Nonabelian parafermions and their dimensions
aa r X i v : . [ h e p - t h ] M a r March, 2010
Nonabelian Parafermions and their Dimensions
Roman Dovgard and Doron Gepner
Department of Particle PhysicsWeizmann InstituteRehovot 76100, Israel
ABSTRACT
We propose a generalization of the Zamolodchikov-Fateev parafermions whichare abelian, to nonabelian groups. The fusion rules are given by the tensor productof representations of the group. Using Vafa equations we get the allowed dimensionsof the parafermions. We find for simple groups that the dimensions are integers.For cover groups of simple groups, we find, for n.G.m , that the dimensions are thesame as Z n parafermions. Examples of integral parafermionic systems are studiedin detail.onformal field theory in two dimensions has been a source of numerous resultsowing to its solvability and its rich structure. It has been successfully applied tostatistical mechanics and string theory.One line of such ideas is a generalization of the Ising model to so calledparafermions, first put forwards by Zamolodchikov and Fateev [1]. These are an-alytic currents with nonintegral spin. The known examples, so far, are based onthe cyclic group Z n , or products of cyclic groups, that is, abelian groups. Theparafermions appear in nature as fixed points of some models of magnets, e.g.,the Andrews Baxter Forrester models [2] and Z n clock models [1]. Also, theyare vital components in string compactification, since they are closely related to N = 2 superconformal theories, yielding solvable realistic string theories, featuringfor example, in the models of ref. [3].Our idea is to generalize the parafermions to nonabelian groups. We can writea parafermionic system for any group G . This we do by assuming that the currentsfall into representations of the group G . I.e., we have a parafermionic multiplet foreach representation of the group, G , and for each vector of the representation. Wethen postulate that the OPE of the parafermionic system obey the group symmetry.This is a straight forwards generalization of the notion of parafermions and, in theabelian case, it gives the usual results of Zamolodchikov and Fateev.To be specific, assume that I ranges over the representations of the group G and that the index i ranges over the vectors in each representation. We introducea parafermion which is a field ψ Ii ( z ), which is an a holomorphic field of dimension∆ I . Let f IJKijk denote the Clebsh-Gordon coefficient of the group. I.e., for eachelement of the group g ∈ G we have the relation, X i ′ ,j ′ ,k ′ Φ Iii ′ ( g )Φ Jjj ′ ( g )Φ Kkk ′ ( g ) f IJKi ′ j ′ k ′ = f IJKijk , (1)where Φ Iii ′ ( g ) is a matrix in the I th representation of the group G . e further define, f IJ ij = b IJij , (2)and X k b K ¯ Kk ′ k f k ( IJ ¯ K ) ij = f IJKijk ′ . (3)We then postulate the following operator product expansions (OPE) for theparafermions, ψ Ii ( z ) ψ Jj ( w ) = b IJij C I ( z − w ) I + (4) X k,K f k ( IJK ) ij ( z − w ) − ∆ I − ∆ J +∆ K (cid:20) ˜ C IJK ψ Kk ( w ) + ˜˜ C IJK ( z − w ) ∂ψ Kk ( w ) (cid:21) + h . o . t ., where h . o . t . stands for higher order terms. The constants C , ˜ C and ˜˜ C are deter-mined by the associativity of the OPE above, once the dimensions ∆ I are deter-mined. We postulate also ψ ( z ) = T ( z ), the stress tensor, ∆ = 2, C = c/ c is the central charge. Thus the algebra contains the Virasoro algebra andwe demand, accordingly, that, ψ Ii ( z ) is a primary field, ˜ C II = ∆ I , and, ˜˜ C II = 1.The fusion rules, i.e., the way operators fuse in the operator product expansion,are then given simply by the tensor product algebra of the representations of thegroup. This is easily calculated in specific examples by means of the charactertables of specific groups. Denote by χ I ( g ) the character in the representation I ofthe group, g ∈ G , χ I ( g ) = X i Φ Iii ( g ) . (5)Then the fusion coefficients of the parafermions are given by, N KIJ = 1 O ( G ) X g ∈ G χ I ( g ) χ J ( g ) χ K ( g ) ∗ , (6)which is reminiscent of the Verlinde formula [4]. Here, O ( G ) is the order of thegroup. hus, we can use Vafa’s equations [5] to calculate the dimensions of the parafermions,which are found up to some arbitrary integer multiplicative factor. Denote by α I = e πi ∆ I . (7)Then Vafa equations are( α I α J α K α L ) N IJKL = Y R α N IJKL ; R R , (8)where N IJKL = X R N IJR N RKL , (9)and N IJKL ; R = N IJR N RKL + N IKR N RJL + N ILR N RJK , (10)where we define N IJK = N ¯ KIJ .When G is abelian, we are back in the case of Zamolodchikov and Fateev. For a G = Z N group, we denote the I parafermion for the representation Φ I ( e πir/N ) = e πirI/N , for any I and r modulo N . The dimension of the I th parafermion is ∆ I and ∆ I = ∆ N − I since it is the complex conjugate field. We find from eq. (6) thatthe structure constant is N KIJ = 1 if K − I − J = 0 mod N and is zero otherwise.Here Vafa’s equations become,∆ I + ∆ J + ∆ K + ∆ L = ∆ K + L + ∆ K + I + ∆ K + J mod Z, (11)where I + J + K + L = 0 mod N are any. This equation, already appears in ref.[1], eq. (A4) there, derived from the mutual semilocality. This eq. (11) implies, in articular, by taking I = J = 1, that2∆ K +1 − ∆ K − ∆ K +2 = β mod Z, (12)where β = 2∆ − ∆ . Thus, ∆ K = − βK / Z is the unique solution to eq.(11), which satisfies, ∆ =integer and ∆ = ∆ N − . It follows that∆ I = M I + mI / ( sN ) , (13)where M I and m are arbitrary integers and s = 1 for odd N and s = 2 for even N .We set ∆ r = ∆ N − r . Thus, this method is consistent with the known abelian case.Thus, for each group we simply substitute the characters into Vafa equationsto find the dimensions of the parafermions.Let us introduce some basic notions of group theory. The group G is calledsimple if the only normal subgroups are itself or the trivial one. An automorphismis a one to one and onto map σ : G → G such that σ ( gh ) = σ ( g ) σ ( h ) , (14)where g, h ∈ G . An internal automorphism is the map, σ h ( g ) = hgh − , (15)where h is a fixed element of the group. We denote the automorphism groupby Aut( G ), which is a group under decompositions. We denote by Int( G ) theinternal automorphism subgroup of Aut( G ), which is a normal subgroup. Theouter automorphism group, Out( G ) is defined as the quotient group,Out( G ) = Aut( G )Int( G ) . (16)The group G itself is a subgroup of Aut( G ) by identifying it with Int( G ) (weassume that G is centerless, see below. If we denote the center by Z ( G ) then, nt( G ) ≈ G/Z ( G )). A group, H , is called almost simple, if there exists a simplegroup G such that G ⊂ H ⊂ Aut( G ) . (17)We call H a cover group by automorphism of the group G .We call the parafermion system integral if the only solution to Vafa equationis that all the dimensions are integral. Our result about this can be phrased as:Conjecture (1): The parafermion system of the group H is integral if and onlyif H is a nonabelian almost simple group or the trivial group.Another type of cover group is an extension by a center. The center of a group H is the subgroup of elements that commute with every group member. The centerof the group H is a normal subgroup denoted by Z ( H ), h ∈ Z ( H ) if and only if, gh = hg for all g ∈ H . Of course the center of a nonabelian simple group istrivial. If H is a group such that G = H/Z ( H ), where G is a simple group, wecall H a cover group of G by a center. Other type of center group is mixed bothby an automorphism and by a center, where W = H/Z ( H ) is an almost simplegroup of the simple group G , G ⊂ W ⊂ Aut( G ). We denote such a cover group as H = n.G.m , where the center is a Z n group and the outer automorphism is a Z m group, i.e., the quotient of the almost simple group to its simple subgroup is a Z m group. Of course, the outer automorphisms group can be nonabelian (for a simplegroup it is always solvable).Now, suppose that H is a Z n cover group of an almost simple group G , G = H/Z ( H ), where Z ( H ) ≈ Z n . Let I be an irreducible representation of H . By Schure’s lemma, since the center commutes with all elements of H , it is anirreducible representation of Z n . These representations are denoted by their chargemodulo n . So let d I mod n be the charge of the I th representation. Our result canthen be quoted as:Conjecture (2): The parafermion system of a cover group of the type H = n.G.m , where G is a nonabelian simple group, gives the same dimensions as Z n arafermions. I.e., the dimension of the I th representation is,∆ I = M I + md I / ( sn ) , (18)where s = 1 ( s = 2) for odd (even) n , and M I and m are some integers.The same applies to a center which is a product of cyclic groups. One justadds up the dimensions.We can give a sort of proof for this conjecture. Let d I = 0 then it is anirreducible representation of H , where G = H/Z ( H ), G is almost simple. Since therepresentation I acts trivially on the center, it can be lifted to a representation of G .Thus, by conjecture (1), the corresponding parafermion, ψ Ii , has integral dimension,and we can include it in the chiral algebra. So, all representations with d I = 0can be collected into an extended chiral algebra. Similarly, all representations withthe same d I = 0 can be collected into one representation of this extended algebra.Thus, the theory is, in fact, a Z n parafermionic system with this extended algebraand, using Vafa’s equations, conjecture (2) follows.The characters tables for the simple groups and their cover groups are enu-merated in the atlas of finite groups [6]. We wrote a computer program to solveVafa’s equations for each case. This way we found the dimensions for all the simplegroups with up to 14 conjugacy classes, along with most of their cover groups. Wefind that, indeed, conjectures (1) and (2) are obeyed and for any group of the type n.G.m , where G is simple, the dimensions are as for Z n parafermions in accordancewith conjecture (2).Our objects here is to study the nonabelian parafermions. There is one specialcase which is simpler to analyze. This is the case where all the dimensions areintegral. We have such a solution for any group. Of course, if the group is almostsimple and nonabelian, this is the only solution. For simplicity we assume that allthe dimensions are two, ∆ I = 2, for all the representations I . Then, the OPE, eq. ψ i ( z ) ψ j ( w ) = b ij c/ z − w ) + f kij (cid:20) ψ k ( w )( z − w ) + ∂ψ k ( w )( z − w ) (cid:21) , (19)up to regular terms, where for simplicity of notation we denote ψ Ii ( w ) by one index ψ i ( w ). Of course we can have such an algebra for any b ij and any f kij provided theyobey the associativity of the OPE. To check the latter, it is convenient to writethe algebra in modes, ψ i ( w ) = X n L ni w − n − , (20)and we get the commutator relations,[ L ni , L mj ] = b ij ( c/ n − n ) δ n + m, + X k f kij ( n − m ) L n + mk . (21)We take, of course, ψ ( w ) = T ( w ), the stress tensor, and b = f = 1. Theassociativity of the OPE is equivalent to the Jacobi identity,[ A, [ B, C ]] + [ B, [ C, A ]] + [ C, [ A, B ]] = 0 , (22)which, in this case, is obeyed if and only if the following relations hold, X k b ik f kjl = f ijl , (23)and f ijl is fully symmetric in all indices, along with X l f lik f fjl = X l f lij f flk , (24)for all i, j, k, f . With these relations the OPE is associative and we have a conformalfield theory with this extended algebra. In addition, unitarity requires that( f kij ) ∗ = f ¯ k ¯ i ¯ j . (25)Let us consider the case where G = Z N , that is, abelian parafermions, withthe algebra eq. (21). Here we assume that the representation weights are i , j and , defined modulo N , and that, f i + ji,j = 0 , f ki,j = 0 where k = i + j mod N (26)Also, b i,j = δ N ( i + j ) , (27)where δ N ( x ) = 1 if x = 0 mod N and is zero otherwise. The unitarity requirementbecomes, f N − i − jN − i,N − j = ( f ki,j ) ∗ . (28)The solution to the structure constants obeying eqs. (23-28) can be seen to begiven by f ii = 1 , (29) f i + ji,j = a i a i +1 . . . a i + j − a a . . . a j − , (30)where a i = f i +1 i, , (31)and a N = a N − = 1 , a r = e iφ r , (32)where φ r is a real phase obeying, φ r = φ r + N = φ N − − r . (33)This can be seen to be the most general solution, obeying the Jacobi identity andunitarity for any values of the real parameters φ r , r = 1 , , . . . , [( N − / x ] is the smallest integer bigger or equal to x . Thus we have a multiparameteralgebra depending on [( N − /
2] real angles. o study the unitarity of these field theories we can use the Kac determinantmethod. This is left for future work. There is one simple case, though. This iswhen all the angles vanish, f i + jij = 1 , (34)for all i and j . The algebra then assumes the form[ L in , L jm ] = c
12 ( n − n ) δ n + m δ N ( i + j ) + ( n − m ) L i + jn + m , (35)where n, m are integers, and i, j integers modulo N . We can define a new basis forthe algebra as, M rn = 1 N N − X s =0 e πisr/N L sn , (36)where r is defined modulo N . It is easy then to see that the M ’s form a set ofcommuting Virasoro algebras, all with the same central charge,[ M in , M js ] = δ i,j h c N ( n − n ) δ n + s + ( n − s ) M in + s i . (37)So the theory is a tensor product of Virasoro algebras with the central charge c/N .We conclude that the unitary minimal models of this algebra are for c < N andthe central charge and dimensions are given by, c = N (cid:18) − m ( m + 1) (cid:19) , m = 2 , , , . . . , (38) h = N X i =1 ( mp i − ( m + 1) q i ) − m ( m + 1) , (39)where 1 ≤ p i ≤ m and 1 ≤ q i ≤ m − he possible applications of this work are, as indicated in the introduction, itsimmediate importance to both statistical systems and string theory. For the latter,many parafermionic systems can be completed to an N = 2 superconformal fieldtheories, which can then yield space–time supersymmetric string theories in fourdimensions [3]. It is an obvious step to study the spectrum and realism of suchstring theories. In particular, if ∆ I < / G + ( z ) = r c + 23 ψ Ii ( z ) : e iγφ ( z ) : , (40) G − ( z ) = r c + 23 ψ ¯ I ¯ i ( z ) : e − iγφ ( z ) : , (41)where φ ( z ) is a canonical free boson and γ / I = 3 /
2, we normalized b I ¯ Ii ¯ i C I = 1,and c is the central charge of the parafermionic theory. The N = 2 superconformalalgebra will then be obeyed provided the central charge and the dimension arerelated by c = 2∆ I − I , (42)which is obeyed in many cases. EFERENCES
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