Nonabelian parafermions and their dimensions
ABSTRACT We propose a generalization of the Zamolodchikov-Fateev parafermions which are abelian, to nonabelian groups. The fusion rules are given by the tensor product of representations of the group. Using Vafa equations we get the allowed dimensions of 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 the same as $Z_n$ parafermions. Examples of integral parafermionic systems are studied in detail. Comment: 12 pages.
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ABSTRACT: Characterisations of finite groups in which normality is a transitive relation are presented in the paper. We also characterise the finite groups in which every subgroup is either permutable or coincides with its permutiser as the groups in which every subgroup is permutable.Journal of the Australian Mathematical Society 09/2003; 75(02):181 - 192. · 0.45 Impact Factor
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arXiv:1003.2342v1 [hep-th] 11 Mar 2010
Nonabelian Parafermions and their Dimensions
Roman Dovgard and Doron Gepner
Department of Particle Physics
Rehovot 76100, Israel
We propose a generalization of the Zamolodchikov-Fateev parafermions which
are abelian, to nonabelian groups. The fusion rules are given by the tensor product
of representations of the group. Using Vafa equations we get the allowed dimensions
of 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 the
same as Znparafermions. Examples of integral parafermionic systems are studied
Conformal field theory in two dimensions has been a source of numerous results
owing to its solvability and its rich structure. It has been successfully applied to
statistical mechanics and string theory.
One line of such ideas is a generalization of the Ising model to so called
parafermions, first put forwards by Zamolodchikov and Fateev . These are an-
alytic currents with nonintegral spin. The known examples, so far, are based on
the cyclic group Zn, or products of cyclic groups, that is, abelian groups. The
parafermions appear in nature as fixed points of some models of magnets, e.g.,
the Andrews Baxter Forrester models  and Zn clock models . Also, they
are vital components in string compactification, since they are closely related to
N = 2 superconformal theories, yielding solvable realistic string theories, featuring
for example, in the models of ref. .
Our idea is to generalize the parafermions to nonabelian groups. We can write
a parafermionic system for any group G. This we do by assuming that the currents
fall into representations of the group G. I.e., we have a parafermionic multiplet for
each representation of the group, G, and for each vector of the representation. We
then 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 the
abelian 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 introduce
a parafermion which is a field ψI
i(z), which is an a holomorphic field of dimension
denote the Clebsh-Gordon coefficient of the group. I.e., for each∆I. Let fIJK
element of the group g ∈ G we have the relation,
i′j′k′ = fIJK
ii′(g) is a matrix in the Ith representation of the group G.
We further define,
We then postulate the following operator product expansions (OPE) for the
(z − w)2∆I+(4)
(z − w)−∆I−∆J+∆K
K(z − w)∂ψK
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 ∆Iare deter-
mined. We postulate also ψ1
1(z) = T(z), the stress tensor, ∆1 = 2, C1 = c/2,
where c is the central charge. Thus the algebra contains the Virasoro algebra and
we demand, accordingly, that, ψI
i(z) is a primary field,˜C1I
I= ∆I, and,˜˜C
I = 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 the
group. This is easily calculated in specific examples by means of the character
tables of specific groups. Denote by χI(g) the character in the representation I of
the group, g ∈ G,
Then the fusion coefficients of the parafermions are given by,
which is reminiscent of the Verlinde formula . Here, O(G) is the order of the
Thus, we can use Vafa’s equations  to calculate the dimensions of the parafermions,
which are found up to some arbitrary integer multiplicative factor. Denote by
αI= e2πi∆I. (7)
Then Vafa equations are
where we define NIJK= N¯ K
When G is abelian, we are back in the case of Zamolodchikov and Fateev. For a
G = ZNgroup, we denote the I parafermion for the representation ΦI(e2πir/N) =
e2πirI/N, for any I and r modulo N. The dimension of the Ith parafermion is ∆I
and ∆I= ∆N−Isince it is the complex conjugate field. We find from eq. (6) that
the structure constant is NK
IJ= 1 if K − I − J = 0modN and is zero otherwise.
Here Vafa’s equations become,
∆I+ ∆J+ ∆K+ ∆L= ∆K+L+ ∆K+I+ ∆K+JmodZ,(11)
where I + J + K + L = 0modN are any. This equation, already appears in ref.
, eq. (A4) there, derived from the mutual semilocality. This eq. (11) implies, in
particular, by taking I = J = 1, that
2∆K+1− ∆K− ∆K+2= β modZ, (12)
where β = 2∆1− ∆2. Thus, ∆K= −βK2/2modZ is the unique solution to eq.
(11), which satisfies, ∆0=integer and ∆1= ∆N−1. It follows that
∆I= MI+ mI2/(sN),(13)
where MIand 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 equations
to find the dimensions of the parafermions.
Let us introduce some basic notions of group theory. The group G is called
simple if the only normal subgroups are itself or the trivial one. An automorphism
is 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,
σh(g) = hgh−1,(15)
where h is a fixed element of the group. We denote the automorphism group
by Aut(G), which is a group under decompositions. We denote by Int(G) the
internal automorphism subgroup of Aut(G), which is a normal subgroup. The
outer automorphism group, Out(G) is defined as the quotient group,
The group G itself is a subgroup of Aut(G) by identifying it with Int(G) (we
assume that G is centerless, see below. If we denote the center by Z(G) then,