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ON THE POSSIBILITY OF ZN EXOTIC SUPERSYMMETRY IN

arXiv:hep-th/9109057v1 30 Sep 1991ON THE POSSIBILITY OF ZN EXOTIC SUPERSYMMETRY INTWO DIMENSIONAL CONFORMAL FIELD THEORYF. RavaniniService de Physique Th´eorique, C.E.A.

- Saclay ∗Orme des Merisiers, F-91190 Gif-sur-Yvette, FranceandI.N.F.N. - Sez.

di Bologna, ItalyAbstractWe investigate the possibility to construct extended parafermionic con-formal algebras whose generating current has spin 1 + 1K , generalizing thesuperconformal (spin 3/2) and the Fateev Zamolodchikov (spin 4/3) algebras.Models invariant under such algebras would possess ZK exotic supersymme-tries satisfying (supercharge)K = (momentum). However, we show that forK = 4 this new algebra allows only for models at c = 1, for K = 5 it is atrivial rephrasing of the ordinary Z5 parafermionic model, for K = 6, 7 (and,requiring unitarity, for all larger K) such algebras do not exist.

Implicationsof this result for existence of exotic supersymmetry in two dimensional fieldtheory are discussed.Saclay preprint SPhT/91-121August 1991Submitted for publication to Int.J.Mod.Phys. A∗Laboratoire de la Direction des Sciences de la Mati`ere du Commissariat `a l’Energie Atomique0

1IntroductionConformal Field Theory (CFT) in two dimensions (2D) [1, 2] can give informationon the general structure of the space of all 2D Quantum Field Theories (QFT).Indeed, each reasonable QFT must possess ultraviolet (UV) as well as infrared (IR)fixed points for which the Callan-Symanzik β function is zero, thus showing scaleinvariance. Following Polyakov [3] it is conceivable that all scale invariant QFT alsopossess conformal invariance.

Hence the UV and IR limits of any 2D QFT must bedescribed by suitable CFT’s.Moreover, if a QFT has some particular symmetry preserved all along the renor-malization flow, this symmetry should exhibit itself at the UV and IR points too.For example consider an N = 1 supersymmetric theory. Its action is invariant un-der transformations by a spin 1/2 charge Q (the so called supercharge) such thatQ2 = P, where P is the (conserved) total momentum.

Corresponding to this chargeQ there is a conserved current of spin 3/2 that in the UV limit becomes the wellknown G(z) current enlarging the conformal symmetry to an N = 1 superconformalone. Thus the UV limit of such a theory must be a superconformal model, minimalor not.

In a certain sense, we could say that existence of a superconformal algebraguarantees the existence of reasonable UV limits for 2D N = 1 supersymmetricQFT’s.One can ask if such a structure can be generalized to models having, say, Z3graded supersymmetry, i.e. models whose action is invariant under transformationsQ, Q† such that Q3 = Q† 3 = P. If so, the UV limit of these Z3 supersymmetricmodels is described by appropriate “Z3 exotic” superconformal models, invariantunder an algebra that generalizes the superconformal one to the case of Z3 gradation.As P has dimension 1, Q and Q† must have dimension 1/3, and the correspondingconserved current must be of dimension 4/3.

Such an algebra extending the Vira-soro algebra by means of a couple of conjugated currents A(z), A†(z) of spin (anddimension) 4/3 has been investigated by Fateev and Zamolodchikov [4].1

These Z2 (Susy) and Z3 (spin 4/3) algebras, that we shall call in the followingSZ2 and SZ3 respectively, show some well known common structure. First of all,both allow for a series of unitary minimal models, accumulating to c = 3/2 and toc = 2 respectively [4, 5].

For c larger than these values there is still a continuumof non-minimal models. The set of all SZ2 ⊗SZ2 invariant models, minimal ornot, is the set of all UV or IR fixed point of 2D QFT invariant under N = 1supersymmetry; the same set for SZ3 represents all fixed points of QFT invariantunder Z3 exotic supersymmetry.Each minimal model in the two series can beperturbed by some relevant scalar operator contained in its Kac table.

The leastrelevant operator (that with conformal dimension closer to 1) leads to two differentbehaviours: for negative values of the perturbing coupling constant the model ismassive and integrable, and its scattering matrix is known [6]; for positive couplingconstant the perturbation defines a massless flow that has been shown, at least byperturbative arguments, to have a non trivial IR limit also belonging to the sameseries [7, 8]. The Z2 or Z3 supersymmetry is preserved along the flow, i.e.

one candefine non-local charges Q in the perturbed model, such that Q2 = P or Q3 = Prespectively.In particular, when one perturbs by the least relevant operator the model in theminimal series having the lowest central charge c, one can show that still there is aconserved current Q in the perturbed model with Q2 = P or Q3 = P. For negativecoupling constant the scattering matrix of the massive model is known [9] and isindeed invariant under such a charge Q. For positive coupling the models are knowto flow to usual Z2 (Ising) and Z3 (Potts) models respectively.

As these IR limitsdo not have Q invariance any more, one concludes [7, 9] that along the flow thereis spontaneous supersymmetry breaking. In the Z2 case, it has been shown [7] thatthe resulting goldstino is a field that in the IR limit evolves in the spin 1/2 fermionof the Ising model.

Similarly, in the Z3 case, Zamolodchikov [9] has argued that thegoldstini fields corresponding to the broken A(z), A†(z) currents should become, inthe IR limit, the couple of 3-state Potts parafermions of spin 2/3. Notice that in2

both cases the spin of the broken current and that of the resulting goldstino sumup to 2 (any better understanding of this fact should be welcome). It has also beenobserved that in both cases the scattering matrix of the negative coupling massivemodel coincides with the Boltzmann weights of the corresponding Z2 Ising or Z3Potts models [9].In this paper we address the problem to generalize these examples to ZK gradedexotic superconformal algebras, SZK for short.

If such algebras exist, they canbe the base for the construction of SZK ⊗SZK invariant models. Then one canaddress the problem of perturbing these models by their least relevant operator thusgetting examples of massive and massless non-conformal ZK exotic supersymmetricmodels.

In particular, the picture valid for Z2 and Z3 should naively suggest thatsuitably perturbing the SZK model with lowest central charge, one could get acase of spontaneously broken ZK exotic supersymmetry, whose goldstino, in the IRlimit, could describe the usual ZK parafermion of the ZK Ising model. Boltzmannweights for the ZK Ising models are known [10].

Assuming that they can be as wellused as scattering matrices of some 2D QFT, Bernard and Pasquier [11] have shownthat they are indeed invariant under transormations Q such that QK = P. Thus,they are the natural candidates of an eventual massive model obtained by suitablydeforming the lowest c SZK model.If conversely the SZK models do not exist, namely because the SZK algebrasare inconsistent for some K, then no UV limit can be defined for a ZK exoticsupersymmetric theory, and, as a reasonable QFT must have an UV limit, wecan conclude that no ZK exotic supersymmetry exists for that K at all, and theBoltzmann weights of [11] cannot be used as scattering matrices of a QFT.The interest of searching possible SZK algebras is even more general: in ap-pendix A of [12], Fateev and Zamolodchikov describe the most general ZK symmet-ric parafermionic algebra, of which the usual ZK parafermions are a particular case.All ZK-invariant conformal models should have some realization of this generalZK algebra somewhere hidden in their operator product expansion (OPE) algebra.3

Thus, knowledge of possible associative ZK symmetric algebras (and of their rep-resentations) should help in the classification of all conformal models having ZKsymmetry. SZK algebras explored in the present paper are one among the manypossibilities described in [12].2The ZK-superconformal algebrasWe begin our investigation by giving the general form of the SZK algebras we areinterested in.

We proceed by direct generalization of the known Z2 and Z3 cases.The SZ2 algebra has the simple Z2-graded fusion rules ψψ = 1, and the SZ3 oneis described by Z3-graded fusion rules ψψ = ψ†, ψψ† = 1. Requirement of ZK-gradation of the SZK algebra means considering a set of ZK symmetric fusion rulesfor the currents:ψiψj = ψi+j,(ψ0 = 1,ψ†i = ψK−i)(1)Here and in the following i, j, k, ... indices are always to be taken modulo K andwe introduce the notation ˆı = K −i.

We are interested in N = 1 supersymmetry,i.e.we require that for a given conformal dimension ∆k there can be only onecouple of currents ψk and ψ†k. Furthermore, we require that no currents of spin oneare present as secondaries in the family of the identity, otherwise they would formthe Kac-Moody algebra of a continuous internal symmetry, while we are interestedin the case where no symmetry additional to the ZK-susy is postulated.

We alsorequire that the only current of dimension two appearing in the identity family is thestress-energy tensor. Currents of higher spin 3,4,5... are allowed, which means thatthe identity family is not necessarily that of pure Virasoro algebra, it can as wellcontain currents generating some W-algebra.

All these requirements fix the form ofthe operator product expansion (OPE) algebra to be the following (z12 = z1 −z24

and ˆ̸= iψi(z1)ψj(z2) = Cijzαij12hψi+j(z2) + z12αij+2∆i2∆i+j ∂z2ψi+j(z2) + O(z212)iψi(z1)ψ†i (z2) = z−2∆i121 + z2122∆ic T(z2) + O(z312)T(z1)ψi(z2) = ∆iψi(z2)z212+ ∂z2ψi(z2)z12+ O(1)T(z1)T(z2) = c/2z412+ 2T(z2)z212+ ∂z2T(z2)z12+ O(1)(2)where we introduced the useful notation αij = ∆i+j −∆i −∆j.The last twoequations show that Virasoro algebra with central charge c is a subalgebra of SZKalgebra and that ψi’s are Virasoro primary fields of left conformal dimension ∆i. Asthey are to be conserved currents, their right conformal dimension ¯∆i must be zero.Hence the spin of the current ψi, as well as its full conformal dimension, is given by∆i.

Of course there will be a “right” algebra of currents ¯ψi(¯z) and ¯T(¯z) pertainingthe right chiral part of the models and commuting with the {ψi, T} algebra, so thatthe models will be invariant under SZK ⊗SZK symmetry. All the considerationsin the following will be done for the left algebra and apply as well to the right one.The spins ∆k can not take arbitrary values.

As explained in Appendix A of [12],or equivalently using the techniques of [13, 14], it is possible to show that the mostgeneral value for ∆k compatible with the fusion rules (1) is given by∆k = pk(K −k)K+ Mk(3)where Mk ∈Z, Mˆk = Mk, M0 = 0 and p can be integer if K is odd and integer orhalf-integer if K is even. Furthermore, in the case of SZK algebra, we must requirethat one of the currents ψk, say ψ1, has spin ∆1 = 1 + 1/K, in order to be ableto define a conserved charge Q such that QK = P. This fixes M1 = Mˆ1 = 2 andp = −1, hence the formula for ∆k we shall assume in the following is∆k = Mk −k(K −k)K,M0 = 0M1 = 2Mˆk = Mk(4)where the integers Mk have to be constrained by the allowed behaviours of correla-tion functions near their singularities (see below).5

The structure constants Cij are chosen such that all non-zero two point functionsare normalized to 1 (Ciˆı = 1), and enjoy full symmetry, i.e. definingCij = Qi+ji,j = Qi,j,N−i−j(5)full symmetry of the symbols Qi,j,k must be required.

This restricts the number ofindipendent structure constants. Moreover charge conjugation symmetry impliesCij = C∗ˆıˆ.3AssociativityThe most important requirement on SZK algebras is their associativity or, equiva-lently, duality of the 4-point correlation functions of fields ψi.

As fields ψi(z) do notdepend on ¯z, their 4-point functions will have dependence on z1, ..., z4 only, and noton ¯z1, ..., ¯z4. Invariance under the projective group SL(2,C) implies that one canchoose 3 of the 4 positions in any 4-point function as 0,1 and ∞so that it dependsessentially only on one projective invariant variable x, the so called anharmonicratio⟨0|ψl(∞)ψk(1)ψi(x)ψj(0)|0⟩= Gklij(x)(6)ZK invariance forces this correlation function to be 0 if i + j + k + l ̸= 0 mod K.The requirement of duality on the 4-point function can be put as conditions onGklij(x)Gklij(x) = Gjlik(1 −x) = eπiαikx−2∆iGkjil (1/x)(7)The phase in the last equality comes from the braiding of the (semilocal) fields ψiand ψk necessary to bring x close to ∞.

Due to the mutual semilocality of fields [12],Gklij is not in general a single valued function. Its expansion in blocks must reproduceits monodromy properties, which in turn can be read from the OPE’s (2).

Obviously,blocks for the right chiral part of the correlation function are trivially equal to 1.Moreover, in each channel there is only one possible exchanged family and therefore6

only one block:Gklij(x)=CijCklF klij (x)Gjlik(1 −x)=CikCjlF jlik(1 −x)Gkjil (1/x)=CilCkjF kjil (1/x)(8)The behaviour of the blocks at x = 0, 1, ∞can be easily inferred from the OPE’sF klij (x)∼x→0xαij∞Xn=0cnxnF jlik(1 −x)∼x→1(1 −x)αik∞Xn=0dn(1 −x)nF kjil (1/x)∼x→∞1xαil∞Xn=0hnx−n(9)The series expansion are convergent in a neighborhood of 0, 1, ∞respectively. Someof the coefficients cn, dn, hn can be computed from the information contained in theOPE’s (2).

In particular c0, d0, h0 are always guaranteed to be equal to 1. Moreover,if i ̸= ˆF klij (x) = xαij 1 + (αij + 2∆i)(αij + 2∆j)2∆i+jx + O(x2)!

(10)while if i = ˆF kˆkiˆı (x) = x−2∆i1 + 2∆i∆kcx2 + O(x3)(11)The behaviour for the blocks in eq. (9) reproduces the correct monodromy propertiesof the multivalued correlation function.

The blocks are (up to the branch singularityin the leading factor) locally holomorphic functions of x. They can be analyticallycontinued in the whole plane, excluding the branch points.

Closure under analyticcontinuation requires [13, 15]αij + αik + αil −2∆i = −R(12)where R is a non negative integer. This in turn fixes the general form of blocks tobeF klij (x)=xαij(1 −x)αikP(x)F jlik(1 −x)=(1 −x)αikxαijQ(1 −x)F kjil (1/x)=x−αil(1 −x−1)αikT(1/x)=e−πiαikx2∆ixαij(1 −x)αikxRT(1/x)(13)7

where P(x) =PRn=0 Pnxn, Q(x) =PRn=0 Qnxn and T(x) =PRn=0 Tnxn are polyno-mials of degree R in x. In the last equality eq.

(12) has been used. To avoid heavynotation we dropped indices i, j, k, l from the polynomials, but it must be bared inmind that they, as well as the integer R, are specific of the particular correlationfunction Gklij(x).

Normalization of the blocks implies P0 = Q0 = T0 = 1.The duality requirement (7) can then be simply translated in conditions on thesepolynomialsst-duality:⇒CijC∗klP(x) = CikC∗jlQ(1 −x)su-duality:⇒CijC∗klP(x) = CilC∗kjxRT(1/x)(14)These equations can take a particularly simple form when they are considered forsome special cases. Here and in the following we use the notation Gklij = ⟨i j k l⟩.Remembering that Ci,ˆı = 1, we first consider the correlation functions ⟨i ˆı k ˆk⟩.The constraints2∆i∆kc= P2 −12(α2ik + αik)(15)andP1=αikQ1=(αik + 2∆i)(αik + 2∆k)2∆i+k+ αij(16)T1=(αiˆk + 2∆i)(αiˆk + 2∆k)2∆i−k+ αikobtained comparing the blocks with the expansions (10,11) can be convenientlyused, together with the duality constraints|Cik|2 = P(1) =RXn=0Pn,Qn =1|Cik|2RXp=nnpPp(17)|Ci,ˆk|2 = Pn,Tn =1|Ci,ˆk|2PR−n(18)to fix as much as possible the parameters.

In particular for R = 0 it should beαik = 0 and then c = ∞. Hence if at least one of the correlation functions ⟨i ˆı i ˆı⟩has R = 0 the whole algebra is unconsistent.8

In the case i = k, the correlation function ⟨i ˆı i ˆı⟩has even more constraints.Indeed, here su-duality brings again to the same block, hence the polynomial T(x)is equal to P(x), and we have the conditionP(x) = xRP(1/x) ⇒PR−n = Pn(19)It is easy to convince oneself that if R ≤3 the polynomial is completely fixed. ForR = 1 the only way to avoid unconsistency is to have αik = 1.

In the R = 2, 3 caseseq. (15) can be conveniently used to fix c. For higher values of R (15) can be usedto express the coefficient P2 in terms of c.Another case of strongly constrained correlation function is that of four equalfields ⟨i i i i⟩.

In this case the blocks in all the three channels coincide and thereis only one polynomial whose coefficients are constrained by the equations P(x) =xRP(1/x) and P(x) = P(1 −x). They have no solution for R odd, for R even thecoefficients Pn are fully determined if R < 6.A last remark is in order about conformal dimensions: we have seen in theprevious section that they are determined up to integers Mk.Now, conformaldimensions enter in the determination of R, eq.(12).

The fundamental requirementthat for all 4-point functions R must be a non negative integer can then be translatedinto a set of inequalities to be satisfied by the integers Mk, thus strongly selecting,as we shall see in the next section, the possible choices of conformal dimensions.The strategy to study a given algebra will then be the following:1. identify the parameters in the algebra.These are c and the independentstructure constants. Moreover, spin of fields will eventually depend on a setof integers Mk as in eq.(4).2.

list all the non-zero 4-point functions and compute R for each of them. Im-posing R ≥0 the integers Mk can be selected to a few possible choices.3.

For each choice consider the functions ⟨i ˆı k ˆk⟩. If at least one of them has R =0, discard the choice.

Else, identify the ⟨i ˆı k ˆk⟩functions with lower values9

of R and, after having constrained as much as possible the coefficients of thepolynomials using duality, try to fix c and/or some Cij through eqs. (15,17,18).If different correlation functions provide unconsistent values of c or Cij, discardthe choice.4.

Otherwise use the values so obtained to fix as much as possible the other cor-relation functions and check all the possible duality constraints. If somewheresome unconsistency appears, discard the choice.

If instead the choice passesall the checks it defines a consistent associative algebra. If some parameter isleft free, the algebra allows for an infinity of associative realizations, one foreach value of the parameter.5.

redo steps 3 and 4 for all the choices selected by step 2.Next section will illustrate this procedure on some simple examples.4SZK algebras for K ≤7To illustrate the general theory of the previous section, let us study some particularcase in detail. We shall discover that even for low values of K some interestingsurprises arise.SZ2 algebra - In the case K = 2, i.e.

usual N = 1 superconformal algebrathere is only one fermionic field ψ1 of spin 3/2. No non trivial structure constantappear and the only parameter in the algebra is c. There is only one non-trivial4-point function ⟨1 1 1 1⟩with R = 6.

Crossing symmetry fixes it up to a freeparameter:⟨1 1 1 1⟩= x−3(1 −x)−3(1 −3x + Px2 + 9 −2P3x3 + Px4 −3x5 + x6)(20)that can be re-expressed in terms of c as P = (15c −3)/2c. No other restrictioncan be found on the algebra.

Therefore we conclude that SZ2 is associative for anyvalue of c. Unitarity [5] will then restrict to c = 321 −8m(m+2)or c ≥3/2.10

SZ3 algebra - The case K = 3, namely the spin 4/3 algebra of Fateev andZamolodchikov [4], has two free parameters: c and the structure constant C1,1 = λ.On the other hand, there is only one non trivial 4-point function ⟨1 ˆ1 1 ˆ1⟩, forwhich R = 4. Crossing symmetry then determines the 4-point function up to a freeparameter P⟨1 ˆ1 1 ˆ1⟩= x−8/3(1 −x)−4/3(1 −43x + Px2 −43x3 + x4)(21)Both c and λ can be computed in terms of P by use of eqs.(15,17).

Eliminating Pin the result we reobtain the well known relation computed by Fateev and Zamolod-chikov 9c|λ|2 = 4(8 −c). Hence the SZ3 algebra has only one free parameter, thatcan be chosen to be c. Unitarity [16] will fix it to c = 21 −12m(m+4)or c ≥2.SZ4 algebra - The first new case K = 4 has parafermions ψ1, ψ2 and ψ3 = ψ†1of dimensions ∆1 = ∆3 = 5/4 and ∆2 = M2 −1 respectively.

The algebra hastwo parameters: c and the structure constant C1,1 = λ. There are four non trivial4-point functions.

The requirement that for each of them R ≥0 amounts to a setof inequalities in M2:from ⟨1 ˆ1 1 ˆ1⟩⇒M2 ≤6from ⟨2 2 2 2⟩⇒M2 ≥1from ⟨1 ˆ1 2 2⟩⇒M2 ≥1from ⟨1 1 1 1⟩⇒M2 ≤2We see that these, together with the fact that ∆2 must be different from zero toavoid doubling of the vacuum, imply M2 = 2 and therefore ∆2 = 1. The correlationfunction ⟨2 2 2 2⟩with R = 4 is completely fixed by crossing symmetry⟨2 2 2 2⟩= x−2(1 −x)−2(1 −2x + 3x2 −2x3 + x4)(22)and can be conveniently used to fix c = 1 by use of eq.(15).

The other correlationfunction ⟨1 ˆ1 1 ˆ1⟩has R = 4. Duality should fix it up to a parameter depending on11

c. The value c = 1 fixes this parameter, and the correlation function reads⟨1 ˆ1 1 ˆ1⟩= x−5/2(1 −x)−3/2(1 −32x + 72x2 −32x3 + x4)(23)Eq. (17) then allows to compute the structure constant (up to an inessential phasethat we fix to 1) as λ =q5/2.One can check that the remaining correlationfunctions are compatible with this fixing of the parameters.The surprising result here is that c is no more free.

SZ4 does not allow for a seriesof minimal models at different c plus continuum, but rather for a series of models atc = 1, that can be indeed easily identified with points on the gaussian and orbifoldlines for suitable values of the compactification radius. In a certain sense, the seriesof models present in SZ2 and SZ3 cases are here “squeezed” to the c = 1 lines.

Themodels can still “flow” from one another, but the perturbing operator is now thelimiting case of least relevant operators, namely the marginal operator that allowsto move along the c = 1 lines. All the SZ4 invariant models are connected by thismarginal operator to one of the two modular invariant solutions of Z4 parafermion(the diagonal one lying on the orbifold line).

One could then still speculate aboutSZ4 spontaneous symmetry breaking with goldstini given by Z4 parafermions. Thispicture is however somewaht delicate here: the current ψ1 of spin 5/4 seems tobreak nicely to give a paragoldstino of spin 3/4 (again 5/4 + 3/4 = 2), but theother current of spin 1 cannot break spontaneously due to Coleman theorem [17],and indeed we find it again in the Z4 usual parafermion.As the stress-energytensor can be related to this current via U(1) Sugawara construction, this impliesalso that conformal symmetry cannot be broken along this “flow” and the centralcharge should not change, as indeed is the case.

All these strange features indicatethat K = 4 is somewhat a “limiting” case for ZK exotic superconformal algebras.SZ5 algebra - In this case there are four parafermions of spin ∆1 = ∆4 = 6/5,∆2 = ∆3 = M2 −6/5 respectively. The algebra has 3 parameters: c, λ = C1,1 andµ = C1,2.

The requirement that R ≥0 for all 4-point functions translates again intoa set of inequalities for M2 that can be simultaneously satisfied only if M2 = 2, 3.12

In the first case M2 = 2 we have ∆1 = 6/5, ∆2 = 4/5 and we reobtain the usualZ5 parafermionic model of Fateev and Zamolodchikov [12], merely with the nameof the two parafermions reversed. The other case is ∆1 = 6/5, ∆2 = 9/5.

Usingthe correlation function ⟨1 ˆ1 1 ˆ1⟩, with R = 3, we can fix c = −6 and λ =q4/5.The ⟨1 ˆ1 2 ˆ2⟩correlation function, with R = 3, reads⟨1 ˆ1 2 ˆ2⟩= x−12/5(1 −x)−6/51 −65x −35x2 + 45x3(24)where we have fixed the coefficient of x2 in the polynomial inserting c = −6 ineq. (15), and that of x3 by imposing su-duality, eq.(18).

st-duality then requiresthat the sum of coefficients in the polynomial P(x) is equal to |µ|2. But this sumis zero, thus implying µ = 0 in contradiction with the fusion rules.

This argumentrules out the M2 = 3 case.SZ5 can then only coincide with the Z5 parafermionic algebra with c = 8/7,λ =q8/5, µ =q9/5. This is a very particular case: it is easy to check that noother ZK parafermionic model contains a current of spin 1+1/K.

See also appendixA for a more general result on this point.SZ6,7 algebras - For K = 6 we have 5 parafermions of spin ∆1 = ∆5 = 7/6,∆2 = ∆4 = M2 −4/3, ∆3 = M3 −3/2. The parameters in the algebra are c,λ = C1,1, µ = C1,2 and ρ = C2,2.

The requirement R ≥0 for all 4-point functionsrestricts M2 and M3 to the following chioces:A.M2 = M3 = 2∆1 = 7/6,∆2 = 2/3,∆3 = 1/2B.M2 = 2, M3 = 3∆1 = 7/6,∆2 = 2/3,∆3 = 3/2C.M2 = M3 = 3∆1 = 7/6,∆2 = 5/3,∆3 = 3/2Case A is immediately ruled out because {1, ψ2, ψ4} form in this case a Z3 para-fermion subalgebra with c = 4/5 while {1, ψ3} form a Z2 algebra with c = 1/2,incompatible with the former. Case B also is nonsense, as {1, ψ2, ψ4} still formsa Z3 subalgebra with c = 4/5, but we know that at c = 4/5 there is no room forVirasoro primary fields of dimensions 7/6 or 3/2.

We are left with case C, where13

the correlation function ⟨1 ˆ1 1 ˆ1⟩(with R = 3) fixes c = −49/10. Besides the factthat this automatically excludes possibility to build up any unitary model, the SZ6algebra of case C is inconsistent even for this negative value of c. Indeed it can bechecked that the correlation functions ⟨1 ˆ1 2 ˆ2⟩and ⟨1 ˆ1 3 3⟩give two incompatible(and negative!) values of |µ|2.Thus there is no possibility to realize an associative SZ6 algebra.

The samehappens for K = 7, where the only possibility ∆1 = 8/7, ∆2 = 11/7, ∆3 = 9/7 givesa negative value of c and is ruled out by arguments similar to those of K = 6. Forhigher values of K the analysis is in principle still possible, but the number of cor-relation functions to analyze increases and computations can become cumbersome.To get more general results we have to turn to a closer analysis of some particularcorrelation function and restrict more the problem by some new input.5General unitary case - Proof of unconsistencyfor K > 5If we are content to explore the possibility to have unitary ZK exotic superconformaltheories, then the additional requirements c > 0 and ∆i > 0 help to get a generalanswer.

This can be given in form of aTheorem - With the constraints c > 0 and ∆i > 0, i = 1, 2, 3, 4, there is noassociative SZK algebra for K > 5.In the proof of this statement, we make use of three conformal dimensions∆1 = 1 + 1K,∆2 = M2 −2 + 4K,∆3 = M3 −3 + 9K(25)We already know from previous section that SZK algebras exist for K ≤5 and areabsent for K = 6, 7. Here we shall consider values of K ≥6.

Reqirement ∆i > 0bounds Mi toM2 ≥2,M3 ≥2for K ≤83for K ≥9(26)14

Let us first consider the correlation function ⟨1 ˆ1 1 ˆ1⟩, for which R = 4∆1−∆2 = 6−M2 ≥0 reqires M2 ≤6. The case M2 = 6 leads to R = 0 and, as explained in section3, is unconsistent, the same happens for M2 = 5, R = 1 as α11 = M2 −4 + 2K ̸= 1.The case M3, R = 3 always gives, through eq.

(15) negative values of c, hence it isdiscarded too. We are left with two possibilities:A.M2 = 2, R = 4 ⇒P(x) = 1 −2 K+1K x + P2x2 −2 K+1K x3 + x4B.M2 = 4, R = 2 ⇒P(x) = 1 −2Kx + x2 ⇒c = 2(K+1)K−2Case A - To go on we have to resort to another correlation function, namely to⟨1 ˆ1 2 ˆ2⟩, that, for M2 = 2, has R = 4 −M3.

The reqirement R > 0 (R = 0 againis unconsistent) then restrict M3 ≤3. Thus for K ≥9, M3 = 3, R = 1 is the onlypossible value.

In this case P(x) = 1 + 4−KK x and |C11|2 = P1 = 4−KKis negativefor all K > 4. This rules out this case.

There is still the possibility of M3 = 2 forK = 6, 7, 8. The cases K = 6, 7 are ruled out by the results of the previous section.K = 8 is the only case where we have also to consider ∆4 = M4 −2.

∆4 > 0 impliesM4 > 2 while the correlation function ⟨2 2 2 2⟩requires M4 ≤2. Also this case isruled out.Case B - The function ⟨1 ˆ1 1 ˆ1⟩previously considered is completely fixed in thisR = 2 case and yeldsc = 2(K + 1)K −2,|C11|2 = 2 + 2K(27)In this case the correlation function ⟨1 ˆ1 2 ˆ2⟩has R = 8 −M3, hence it mustbe M3 < 8.Here it is convenient to resort to the function ⟨1 1 1 ˆ3⟩, havingR = M3 −3M2+6 = M3−6, that reqires M3 ≥6.

Hence M3 = 6, 7. Consider againthe function ⟨1 ˆ1 2 ˆ2⟩.

For M3 = 7, R = 1 it is possible to compute |C11|2 = 1 + 4K,in contradiction with (27). For M3 = 6 the value of |C11|2 in (27) helps to fix thecoefficient P2 from which c can be evluated back.

We obtain c = 4(K+1)(K+2)K2−4K+8 , incontradiction with (27).All the possible cases are then ruled out by simply considering a set of fewcorrelation functions for some of the fields in the algebra. We believe that this15

result, surely valid for unitary theories, is in fact absolutely general: there are noZK exotic (in the sense of QK = P) superconformal algebras for K > 5.6Conclusions and implicationsThe main result of this paper is the impossibility to construct ZK exotic N = 1superconformal algebras for K > 5. For K = 5 the result is trivial, for K = 4is c = 1 theory, and finally we are “seriously” left only with the already knowncases K = 2, 3, i.e.ordinary N = 1 superconformal algebra and the spin 4/3algebra.

What is established is that for K > 5 it is not possible to realize at theconformal point, an algebra of currents such that it can define a charge Q satisfyingQK = P. This does not mean that “more exotic” supersymmetric algebras cannot be constructed in two dimensions: for example, Fateev [18] has recently shownthat in some parafermionic models it is possible to consider a charge Q such thatQK = Ps, where Ps is an appropriate higher spin local conserved charge. Thesemodels are in connection with the parafermionic algebras introduced in the appendixA of [12], and further studied, from the unitarity point of view, in [16].

In fact, SZ3is also a particular case of these algebras.As QK = P can not be realized at criticality, it is presumably impossible alsoin perturbations of CFT’s. It then becomes problematic to identify the Boltzmannweights of [11] with the scattering matrices of a 2D QFT.

It can happen that sucha QFT does not exist, but also that it exists and its UV limit is quite tricky. Also,the spontneous symmetry breakdown and the goldstino problem needs more under-standing.

An intriguing observation in this connection is that the goldstino picturebreaks down at K = 4. Palla [19] has studied perturbations of ZK parafermionicmodels that can convert some of the parafermionic currents into conserved quanti-ties.

Now, this is possible exactly starting from K = 4. Many indications point tothe fact that the behaviour of parafermionic theories should be quite different forK < 4 and K > 4, with K = 4 as a limiting case.

If this is related or not to more16

fundamental issues like Galois theory is to be understood.Acknowledgements - I am grateful to M.Bauer, D.Bernard, V.Pasquier andJ.B.Zuber for many useful discussions. I thank the Service de Physique Th´eoriqueof C.E.A.

- Saclay for the kind ospitality. The Theory Group of I.N.F.N.

- Bolognaand the Director of Sez. di Bologna of I.N.F.N.

are acknowledged for the financialsupport allowing me to spend this year in Saclay.AppendixIt is interesting to ask if there are SZP algebras hidden in the usual ZK parafer-mionic models, even for P ̸= K, generalizing the curious phenomenon observed forZ5. We shall show here that no such case is possible for P > 5, in agreement withthe results of the main part of the paper, and that the only other cases can alwaysbe traced back to the well known SZ2 or SZ3 algebras.First of all let us prove that for P > 5 there is no parafermion of spinP +1Pcontained in any ZK parafermionic algebra, for all K.To do that, we have toconsider the equationP + 1P= k(K −k)K(28)where the expression on the r.h.s., for k = 1, ..., K −1 gives the most general spinfor a parafermion in ZK.

The case k = 1 is clearly impossible, for all K and all Pas it equates a l.h.s. greater than 1 with a r.h.s.

less than 1. So consider k ≥2.Solving (28) for K one getsK = k + 1 + k + P + 1Pk −P −1(29)As K must be an integer, we have to reqire k + P + 1 ≥Pk −P −1, i.e.k ≤2P + 1P −1(30)17

For P > 5 this implies k ≤2, hence the only possibility is k = 2. Substituting in(29) we getK = 4 +4P −1(31)which can never be integer if P > 5.

This proves that no SZP is contained in ausual parafermionic algebra for P > 5.For P ≤5 use of (29) and (30) allows to list all the possible occurrencies.• For P = 5 we have K = 5, k = 2, 3. This is the result noticed in the paperthat SZ5 coincides with the Z5 parafermion.• For P = 4 no solution appears.• For P = 3 we have K = 6, k = 2, 4, thus showing that a realization of SZ3is contained in the Z6 parafermionic algebra.

This is not surprising as the Z6model is known to belong to the unitary minimal series of spin 4/3 algebra,namely for m = 4.• For P = 2 there are two solutions: K = 6, k = 3 says that the Z6 model isalso supersymmetric, (it belongs indeed also to the superconformal minimalseries for m = 6) while K = 8, k = 2, 6 shows two fields of spin 3/2 for the Z8model.• For P = 1 the solution K = 8, k = 4 completes the result for P = 2: the twospin 3/2 curents are associated to a current of spin 2, thus {T, ψ2, ψ4, ψ6 formin this case two copies of the N = 1 superconformal algebra: the Z8 modelis doubly N = 1 supersymmetric. Another solution appears for P = 1 whenK = 9, k = 3, 6.This exhausts all possible realizations of SZP algebras in ZK parafermionic models.References18

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