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New N=1 Extended Superconformal

arXiv:hep-th/9207072v1 22 Jul 1992Universit¨at BonnPhysikalisches InstitutNew N=1 Extended SuperconformalAlgebraswith Two and Three Generators(to be published in Int. Jour.

of Mod. Phys.

A)R. Blumenhagen, W. Eholzer, A. Honecker, R. H¨ubelAbstractIn this paper we consider extensions of the super Virasoro algebra by one and two superprimary fields. Using a non-explicitly covariant approach we compute all SW-algebraswith one generator of dimension up to 7 in addition to the super Virasoro field.Incomplete analogy to W-algebras with two generators most results can be classified us-ing the representation theory of the super Virasoro algebra.

Furthermore, we find thatthe SW( 32, 112 )-algebra can be realized as a subalgebra of SW( 32, 52) at c = 107 . We alsoconstruct some new SW-algebras with three generators, namely SW( 32, 32, 52), SW( 32, 2, 2)and SW( 32, 2, 52).Post address:BONN-HE-92-02Nußallee 12hep-th/9207072W-5300 Bonn 1Bonn UniversityGermanyJanuary 1992e-mail: unp039@ibm.rhrz.uni-bonn.deISSN-0172-8733

1. IntroductionExtended conformal and superconformal algebras have gained intensive interest in theo-retical physics because they can be considered as symmetry algebras of conformal quantumfield theories (CFT).

One hopes to describe all rational conformal quantum field theories(RCFT) as minimal models of extended chiral algebras [1], so-called W-algebras.The first W-algebras were constructed by Zamolodchikov in 1985 [2]. He extended theVirasoro algebra by primary fields up to dimension three.

Two of these W-algebras havethe typical property that normal ordered products appear. We denote the extension of theVirasoro algebra by primary fields of conformal dimensions δ1, .

. ., δn by W(2, δ1, .

. ., δn).In the last years, several methods have been developed to find new W-algebras.

One ap-proach uses the free field construction based on Lie algebras and their infinite dimensionalextensions, the Kac-Moody algebras [3][4][5].In this paper we use a different method, constructing the algebra explicitly. Using thismethod a whole bunch of W-algebras has been constructed [6][7][8][9][10][11].

In [10][11] aconstructive algorithm has been presented which is based on a structural theorem aboutchiral SU(1, 1) invariant algebras [12]. It has been shown in [9] that W-algebras with twogenerators can only exist for generic values of the central charge, if the conformal dimensionof the additional primary field is contained in { 12, 1, 32, 2, 3, 4[7][6], 6[8]}.

In [10][11] it hasbeen shown that several other W(2, δ)-algebras exist for finitely many values of the centralcharge c. The possible c values are essentially determined by the representation theory ofthe Virasoro algebra.RCFT’s describe two dimensional statistical models at second order phase transitions.Furthermore there are statistical models which possess superconformal invariance [13] atcriticality. These models are described by minimal models of the super Virasoro algebra.Similar to the Virasoro algebra, the minimal series is bounded from above by c =32.In order to get minimal models with higher values of the central charge one investigatesextensions of the super Virasoro algebra, called super W-algebras.In this paper we consider extensions of the super Virasoro algebra by one and two superprimary fields.

In general we denote the extension of the super Virasoro algebra by superprimary fields of dimension d1, . .

., dn by SW( 32, d1, . .

., dn). Some SW( 32, d)-algebras havealready been calculated in [14][15][16][17].

As far as one knows SW( 32, d)-algebras existgenerically only for d ∈{ 12, 1, 32, 2}. For d ̸= 2 these algebras are super Lie algebras.

In[17] all SW( 32, d)-algebras up to d = 72 have been constructed using the super conformalbootstrap method. Here we extend this series of algebras up to d = 7 using the same meth-ods as presented in [10].

Superconformal covariance is implemented by considering Jacobiidentities involving the generator G(z) of pure super transformations. This algorithm dif-fers from the explicitly covariant one presented in [18].

Nevertheless these two algorithmsseem to be equivalent. For d ≥52 the SW( 32, d)-algebras exist only for discrete valuesof c. Most of these values can be classified in complete analogy to the W(2, δ)-algebras,essentially using representation theory of the super Virasoro algebra.

Furthermore, we findthat the SW( 32, 112 )-algebra can be realized as a subalgebra of the SW( 32, 52)-algebra forc = 107 . A similar construction has been carried out in [19] where it has been shown thatSW( 32, 72) is contained in SW( 32, 32) for c = 75.1

This paper is organized as follows: In the next chapter we review a general theorem aboutchiral SU(1, 1) invariant algebras. In the third chapter we give the general outline forthe construction of SW-algebras in the non-explicitly covariant approach.

Chapter fourcontains our results on SW( 32, d)-algebras. Afterwards we discuss these results in chapterfive.

We continue with our results concerning SW( 32, d1, d2)-algebras in chapter six. Inchapter seven we draw some conclusions from the results obtained.2.

General theorems about SU(1,1) invariant algebrasThis chapter is devoted to a short review of well known results concerning algebras of localchiral fields. We will only give a brief summary; for proofs as well as for details we referto [20][10].Let F be the algebra of local chiral fields of a conformal field theory defined on 2-dimensional spacetime (with compactified space).

Because of SU(1, 1)-invariance F carriesa natural grading by the conformal dimension and is spanned by the non-derivative (i. e.quasiprimary) fields together with their derivatives. In F the operation of building normalordered products (NOPs) is defined (see below).We define the Fourier decomposition of a left chiral field by φ(z) = Pn−d(φ)∈ZZ zn−d(φ)φn .We call the Fourier-components φn the ‘modes’ of φ.Denote the vacuum of the theory by |v⟩.

Requiring the regularity of φ(z) |v⟩at the originimpliesφn |v⟩= 0∀n < d(φ)(2.0)It is well known that the modes of the energy momentum tensor satisfy the Virasoroalgebra[Lm, Ln] = (n −m)Lm+n + c12(n3 −n)δn+m,0(2.1)with central charge c. A primary field φ of conformal dimension d is characterized by thecommutator of its modes with the Virasoro algebra (2.1):[Lm, φn] = (n −(d −1)m)φn+m(2.2)Iffa field φ satisfies (2.2) for m ∈{−1, 0, 1} this field is called ‘quasiprimary’.As was shown by W. Nahm [20][12] locality and invariance under rational conformal trans-formations already impose severe restrictions on the commutator of two quasiprimary chiralfields:Let {φi | i ∈I} be a set of non-derivative fields of integer or half integer dimensionsd(φi) = di, which together with their derivatives span F. Then the mode algebra of theFourier components of left chiral fields has the form[φi,m, φj,n]± =Xk∈ICφkφiφj pijk(m, n)φk,m+n + dij δn,−mn + di −12di −1(2.3)2

Here the dij describe the normalization of the two point functions and the Cφkφiφj are thecoupling constants between three quasiprimary fields. The pijk are universal polynomialsdepending only on the dimensions of the fields φi, φj and φk; their explicit form is presentedin [10].In addition to their Lie bracket structure the algebra F admits another important opera-tion, namely forming normal ordered products (NOPs) of chiral fields.

Usually the NOPof two chiral fields φ, χ is defined in terms of Fourier components as followsN(φ, χ)n := ǫφχXk

This definition yields a quasiprimary field ofconformal dimension di + dj + n.Since the field content of F is infinite one introduces the notion of non-composite, ‘simple’fields. To be more precise, if a basis for F can be obtained from a set of fields {φi}i∈Jusing the operations given above (forming normal ordered products and derivatives) wewill say that the fields {φi}i∈J generate F. If all fields in {φi}i∈J are quasiprimary andorthogonal to all normal ordered products we will call these fields ‘simple’.

A set of simplefields will be called a simple set.The commutators of normal ordered products are completely determined by the commu-tators of the simple fields involved. This means that the whole Lie algebra structure ofthe mode algebra of F is already fixed by the commutation relations of the simple fields.The coupling constants of the simple fields determine all other coupling constants.In general, W-algebras are not Lie algebras since the commutators do not close linearly inthe fields.

If the OPE corresponding to the mode algebra is required to be associative, themode algebra has to fulfil all possible Jacobi identities. In order to ensure the validity ofall Jacobi identities it is sufficient to verify that only those involving simple fields (besidesL) are satisfied.

Looking at the corresponding OPE one notices that it is sufficient to3

show that the factors in front of primary fields vanish. This leads to equations among thecoupling constants connecting three simple fields.3.

General definitions and theorems about SSW-algebrasIn this chapter we adapt the results of the conformal case to the superconformal one.To this end, we recall the definitions of super fields, super primaries and super Jacobiidentities. Furthermore, we set up some conventions and establish some simple relationsbetween various structure constants arising from superconformal covariance.The super Virasoro algebra is the extension of the Virasoro algebra by a primary field Gof dimension 32.

Using the normalization dGG = 2c3 , the commutation relations take thefollowing form:[Lm, Ln] = (n −m)Lm+n + c12(n3 −n)δn+m,0[Lm, Gn] = (n −12m)Gm+n[Gm, Gn]+ = 2Lm+n + c3(m2 −14)δm+n,0,(3.1)which can also be obtained from the theorem of chapter 2. This algebra is known as thecentral extension of the algebra formed by the generators of superconformal transforma-tions in super space Z consisting of the points (z, θ), where θ is a Grassmannian variable.The two fields can be composed to the super Virasoro field L = 12G + θL, defined on Z.A field Φ = φ+θψ is called super primary of dimension d = d(Φ), iffφ and ψ are Virasoroprimaries of conformal dimension d and d + 12 respectively and[Gn, φm]± = CψGφp 32 ,d,d+ 12 (n, m) ψn+m = CψGφψn+m[Gn, ψk]± = CφGψp 32 ,d+ 12 ,d(n, k) φn+k =CφGψ2d (k −(2d −1)n)φn+k,(3.2)with φ and ψ super partners of each other; here CψGφ and CφGψ are certain structure con-stants and we have inserted the universal polynomials pijk.In perfect analogy to thequasiprimary case a field φ + θψ is called super quasiprimary iffφ and ψ are Virasoroquasiprimaries and (3.2) holds for n ∈{ 12, −12}.

This definition is equivalent to the covari-ant definition of super primary fields [21].Let A = {G, L, φ1, ψ1, ..., φn, ψn} be a simple set with super primaries Φi = φi + θψi.We will call the algebra generated by A a SW( 32, d(Φ1),...,d(Φn)) and the fields Φ1, ..., Φnadditional fields.To ensure the existence of SW( 32, d1, ..., dn) super Jacobi identities have to be checkedwhich are of the general form:ǫχiχk[[χi,p, χj,q]±, χk,r]±cycl. = 0,(3.3)4

where χi, χj and χk are components (φ or ψ) of super fields. In the sequel we will denote thefactor in front of the field χl,p+q+r on the left hand side of (3.3) by (χi,p, χj,q, χk,r, χl,p+q+r).The simplest example for SW-algebras is the super Virasoro algebra (3.1) since it obeysthe super Jacobi identities (3.3), which is also a super Lie algebra.The identities (3.3) will lead to relations involving different structure constants.

Some ofthese identities hold in general, already determined by the supersymmetric structure. Thisillustrates the fact that we are working with a non-explicitly covariant approach, dealingwith components of super fields rather than with super fields themselves.These relations between various structure constants can be derived from simple Jacobiidentities involving the component G of the super Virasoro field.

In order to give someexamples we fix the normalization of the fields first:dφiφi = (−1)2d(φi)+12d(φi) + 1cd(φi)dψiψi =cd(ψi)(3.4)Note that in the covariant approach this leads to the standard normalization of super fields[18].Then we can e.g. determine the structure constants CψGφ and CφGψ in (3.2):CψiGφi =12diCφiGψiandCψiGφi2 = 1(3.5)The first of these relations between the different structure constants is a direct consequenceof the normalization of the fields φi and ψi and the fact that they are simple fields.

Theother relation is easily seen for di ∈IN by the following argument (the case di ∈IN + 12 istreated similarly): Consider the following Jacobi identity:[[Gr, Gs]±, φi,m]±cycl. = 0Because (Gr, Gs, φi,m, φi,r+s+m) has to be zero we obtain:CLGGCφiLφip2,di,di(r + s, m)−CψiGφiCφiψiGp 32 ,di,di+ 12 (s, m)pdi+ 12 , 32 ,di(s + m, r) + pdi, 32 ,di+ 12 (m, r)pdi+ 12 , 32 ,di(m + r, s)= 0using CφiψiG = 2diCψiGφi, CLGG = 2 and CφiLφi = di, one obtains:CψiGφi2 = 1In this chapter we will choose the positive root, so that CψiGφi = 1.Furthermore, all possible coupling constants between three super primary fields (φi, ψi) ×(φj, ψj) →(φk, ψk) are determined by a single coupling constant of the components:5

i) di + dj + dk ∈IN + 12:Cφkψiφj = 2dkCkij ,Cφkφiψj = (−1)2di+12dkCkij ,Cψkψiψj = (−1)2di+1(σijk + 12)Ckij(3.6a)with Ckij = Cψkφiφjii) di + dj + dk ∈IN:Cψkψiφj = hikj2dkˆCkij ,Cφkψiψj = (−1)2di+1hijkˆCkij ,Cφkφiψj = (−1)2di hjki2dkˆCkij(3.6b)with ˆCkij = Cφkφiφjand σijk = di + dj + dk −1 , hijk = di + dj −dkFor a proof consider the condition(Gl, φi,m, φj,n, ψk,l+m+n) = 0.This yields an equation in the coupling constantsCψkψiφj,Cψkφiψj,Cφkφiφj.Setting l = 12 , m = dj −dk , n = 1 −dj and l = −12 , m = 1 −di , n = di −dk oneobtains two equations. Combining these equations proofs the first two relations.

The otherrelations are obtained by studying(Gl, φj,m, ψi,n, φk,l+m+n) = 0(Gl, ψi,m, ψj,n, φk,l+m+n) = 0(Gl, ψj,m, φi,n, ψk,l+m+n) = 0.Remark 3.1: If one assumes that a basis of super quasiprimary fields exists (e.g. inthe covariant approach) the formulae given above hold also for super quasiprimary fieldssince only l ∈{ 12, −12} is needed for the proof.

The same formulae can be deduced in theexplicitly covariant approach [18].Remark 3.2: Because of Ckij = (−1)[di]+[dj]+[dk+ 12 ]Ckji and ˆCkij = (−1)[di]+[dj]+[dk]ˆCkji oneobtains for (φi, ψi) = (φj, ψj):i) ˆCkii ̸= 0⇒dk ∈2INii) Ckii ̸= 0⇒dk + 12 ∈2INIn the explicitly covariant formulation [18] it is evident that in the singular part of theOPE of a super primary field Φi with Φj no normal ordered products of these super6

fields appear. In the non-explicitly covariant approach this may happen only if the fieldN (φi, φj) appears in the commutator [ψi, ψj]±.

In this case either N (φi, φj) must havea vanishing primary projection and can be replaced by a linear combination of the otherfields of dimension di + dj or the relevant coupling constant has to vanish. Denote theprimary projection of φ by Pφ.

Then one hasPN (φi, φj) ̸= 0⇒CN(φi,φj)ψiψj= 0(3.7)Since[ψi,m, ψj,n]± = CN(φi,φj)ψiψjN (φi, φj)m+n + ... ,applying G 12 to [ψi,m, ψj,n]± yieldsCN(φi,ψj)GN(φi,φj)CN(φi,φj)ψiψjN (φi, ψj)n+m+ 12 + CN(ψi,φj)GN(φi,φj)CN(φi,φj)ψiψjN (ψi, φj)n+m+ 12 + ...= [G 12 , [ψi,m, ψj,n]±]±= (m −di + 12)[φi,m+ 12 , ψj,n]± + (n −dj + 12)[ψi,n+ 12 , φj,m]±,where the dots stand for fields which are linearly independent from N (φi, ψj) and N (ψi, φj).Because CN(φi,ψj)GN(φi,φj) and CN(ψi,φj)GN(φi,φj) are nonzero and in the commutators in the last linethe field N (φi, ψi) does not occur CN(φi,φj)ψiψjhas to vanish.Remark 3.3: We use the following normalization in our calculations instead of (3.4):dφiφi =cd(φi)anddψiψi =cd(ψi)(3.4′)While the normalization (3.4) is more convenient for proofs this one is more convenientfor practical calculations (in this normalization the d-matrices almost factorize completlyinto linear factors (cf. appendix A and C) ).Now equations (3.5) and (3.6) have to be modified slightly.

One obtains e.g. :CψiGφi = 2di + 12di(−1)2di+1CφiGψiandCψiGφi2 = (−1)2di+1(2di + 1)(3.5′)Cφkψiφj2 = (2dk)2 (−1)2di+2dk(2dk + 1)2di + 1Ckij2(3.6′)After this general setup we present an algorithm for the explicit constuction of SW-algebras.Let A = {G, L, φ1, ψ1, ..., φn, ψn} be a simple set and Φi = φi + θψi superprimaries of dimension di.Then one proceeds as follows:First write down all linearly independent NOP’s which may appear in the commutator ofthe simple additional fields.

The algorithm used is based on the following facts. Let φ be7

a quasiprimary NOP of dimension δ. Then the quasiprimary projection of ∂φ = [L1, φ]is zero and leads to a linear equation in quasiprimary NOPs of dimension δ + 1.

Suchequations can be used to reduce the set of all possible NOPs of dimension δ + 1 to a basis.Formally, this can be described by assigning to each quasiprimary field of dimension δ acoloured partition p(δ). The linear equation obtained by the quasiprimary projection canbe written in terms of partitions of δ + 1.

Thus the main step of the algorithm is to deletefor any p(δ) an arbitrary p(δ + 1) that can be obtained by adding 1 to some element ofp(δ). For details of the algorithm as well as for a proof and examples we refer the readerto [22].Secondly, calculate all structure constants appearing in these commutators.

The structureconstants connecting three additional super primary fields remain as free parameters.Finally, one has to check all Jacobi identities with additional super primary fields only.This will in general lead to conditions for the free coupling constants and the central charge.Note that the coupling constants containing one G are determined by (3.5) and the couplingconstants between the superconformal families are determined by a single coupling constant(cf. (3.6)).In order to check the validity of the Jacobi identities it is sufficient to ensure that allfactors in front of the primary fields vanish.

Primary fields include the additional simplefields and perhaps fields built from NOPs of two additional fields and G, L. If the couplingconstants of the additional simple fields vanish, such NOPs cannot turn up in the Jacobiidentities and one has to check the factors in front of the additional simple fields only.If some coupling constants are nonzero one has to calculate which primary projections ofNOPs of additional simple fields do not vanish and to ensure that the coefficients in frontof these fields are zero. This is primarily a practical problem since the calculation of thesefactors is very complicated.

Therefore we checked the factors in front of the simple fieldsonly. In fact, there is no example showing that these conditions are not sufficient (in allpreviously known cases we obtained the same results).

Indeed there are some exampleswhere non-simple primary fields occur in the Jacobi identities and do not lead to furtherrestrictions (see for example SW( 32, 2) in the next chapter).For the special case of SW-algebras with two and three generators this general outlinereduces to the following.While for SW-algebras with two generators there remain two free parameters (c and theself-coupling) there are five free parameters (c and four coupling constants) for SW-algebras with three generators.The coupling constants Cjii vanish if dj ∈2IN + 1 ordj ∈2IN + 12 (cf. remark 3.2).

In almost all calculations we verified that the Jacobi iden-tities with L or G involved are satisfied automatically, if one uses (3.5) and (3.6). In one8

superconformal family four Jacobi identities remain to be checked:[[φm, φn]±, φk]±,cycl. = 0[[φm, φn]±, ψk]±,cycl.

= 0[[ψm, ψn]±, φk]±,cycl. = 0[[ψm, ψn]±, ψk]±,cycl.

= 0For SW( 32, d1, d2)-algebras there are six Jacobi identities involving two fields of one addi-tional superconformal family and one of the second family, such that alltogether 20 Jacobiidentities have to be verified.We denote the coupling of Φi with Φj to Φk by CΦkΦiΦj, whereCΦkΦiΦj =(Cφkφiφj , d(Φk) ∈INCψkφiφj , d(Φk) ∈IN + 12(3.6)If there exist null fields for a possible discrete value of the central charge one has to ensurethat either all further calculations are performed without these null fields and fixed centralcharge or that they are performed generically.4. Explicit results on SSW( 32,d)-algebrasIn this chapter we present our results on explicit constructions of several SW( 32, d)-algebras.All symbolic calculations have been performed in MATHEMATICATM and REDUCEwhile for the time consuming commutation operations and the expansion of NOP’s aspecial C-program had to be used.Since the values of the coupling constants do not depend on the special choice of the basisin the space of quasiprimary fields we will in general omit the coupling constants involvingnon-simple fields.

A special choice of a quasiprimary basis is contained in appendix A andB. In appendix C we list the Kac-determinants of states involving exactly one additionalfield.The cases 32 ≤d ≤72 were treated by J.Figueroa-O’Farrill et al.

[17].SSW( 32, 32)This algebra is a super Lie algebra with the followig commutation relations:[φm, φn]+ = Cψφφψm+n + 2Lm+n + c3(m2 −14)δm+n,0[φm, ψn] = 13(n −2m)Cφφψφm+n + 32Gm+n[ψm, ψn] = n −m2Cψψψψm+n + (n −m)Lm+n + c12(n3 −n)δm+n,0where Cψφφ = Cψψψ and Cφφψ = 34Cψψψ9

Thus SW( 32, 32) exists generically and the coupling constant Cψψψ is a free parameter. Notethat one may choose a new basis, such that the resulting commutators define the directsum of two super Virasoro algebras.

This case was discussed in [17] and is similar to theconformal one treated by Zamolodchikov [2].The cases d = 2, 52 with vanishing self-coupling have been investigated in detail by T.Inami et al. [14].SSW( 32, 2)This is the first case where a SW-algebra does not close linearly.

The commutation rela-tions are given by:[φm, φn] = n −m2Cφφφφm+n + (n −m)Lm+n + c12(n3 −n)δm+n,0[φm, ψn] = p2, 52 , 72 (m, n)CN(φ,G)φψN (φ, G)m+n + CN(L,G)φψN (L, G)m+n+p2, 52 , 52 (m, n)Cψφψψm+n+p2, 52 , 32 (m, n) 35CψGφGm+n[ψm, ψn]+ = p 52 , 52 ,4(m, n)CN(φ,φ)ψψN (φ, φ)m+n + CN(ψ,G)ψψN (ψ, G)m+n+CN(φ,L)ψψN (φ, L)m+n + CN(G,∂G)ψψN (G, ∂G)m+n+CN(L,L)ψψN (L, L)m+n+p 52 , 52 ,2(m, n)Cφψψφm+n + 2Lm+n+2c5n + 324δm+n,0where Cφψψ = 25Cφφφ ,Cψφψ = 12Cφφφ andCφφφ2 =4(5c + 6)2(4c + 21)(15 −c).We have verified explicitly that the coupling constant CN(φ,φ)ψψis zero. In general, the fieldN (φ, φ) cannot be neglected since it contributes to the d-matrix.

Only for c = −65 theself-coupling vanishes and all fields containing φ or ψ can be ignored. In this case the fieldsN (φ, φ) and N (ψ, φ) have vanishing primary projection.

Generically, this algebra containsa super primary field of dimension 4 which is quadratic in the simple super primary field.We have verified that if this field occurs in a Jacobi identity the factor in front of it vanishes(cf. remark at the end of chapter 3).Thus the algebra exists generically, which is not surprising since the classical counterpartis the symmetry algebra of the super Toda theory corresponding to the super Lie algebraOsp(3|2).SSW( 32, 52)The self-coupling vanishes and only two c values are possible:c = −52, 10710

SSW( 32, 3)Here the self-coupling is zero again and the allowed c values are:c = −452 , −277 , 54SSW( 32, 72)This is a case with non-vanishing self-coupling and consistency is obtained for:cCΦΦΦ27514601679135−1711−45009216523393In the sequel we present new SW( 32, d)-algebras with d up to 7.SSW( 32, 4)While for c = −203 the algebra SW( 32, 4) is consistent for vanishing self-coupling, it is alsoconsistent with nonzero self-coupling for four values of c:cCΦΦΦ2−1854355359375161644468−134270634247−212−5083692499−12013−2730437500007804122111−2030Only in the case c = −12013 the field N (φ, φ) has a nontrivial primary projection and evenin this case the coupling constant CN(φ,φ)ψψvanishes. This means that there is a primaryfield of dimension 8 contained in the algebra at c = −12013 .

In this case the mentioned fieldcontributes to the d matrix.SSW( 32, 92)This algebra exists only with vanishing self-coupling for three values of the central charge:c = −692 , −8110, 411SSW( 32, 5)Analogously to SW( 32, 3) the self-coupling has to vanish. Consistency implies here c =−10511 .11

SSW( 32, 112 )The algebra SW( 32, 112 ) is consistent for five values of c:cCΦΦΦ2107844918800345548631140−913767129733514069779−15519394077798400181920009−7058−43222353281253075066706603−5130SSW( 32, 6)This algebra is consistent with non-vanishing self-coupling forcCΦΦΦ2−33263096884483137409−18−1691684121617064005−224120788850108780331059143584629321555025With vanishing self-coupling it exists for:c = −932 , −16213 , 2720The field N (φ, φ) has a non-vanishing primary projection only for c = −18 and −224120 . Inthese cases CN(φ,φ)ψψis zero.SSW( 32, 132 )This algebra exists only for zero self-coupling and c = −19514 .SSW( 32, 7)Finally we discuss the algebra obtained by adding a super primary field of dimension 7.Since 7 is odd the self-coupling vanishes.

For this algebra we have checked the condition(φm, φn, φk, φm+n+k) = 0 only. The only allowed c values are:c = −775 , −138In the next section we will give some arguments that for these values of the central chargethe algebra should indeed exist.12

5. Structure of SSW( 32, d)-algebrasIn this chapter we summarize our results concerning SW-algebras with two generators.We will show that in perfect analogy to W(2, δ)-algebras for most values of the centralcharge a classification into special series is possible.W(2, δ)-algebras can be classified by the values of the central charge.Until now fivedifferent classes appeared:• the generically existing algebras related to simple Lie algebras (see e.g.

[23]),• the algebras related to the ADE classification of A. Cappelli et al. [24][10],• the algebras of the (1,s) series [25],• the parabolic W(2, δ)-algebras related to degenerate representations of the Virasoroalgebra [26] and• some exeptional cases [10][27].For SW(2, d)-algebras these five types also exist as will be shown below.The algebras SW( 32, d) with 32 ≤d ≤7 exist for the following values of the central charge:dc32generic2generic52−52, 1073−452 ,−277 , 5472−1711, 754−1854 ,−13,−212 , −12013 ,−20392−692 ,−8110,4115−10511112−7058 ,−15519 , −513, 1140, 1076−224120 ,−932 , −18,−332 ,−16213 , 2720132−195147−775 ,−138First, the generically existing algebras SW( 32, 32) and SW( 32, 2) are closely related to theLie super algebras Osp(2|1) and Osp(3|2), respectively [14].Secondly, all super minimal c values can be related to the ADE classification of modularinvariant partition functions of A. Cappelli et al.

[24]. To be more specific, we need the13

minimal models of the super Virasoro algebra. Therefore we list the values of the centralcharge and the corresponding superconformal dimensions [28][29][30][31].c(p, q) = 32(1 −2(p −q)2pq) with either p, q ∈IN , p, qcoprime andp + q ∈2INorp, q ∈2IN , p2, q2 coprime and p2 + q2 ̸∈2INh(r, s) = (rp −qs)2 −(p −q)28pq+ 1 −(−1)r+s321 ≤r ≤q −1 , 1 ≤s ≤p −1(5.1)r + s even yields representations in the Neveu-Schwarz sector and r + s odd those in theRamond sector.The divisibility conditions in (5.1) ensure that p, q ∈IN with p + q ∈2IN are chosen assmall as possible.Some simple fusion rule arguments suggest the followingProposition:For any d ∈IN with d = 18(p −2)(q −2),p, q ∈2IN ,p2, q2 coprime andp+q2∈2IN+1 the algebra SW( 32, d) exists for c = c(p, q) with vanishing self-coupling.Since in this case the central charge belongs to the super minimal series and the dimensionof the additional super primary field Φ can be parametrized by d = h(1, p−1) = h(q−1, 1),the well known fusion rules for super Virasoro minimal models may be applied [28][29][30].In terms of superconformal families they read[Φ] × [Φ] = [1]such that the additional field of dimension d is a `simple current´ in the sense of [32].

Inthis case general arguments suggest that this `simple current´ can be added to the algebraof the superconformal family of the identity, although this has been proven only for theunitary case [32][33][34][35].Many examples with zero self-coupling may be explained by the proposition. In fact thetwo c values for which SW( 32, 7) might exist belong to this series, so one may assume thatSW( 32, 7) indeed exists for these two values of the central charge.By considering the chiral part of the explicit form of a modular invariant partition functionan interpretation for all super minimal values of the central charge is possible [24]:14

seriesc(p, q) = (p, q)h(r, s) = (r, s)(D2ρ+2, Aq−1)(4ρ + 2, q)(q −1, 1) = 12ρ(q −2)(E6, Aρ−2)(ρ −1, 12)(7, 1) = ρ−42(5, 1) = ρ−22(E6, Dρ+1)(2ρ, 12)(7, 1) = 2ρ−32(11, 1) = 5(ρ−1)2(11, 1) = ρ−62(E8, Aρ−2)(ρ −1, 30)(19, 1) = 3(ρ−4)2(29, 1) = 7(ρ−3)2Note that all fields corresponding to the dimensions h(r, s) given above are local.Because the partition functions related to the first two rows of the table consist of binomialsonly, the number of simple fields in corresponding RCFTs is at most two.This caseis closely related to SW-algebras with two generators. Indeed the (D2ρ+2, Aq−1) seriesleads exactly to the cases explained by our proposition.

Furthermore, note that in the(E6, Dρ+1) and the (E8, Aρ−2) series the number of simple fields is at most 4. So thereseems to be a connection to SW-algebras with four generators.

Taking into account thatthe commutators of the super primary fields corresponding to the first h values cannotcontain the other fields leads to series of SW( 32, d)-algebras, too. For consistency the othersuper primary fields contained in these theories cannot be simple.There are two examples that fit in the (E6, Aρ−2) series both with CΦΦΦ ̸= 0.

Examples areSW( 32, 72) at c = 75 and SW( 32, 112 ) at 107 . This series is realized only for odd ρ leading to(p, q) = (ρ−1, 12) that fullfil the divisibility condition in (5.1).

These two Algebras can alsobe related to the (E6, Dρ+1) series where the dimension d is given by the second h valuein the table. The (E6, Dρ+1) series is realized for ρ = 7, 11, 13 as SW( 32, 52), SW( 32, 92)and SW( 32, 112 ) with vanishing self-coupling.

Note that only the cases where ρ and 6 arecoprime appear. Since SW( 32, 52) and SW( 32, 112 ) exist for c = 107 and both can be related to(E6, D8) there should be a connection.

We have explicitly verified that SW( 32, 52) containsa super primary non-simple field of dimension112 at c =107 . There is strong evidencethat this yields a realization of SW( 32, 112 ) at the c value mentioned because we havecalculated the self-coupling of one component of the non-simple super primary field.

Itsvalue is exactly that obtained by the direct construction of SW( 32, 112 ). For details werefer to appendix D. A similar construction where SW( 32, 72) is realized as a subalgebra ofSW( 32, 32) at c = 75 has been carried out by K. Hornfeck [19].The (E8, Aρ−2) series is realized only for ρ = 17 as SW( 32, 112 ) at c = 1140.For 12 ≤ρ ≤16 the resulting pair (ρ−1, 30) does not belong to the (E8, Aρ−1) series sincethe divisibility condition in (5.1) is not satisfied.

The next possible value for ρ is 23 andthe corresponding dimension 172 .15

Thirdly, note that SW( 32, d) exists for c = c(1, s), s odd and d = 2s−12. This case is similarto the conformal one treated by H.G.

Kausch for W-algebras with two generators. In [25]a free field construction was presented and one can hope that similar techniques lead to afree field realization in the supersymmetric case.

We will call this series the (1, s)-series.Fourthly, all SW( 32, d) with d =32n or d = 2n , n ∈IN exist for c =32(1 −163 d) =32(1 −8n) = c3d(n) or c =32(1 −2d) =32(1 −4n) = c8d(n) respectively.These casesare related to degenerate representations of the super Virasoro algebra and will be calledparabolic.The central charge and the dimensions of the primary fields for the superVirasoro algebra are given by [29]:c = 32(1 −16α02)α± =rα02 + 12 ± α0hr,s = (rα+ −sα−)24−α02(5.2)Note that hr,r = α02(r2 −1) and hr,−r = α02(r2 −1)+ 12r2. The series described above canbe obtained by adding an additional field of superconformal dimension d = h2,2 or d = h3,3to the superconformal family of the identity.

Note that these fields are local because theyobey the locality conditionǫdd = e2πi(1−r)2α20 = e2πi (1−r)2r2−1 d = ±1if one assumes 2 (r−1)2r2−1 d ∈IN. Taking into account that the fields corresponding to h2,2 orh3,3 are either bosonic or fermionic leads to the possible values of d given above.

For r ≥4one would not get a SW-algebra with only two generators. These algebras lead to rationaltheories with effective central charge ˜c = 32 [36].

For the similar case of W(2, δ)-algebrasthe characters and S-matrices have been calculated and the generalization to the supercase is straightforward [26].Finally, some algebras remain which do not belong to any of the series described above. Allof them have non-vanishing self-coupling.

The investigation of the representation theory ofthese algebras leads to a better understanding. One obtains e.g.

that the effective centralcharge is greater than 32 [36].Finally we list all SW( 32, d)-algebras with 32 ≤d ≤7 and show how they fit into thesepatterns.16

dc = c(p, q)d = h(r, s)seriesCΦΦΦ = 0CΦΦΦ ̸= 032genericgeneric2(10, 4) = −65(3,1)(D6, A3)generic52(3, 1) = −52(3,1)(1, s)(14, 12) = 107(5,1)(E6, D8)(14, 4) = −277(3,1)(D8, A3)3(6, 8) = 54(7,1)(D4, A7)c3d(2) = −452(2,2)parabolic72(10, 12) = 75(7,1)(E6, A9)−1711? (18, 4) = −203(3,1)(D10, A3)c8d(2) = −212(3,3)parabolic4−1854?−13?−12013?

(5, 1) = −8110(3,1)(1, s)92(22, 12) =411(5,1)(E6, D12)c3d(3) = −692(2,2)parabolic5(22, 4) = −10511(3,1)(D12, A3)(26, 12) = −513(5,1)(E6, D14)(14, 12) = 107(7,1)(E6, A13)112(16, 30) = 1140(11,1)(E8, A15)−7058?−15519? (26, 4) = −16213(3,1)(D14, A3)6(10, 8) = 2720(7,1)(D6, A7)c3d(4) = −932(2,2)parabolicc8d(3) = −332(3,3)parabolic−18?−224120?132(7, 1) = −19514(3,1)(1, s)7(30, 4) = −775(3,1)(D16, A3)(6, 16) = −138(15,1)(D4, A15)17

6. Explicit results about SSW( 32, d1, d2)-algebrasIn this charpter we present our results about the construction of SW-algebras with twoadditional generators.

For these algebras one has to check the validity of 20 Jacobi iden-tities. For SW( 32, d1, d2) we will denote the two additional super fields of dimension di byΦi = φi + θψi.SSW( 32, 32, 32)In this case there are five free parameters namely the central charge and four couplingconstants.

Consistency implies that the commutators close linearly in the fields and alsothe following condition:Cψ2φ1φ12+Cψ1φ2φ22−Cψ2φ1φ1Cψ2φ2φ2 −Cψ1φ2φ2Cψ1φ1φ1 −4 = 0In complete analogy to the SW( 32, 32)-algebra one can choose a new basis such that theresulting commutators define the direct sum of three super Virasoro algebras. As describedin [17] this case can be generalized easily to the extension of the super Virasoro algebraby n simple fields of dimension 32.The following algebra has been studied in [18] using the covariant approach.SSW( 32, 32, 2)For this algebra five parameters have to be calculated.

The Jacobi identities imply thattwo sets of solutions exist:1. )Cφ2φ1φ1 = 0Cψ1φ1φ1 =Cψ1φ2φ22−4Cψ1φ2φ2Cφ2φ2φ22=412 + 10c + 3Cψ1φ2φ2224 +Cψ1φ2φ2284 + 16c + 21Cψ1φ2φ2260 −4c + 15Cψ1φ2φ222.

)Cψ1φ2φ2 = 12Cψ1φ1φ1Cψ1φ1φ12= 4Cφ2φ1φ12−4Cφ2φ1φ1Cφ2φ2φ2 −16Cφ2φ1φ1 =8c10c −27Cφ2φ2φ2This means that in both cases there remain two free parameters. While the first solutionis a trivial one as discussed below the second solution could be the symmetry algebra ofthe quantized Toda theory corresponding to D(2|1, α).18

SSW( 32, 32, 52)Since the coupling constant of the superconformal family of dimension 32 to the supercon-formal family of dimension 52 and the self-coupling of the latter are zero there remain threefree parameters. Two types of solutions exist with vanishing Cψ2φ1φ1:1.

)Cψ1φ2φ22= 14c −205Cψ1φ1φ1Cψ1φ2φ2 = 2(7c −20)5c ̸= 107orCψ1φ2φ22= −8c + 205Cψ1φ1φ1Cψ1φ2φ2 = −8(c + 5)5c ̸= −522. )Cψ1φ2φ22= −4Cψ1φ1φ1Cψ1φ2φ2 = −8In both cases the central charge is free.For SW( 32, 32, 2) and SW( 32, 32, 52) the first solutions can be decomposed into a direct sum.In general, one has the following structure.

Assume that SW( 32, d) with generators ˆL andˆΦ exists for ˆc and self-coupling ˆC ˆΦˆΦˆΦ = f(ˆc). Then SW( 32, 32, d) exists for generic centralcharge c and:Cψ2φ1φ1 = 0Cψ1φ1φ12= 4ˆc(c −2ˆc)2c −ˆcCψ1φ2φ22= 4c −ˆcˆcCΦΦΦ = cˆcf4(4 + Cψ1φ2φ2)−1This is easily seen since a change of basis implies for this solution:SW( 32, 32, d) ∼= SW( 32) ⊕SW( 32, d)For the second solution of SW( 32, 32, 52) such a change of basis becomes singular and istherefore not possible.

It is remarkable that one can choose a linear combination of thesuper Virasoro field and the additional super field of dimension 32 such that the resultingfield (anti-)commutes with itself. This field is built up by the sum of the Super Virasorofield andCψ1φ1φ14times the field Φ1.SSW( 32, 2, 2)Consistency implies fixed central charge and the following conditions for the four freecoupling constants:c = 32Cφ1φ2φ2 = −Cφ1φ1φ1Cφ2φ1φ1 = −Cφ2φ2φ2Cφ1φ1φ12+Cφ2φ2φ22= 219

For this solution the fields N (φ1, φ1), N (φ2, φ2) and N (φ1, φ2) have no primary projectionand do not contribute to the algebra.Obviously, this algebra has an inner SO(2) symmetry realized as a rotation in the spaceof the super fields of dimension two. Consequently the solution is determined by relationsthat are invariant under the action of SO(2).

Note that a rotation by an angle α in thespace of super fields yields a rotation of angle 3α in the space of self-couplings.One could speculate that SW( 32, 2, 2) for c = 32 is a subalgebra of SW( 32, 2, 2, 72), whichshould exist generically and is related to Osp(4|4) [37].A very similar case for theW(2, 4, 4)-algebra which is related to SO(8) has been treated in [25].SSW( 32, 2, 52)For this algebra there exist two consistent sets of solutions:1. )c = −15Cφ1φ2φ22= −4013Cφ1φ1φ12= −105865Cφ1φ2φ2Cφ1φ1φ1 = −92132.

)c = 392Cφ1φ2φ22= −88Cφ1φ1φ12= −105811Cφ1φ2φ2Cφ1φ1φ1 = −92For c = −15 the field N (φ1, φ2) can be written as a linear combination of the other fieldsof dimension 92. This is the only case in which null fields appear and must be omitted (cf.the remark at the end of chapter 3).SSW( 32, 52, 52)There is no solution for this algebra.Analogously to W-algebras a SW-algebra withtwo additional super fields and no nonzero coupling constant cannot exist, since a Jacobiidentity of type[[φ1m, φ1n]±, φ2k]±cycl.

= 0 cannot be satisfied [10].That means SW( 32, d1, d2) cannot exist for [d1 + 12], [d2 + 12] ∈2IN + 1.SSW( 32, 52, 72)In this case two coupling constants are free and one obtains only one set of solutions with20

no free parameters:c = 136Cψ2φ1φ12= 5208979Cψ2φ2φ22= 31770960698Cψ2φ1φ1Cψ2φ2φ2 = −5166979This is an algebra predicted by Schoutens et al. in [38] by coset considerations.

It hasbeen shown in [19] by K. Hornfeck that if this SW-algebra has a W(2, 3, 4) as subalgebrait can at most exist for c = 136 . Indeed this value is the only possible one.7.

ConclusionUsing a non-explicitly covariant approach we have been able to construct a whole bunchof new SW-algebras with two generators. Most of the results fit into systematic patternsin complete analogy to W-algebras.For some new SW-algebras namely SW( 32, 4) atc = −1854 , −12013 , −13 and SW( 32, 112 ) at c = −7058 , −15519 and SW( 32, 6) at c = −18, −224120an interpretation has not been found yet.

Even the interpretation of SW( 32, 72) at c = −1711is still an open question although this algebra has been known for some time. A studyof the possible and physically relevant highest weight repesentations of these algebras willlead to a better understanding of these algebras.

The methods explained in [39][27] canbe applied to this case with only small changes and work is in progress [36].In contrast to W(2, δ)-algebras no consistent SW( 32, d)-algebras with irrational values ofthe central charge appeared. Furthermore, there is only one generically existing nonlinearSW( 32, d)-algebra, namely SW( 32, 2).For SW-algebras with three generators we have constructed two new algebras that existfor finitely many c values only.Since the difficulties of the transition from W-algebras to SW-algebras have not been toohard we are confident that similar calculations for the super N = 2 case will be possiblein the near future.AcknowledgementsWe would like to thank W. Nahm, M. Flohr, J. Kellendonk, S. Mallwitz, A. Recknagel,M.

R¨osgen, M. Terhoeven and R. Varnhagen for many useful discussions. The creativeatmosphere in the institute was an important support for this work.It is a pleasure to thank the Max-Planck-Institut f¨ur Mathematik in Bonn-Beuel especiallyTh.

Berger, S. Mauermann and T. H¨ofer since nearly all calculations have been performedin their computer centre.21

Appendix:In the appendices A, B and C we list all fields and determinants (of matrices (dij)) whichoccur in the examples considered in chapter 4. Because fields of different conformal dimen-sions are orthogonal, the matrices have block-diagonal form.

Furthermore, all fields in thesuperconformal family of the identity have vanishing (dij) with fields of the superconformalfamily of the additional super primary field. Thus the determinants factorize in two terms:detDδ = detD[1]δ detD[d]δwhere [d] stands for the superconformal family of the additional super primary field ofdimension d and δ is the dimension considered.

But there is one exception to this rule:if d ∈2IN and δ = 2d the field N (φ, φ) is involved and the determinant detDδ does notfactorize. Because such determinants are very complicated we omit them.Appendix A: A basis of quasiprimary fields up to dimension 252 built up by Ld = 32 : 1 field : GdetD[1]32 = 2c3d = 2 : 1 field : LdetD[1]2= c2d = 72 : 1 field : N (L, G)detD[1]72 =112c(21 + 4c)d = 4 : 2 fields : N (G, ∂G),N (L, L)detD[1]4=160c2(21 + 4c)(−7 + 10c)d = 92 : 1 field : N (L, ∂G)detD[1]92 = −235c(−7 + 10c)d = 112 : 2 fields : N (L, ∂2G),N (N (L, L), G)detD[1]112 =154c2(11 + c)(21 + 4c)(−7 + 10c)d = 6 : 4 fields :N (G, ∂3G),N (N (L, G), ∂G), N (N (L, L), L), N (L, ∂2L)detD[1]6=1378c4(11 + c)(21 + 4c)2(−7 + 10c)2(11 + 14c)d = 132 : 2 fields :N (L, ∂3G),N (N (L, L), ∂G)detD[1]132 =16847c2(21 + 4c)(−7 + 10c)(11 + 14c)d = 7 : 1 field : N (N (L, G), ∂2G)detD[1]7= −245c(21 + 4c)(−7 + 10c)22

d = 152 : 5 fields :N (N (G, ∂G), ∂2G),N (L, ∂4G),N (N (L, L), ∂2G),N (N (N (L, L), L), G), N (N (L, ∂2L), G)detD[1]152 =6427885(−1 + c)c5(11 + c)(21 + 4c)3(135 + 8c)(−7 + 10c)3(11 + 14c)d = 8 : 7 fields :N (G, ∂5G),N (N (L, G), ∂3G),N (N (L, ∂G), ∂2G),N (N (N (L, L), G), ∂G), N (N (N (L, L), L), L), N (N (L, L), ∂2L),N (L, ∂4L)detD[1]8=518400024191167(−1 + c)c7(11 + c)2(21 + 4c)4(135 + 8c)(−7 + 10c)4(11 + 14c)2d = 172 : 4 fields :N (L, ∂5G),N (N (L, L), ∂3G),N (N (N (L, L), L), ∂G), N (N (L, ∂2G), ∂G)detD[1]172 = 3686465065c4(11 + c)(21 + 4c)3(−7 + 10c)3(11 + 14c)d = 9 : 4 fields :N (N (L, G), ∂4G),N (N (L, ∂G), ∂3G),N (N (N (L, L), G), ∂2G),N (N (L, L), ∂3L)detD[1]9= 108343(−1 + c)c4(11 + c)(21 + 4c)3(−7 + 10c)3(11 + 14c)d = 192 : 9 fields :N (N (G, ∂G), ∂4G),N (N (N (L, G), ∂G), ∂2G),N (L, ∂6G),N (N (L, L), ∂4G),N (N (N (L, L), L), ∂2G),N (N (L, ∂2L), ∂2G),N (N (N (N (L, L), L), L), G), N (N (N (L, L), ∂2L), G),N (N (L, ∂4L), G)detD[1]192 = 859963392053202877 (−1 + c)2c9(11 + c)3(21 + 4c)6(114 + 5c)(135 + 8c)(−7 + 10c)6(11 + 14c)3d = 10 : 12 fields :N (G, ∂7G),N (N (L, G), ∂5G),N (N (L, ∂G), ∂4G),N (N (L, ∂2G), ∂3G),N (N (N (L, L), G), ∂3G),N (N (N (L, L), ∂G), ∂2G),N (N (N (N (L, L), L), G), ∂G),N (N (N (L, ∂2L), G), ∂G),N (N (N (N (L, L), L), L), L), N (N (N (L, L), L), ∂2L)N (N (L, L), ∂4L),N (L, ∂6L)detD[1]10 = 1011316948992000000762819593107(−1 + c)2c12(11 + c)4(21 + 4c)8(114 + 5c)(135 + 8c)2(−7 + 10c)8(11 + 14c)4(95 + 22c)d = 212 : 10 fields :N (N (G, ∂G), ∂5G),N (N (N (L, G), ∂G), ∂3G),N (L, ∂7G), N (N (L, L), ∂5G),N (N (N (L, L), L), ∂3G),N (N (L, ∂2L), ∂3G),N (N (N (N (L, L), L), L), ∂G),N (N (N (L, L), ∂2L), ∂G),N (N (L, ∂4L), ∂G),N (N (N (L, L), ∂3L), G)detD[1]212 = 47719920649780592640000000065518365270271857229(−1 + c)2c10(11 + c)3(21 + 4c)7(135 + 8c)(−7 + 10c)7(11 + 14c)4(95 + 22c)23

d = 11 : 9 fields :N (N (L, G), ∂6G),N (N (L, ∂G), ∂5G),N (N (L, ∂2G), ∂4G),N (N (N (L, L), G), ∂4G),N (N (N (L, L), ∂G), ∂3G),N (N (N (N (L, L), L), G), ∂2G),N (N (N (L, ∂2L), G), ∂2G),N (N (N (L, L), L), ∂3L),N (N (L, L), ∂5L)detD[1]11 = −281792804290562215547(−1 + c)2c9(11 + c)3(21 + 4c)7(135 + 8c)(−7 + 10c)7(11 + 14c)4d = 232 : 16 fields :N (N (G, ∂G), ∂6G),N (N (N (L, G), ∂G), ∂4G),N (N (N (L, G), ∂2G), ∂3G)N (L, ∂8G),N (N (N (N (L, L), G), ∂G), ∂2G),N (N (L, L), ∂6G),N (N (N (L, L), L), ∂4G),N (N (L, ∂2L), ∂4G),N (N (N (N (L, L), L), L), ∂2G),N (N (N (L, L), ∂2L), ∂2G),N (N (L, ∂4L), ∂2G),N (N (N (L, L), ∂3L), ∂G)N (N (N (N (N (L, L), L), L), L), G), N (N (N (N (L, L), L), ∂2L), G),N (N (N (L, L), ∂4L), G),N (N (L, ∂6L), G)detD[1]232 = 16543163447903718821855232000000217118368393969(−1 + c)4c16(11 + c)6(21 + 4c)12(115 + 4c)(114 + 5c)(135 + 8c)3(−7 + 10c)12(11 + 14c)6(95 + 22c)d = 12 : 23 fields :N (N (N (G, ∂G), ∂2G), ∂3G),N (G, ∂9G),N (N (L, G), ∂7G),N (N (L, ∂G), ∂6G),N (N (L, ∂2G), ∂5G),N (N (L, ∂3G), ∂4G),N (N (N (L, L), G), ∂5G),N (N (N (L, L), ∂G), ∂4G),N (N (N (L, L), ∂2G), ∂3G),N (N (N (N (L, L), L), G), ∂3G),N (N (N (N (L, L), L), ∂G), ∂2G),N (N (N (L, ∂2L), G), ∂3G),N (N (N (L, ∂2L), ∂G), ∂2G),N (N (N (N (N (L, L), L), G), ∂G), L), N (N (N (N (L, ∂2L), G), ∂G), L),N (N (N (L, ∂4L), G), ∂G),N (N (N (N (N (L, L), L), L), L), L),N (N (N (N (L, L), L), L), ∂2L),N (N (N (L, L), L), ∂4L),N (N (N (L, L), ∂2L), ∂2L),N (N (L, L), ∂6L),N (N (L, ∂2L), ∂4L),N (L, ∂8L)detD[1]12 = 6989311525778920478308485492526951610449920000000000000000020710629462273412554478537721(−1 + c)6c23(11 + c)8(21 + 4c)16(115 + 4c)(114 + 5c)2(135 + 8c)4(−7 + 10c)17(11 + 14c)9(95 + 22c)2(161 + 26c)(−81 + 70c)d = 252 : 19 fields :N (N (G, ∂G), ∂7G),N (N (N (L, G), ∂G), ∂5G),N (N (N (L, G), ∂2G), ∂4G),N (N (N (L, ∂G), ∂2G), ∂3G),N (L, ∂9G),N (N (N (N (L, L), G), ∂G), ∂3G),N (N (L, L), ∂7G),N (N (N (L, L), L), ∂5G),N (N (L, ∂2L), ∂5G),N (N (N (N (L, L), L), L), ∂3G),N (N (N (L, L), ∂2L), ∂3G),N (N (L, ∂4L), ∂3G),N (N (N (L, L), ∂3L), ∂2G),N (N (N (N (N (L, L), L), L), ∂G), L),N (N (N (N (L, L), ∂2L), ∂G), L),N (N (N (L, L), ∂4L), ∂G),N (N (L, ∂6L), ∂G),N (N (N (N (L, L), ∂3L), G), L),N (N (N (L, L), ∂5L), G)detD[1]252 = −3595234824692207317857811466802001511082331892644577280000000000000000000012565355615115842815180391716848789977906099(−1 + c)5c19(11 + c)7(21 + 4c)14(114 + 5c)(135 + 8c)3(−7 + 10c)15(11 + 14c)8(95 + 22c)(161 + 26c)(−81 + 70c)24

Appendix B: A basis of quasiprimary fields up to dimension d(Φ)+6 builtup by one super primary field Φ and Ld = d(Φ) : 1 field :φd = d(Φ)+ 12 : 1 field :ψd = d(Φ)+ 32 : 1 field :N (φ, G)d = d(Φ)+2 : 2 fields :N (φ, L),N (ψ, G)d = d(Φ)+ 52 : 2 fields :N (φ, ∂G),N (ψ, L)d = d(Φ)+3 : 2 fields :N (φ, ∂L),N (ψ, ∂G)d = d(Φ)+ 72 : 3 fields :N (φ, ∂2G),N (N (φ, L), G), N (ψ, ∂L)d = d(Φ)+4 : 5 fields :N (N (φ, G), ∂G),N (N (φ, L), L), N (φ, ∂2L),N (ψ, ∂2G),N (N (ψ, L), G)d = d(Φ)+ 92 : 6 fields :N (φ, ∂3G),N (N (φ, L), ∂G), N (N (φ, ∂L), G),N (N (ψ, G), ∂G), N (N (ψ, L), L), N (ψ, ∂2L)d = d(Φ)+5 : 6 fields :N (N (φ, G), ∂2G),N (N (φ, L), ∂L), N (φ, ∂3L),N (ψ, ∂3G),N (N (ψ, L), ∂G), N (N (ψ, ∂L), G)d = d(Φ)+ 112 : 8 fields :N (φ, ∂4G),N (N (φ, L), ∂2G),N (N (φ, ∂L), ∂G),N (N (N (φ, L), L), G), N (N (φ, ∂2L), G),N (N (ψ, G), ∂2G),N (N (ψ, L), ∂L), N (ψ, ∂3L)d = d(Φ)+6 : 12 fields :N (N (φ, G), ∂3G),N (N (φ, ∂G), ∂2G),N (N (N (φ, L), G), ∂G),N (N (N (φ, L), L), L), N (N (φ, L), ∂2L),N (N (φ, ∂L), ∂L),N (φ, ∂4L), N (ψ, ∂4G),N (N (ψ, L), ∂2G),N (N (ψ, ∂L), ∂G), N (N (N (ψ, L), L), G), N (N (ψ, ∂2L), G)25

Appendix C: Determinants of quasiprimary parts of the Kac-matrixwith one additional super primary field involvedSW( 32,2) :detD[2]72 =115c(6 + 5c)detD[2]4= −1150c2(29 + 2c)(6 + 5c)SW( 32, 72) :detD[ 72 ]5= −121c(21 + 4c)detD[ 72 ]112 =1336c2(53 + 2c)(21 + 4c)detD[ 72 ]6=140(−1 + c)c2(53 + 2c)detD[ 72 ]132 =4121(−1 + c)c2(21 + 4c)SW( 32,4) :detD[4]112 =118c(20 + 3c)detD[4]6= −1324c2(61 + 2c)(20 + 3c)detD[4]132 = −1495c2(61 + 2c)(−7 + 10c)detD[4]7= −1270c2(20 + 3c)(−7 + 10c)detD[4]152 = −116731c3(61 + 2c)(20 + 3c)2(−65 + 44c)detD[4]8=129513484c5(5 + 2c)(61 + 2c)2(20 + 3c)2(−7 + 10c)(377 + 10c)(−65 + 44c)SW( 32, 112 ) :detD[ 112 ]7= −433c(11 + c)detD[ 112 ]152=1198c2(11 + c)(85 + 2c)detD[ 112 ]8=11092c2(85 + 2c)(5 + 13c)detD[ 111 ]172=162925c2(11 + c)(5 + 13c)detD[ 112 ]9= −21155c3(11 + c)2(85 + 2c)(−10 + 7c)detD[ 112 ]192=12203047c5(11 + c)2(85 + 2c)2(−10 + 7c)(5 + 13c)(4717 + 1092c + 20c2)detD[ 112 ]10=520857419c6(11 + c)2(15 + c)(85 + 2c)2(−5 + 4c)(5 + 13c)2(4717 + 1092c + 20c2)detD[ 112 ]212=81920019321000497c6(11 + c)3(15 + c)(85 + 2c)2(−5 + 4c)(−10 + 7c)(5 + 13c)2SW( 32,6) :detD[6]152 =1117c(162 + 13c)detD[6]8= −13042c2(93 + 2c)(162 + 13c)detD[6]172 = −11365c2(93 + 2c)(11 + 14c)detD[6]9= −12912c2(162 + 13c)(11 + 14c)26

detD[6]192 = −1338130c3(93 + 2c)(162 + 13c)2(−27 + 20c)detD[6]10 =11602229005c5(93 + 2c)2(162 + 13c)2(11 + 14c)(−27 + 20c)(5917 + 1188c + 20c2)detD[6]212 = −16186752880639c6(33 + 2c)(93 + 2c)2(162 + 13c)2(11 + 14c)2(−27 + 20c)(5917 + 1188c + 20c2)detD[6]11 = −24461283125c6(33 + 2c)(93 + 2c)2(162 + 13c)3(11 + 14c)2(−27 + 20c)2detD[6]232 = −204886728774951175175c8(93 + 2c)3(162 + 13c)5(11 + 14c)2(−27 + 20c)2(−43 + 85c)(5917 + 1188c + 20c2))detD[6]12 = −8192412159655278351185230025c12(33 + 2c)(93 + 2c)5(162 + 13c)6(11 + 14c)3(−27 + 20c)3(−43 + 85c)(5917 + 1188c + 20c2)2(−837 + 8540c + 140c2)Appendix D: Realization of SSW( 32, 112 ) at c = 107 as a subalgebra of SSW( 32, 52)As mentioned in chapter four and five there is a strange connection between these algebrasat c = 107 . SW( 32, 52) contains a super primary field of dimension 112 .

The two componentsof this field are:˜φ = αN (ψ, φ) + 886929580CψGφN (L, ∂2G) −553725143CψGφN (N (L, L), G)˜ψ = α√2N (ψ, ψ) + 16N (φ, ∂φ) + 857567048N (G, ∂3G)+206143251430N (L, ∂2L) −553725143N (N (L, G), ∂G)) −1107425143N (N (L, L), L)α2 =1760011567918where Φ = φ + θψ is the simple super primary field of dimension 52. The fields ˜φ , ˜ψ areorthogonal to all other fields of dimension 112 , 6 and their normalization is described by(3.3’).

We have calculated the self-coupling:C˜ψ˜φ ˜φ2= 84491880034554863This is exactly the coupling obtained in the direct construction of SW( 32, 112 ) at c = 107 .The two different signs of the coupling are realized by the two possible signs for α.27

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