c = − , µ a, − a = − C a,a C a, − a = 2 C a, − a , C a, − a = − C − a,a ( d a, − a − 32 ) ∂ψ a = ( d − a, a + 12 ) ∂ψ a ≡ d a,a = 1 / d − a, a = − / , d a, − a = 3 / , d a,a = 1 / h sc a = 1 , α a,a = − ( a, a, − a ) = α a,a + 2 h sc a = 2 the consistent con- ditions are c = 2 h sc a − h sc a , α a,a = 2 − h sc a ,C a,a C a, − a = 2 h sc a = 0 C a, − a = C − a,a = µ a, − a = 1( d a, − a − h sc a ) ∂ψ a = ( d − a, a + 1 − h sc a ) ∂ψ a ≡ h sc a − d a,a = 1 / d a, − a =2 − ( h sc a ) − , d − a, a = ( h sc a ) − − ( a, a, − a ) = α a,a + 2 h sc a = 4 the consistent con-ditions are α a,a = 4 − h sc a , C a, − a = C − a,a = µ a, − a = 1 C a,a C a, − a = h sc a (2 h sc a − h sc a c ) = 0( d a, − a − h sc a ) ∂ψ a = ( d − a, a + 1 − h sc a ) ∂ψ a ≡ h sc a − d a,a − / 2) = 0 (F32)For ∆ ( a, a, − a ) = α a,a + 2 h sc a = 6 there is only 1useful GJI for ( A, B, C ) in a certain order now, and theconsistent conditions are:(2 h sc a − d a,a − / 2) = 0( µ a, − a − h sc a ) c + ( h sc a − h sc a − ( a, a, − a ) = α a,a + 2 h sc a ≥ e. { A, B, C } = { ψ n/ , ψ n/ , ψ n/ } , n = even This section is exactly the same as section V B 4 sincewe still have N AB = N BC = N AC = 2 if A = B = C = ψ n/ . The subleading term in OPE (F1) has no effect onthese GJI’s. f. { A, B, C } = { ψ a , ψ b , σ γ + c } , a + b = 0 mod n Now we have N ψ a ,ψ b = 1 > 0, so there are new usefulGJI’s in this case than in section VI D 2. Therefore wehave more consistent conditions.For ∆ ( a, b, γ + c ) = 0 the consistent conditions are: µ a,b C a,γ + c C b,γ + a + c = C a,b C a + b,γ + c = C b,γ + c C a,γ + b + c ,α a,γ + c = α a + b,γ + c d a,b = α c,γ + a + b d a,b . (F34)For ∆ ( a, b, γ + c ) = 1 the consistent conditions are: µ a,b C a,γ + c C b,γ + a + c = C a,b C a + b,γ + c ( α a,γ + c − d a,b α a + b,γ + c ) ,C b,γ + c C a,γ + b + c = C a,b C a + b,γ + c (1 − α a,γ + c + d a,b α a + b,γ + c ) ,d a,b ( α a + b,γ + c − α c,γ + a + b ) = 0 . (F35)2For ∆ ( a, b, γ + c ) = 2 the consistent conditions are: C b,γ + c C a,γ + b + c − µ a,b C a,γ + c C b,γ + a + c = C a,b C a + b,γ + c ( α a,γ + c − − d a,b α a + b,γ + c ) ,d a,b ( α a + b,γ + c − α c,γ + a + b ) = 0 . 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