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A Unified Scheme for Modular Invariant Partition Functions of WZW Models

a r X i v :h e p -t h /9306072v 2 21 J u n 1993SUTDP/11/93/72

May,1993

A UNIFIED SCHEME FOR MODULAR INV ARIANT

PARTITION FUNCTIONS OF WZW MODELS

M.R.Abolhassan i a , F.Ardala n a,b a Department of Physics,Sharif University of Technology P.O.Box 11365-9161,Tehran,Iran b Institute for studies in Theoretical Physics and Mathematics P.O.Box 19395-1795,Tehran,Iran Abstract We introuduce a uni?ed method which can be applied to any WZW model at arbitrary level to search systematically for modular invariant physical partition functions.Our method is based essentially on modding out a known theory on group manifold G by a discrete group Γ.We apply our method to su (n )with n =2,3,4,5,6,and to g 2models,and

obtain all the known partition functions and some new ones,and give explicit expressions for all of them.

1.Introduction

Conformal?eld theories(CFT’s)play an important role in two dimensional critical statisti-cal mechanics and string theory,and has been extensively studied in the past decade.1?4 Among these theories WZW models have attracted considerable attentions,because as two dimensional rational conformal?eld theories they are exactly solvable,5,6and most known CFT’s can be obtained from them via coset construction.7Moreover these models explicitly appear in some statistical models like quantum chains,and describe their critical behavier.8WZW models in addition to conformal symmetry,have an in?nite dimensional symmetry whose currents satisfy a Kac-Moody algebra?g at some level k.The partition function of a WZW model takes the form

Z(τ,ˉτ)= χλ(τ)Mλ,λ′χ?λ′(ˉτ),(1.1)

whereχλis the character of the a?ne module whose highest weigth(HW)isλ,Mλ,λ′are positive integers which determine how many times the HW representationsλ,λ′in the left and right moving sectors couple with each other,and the sum is over the?nite set of integrable representations(see Subsec. 2.1.for a precise de?nition).For the consistency of a physical theory it is necessary that the partition function(1.1)be invariant under the modular group of the torus.9Construction and classi?cation of partition functions of WZW models has been the goal of a large body of work in the past few years.However,up to now only the classi?cations of su(2),10and su(3)11?13at arbitrary level,and of simple a?ne Lie algebras at level one14,15have been completed.

To?nd moudular invariant partition functions of WZW models a number of methods have been used:Automorphism of Kac-Moody algebras,16,17simple currents,18,19confor-mal embedding,20?22automorphism of the fusion rules of the extended chiral algebra,23,24 lattice method,26,27and?nally direct computer calculations.28In these e?orts many mod-ular invariant partition functions have been found,and may be arranged in three broad categories:

i)Diagonal Series?For every WZW model with a simply connected group manifold, there exists a physical modular invariant theory with diagonal matrix Mλ,λ′=δλ,λ′.9,17 They are often designated as a member of the A series.

ii)Complementary Series?There are some nondiagonal series for every WZW model whose Kac-Moody algebra has a nontrivial centre,16,17associated to subgroups of the centre.They are often designated as members of the D series.

iii)Exceptional Series?In addition to the above two series,WZW models have a number of nondiagonal partition functions which occur only at certain levels.They are called E series.

Some of the known E series have been found by the conformal embedding method (see e.g.Ref.13),some by utilizing the nontrivial automorphism of the fusion rules of the extended algebra,24,25and some others by computer calculations.28Alhough many exceptional partition functions have been obtained by these methods,however they don’t follow from a uni?ed method and prove to be impractical for high rank groups and high levels.Furthermore,these methods don’t answer the question of why there are exceptional partition functions only at certain levels.It must be mentioned that corresponding to any

physical theory,there exists a charge conjugation,c.c.,counterpart,such that M(c.c.)

λ,λ′=

Mλ,C(λ′),where C(λ)is the complex conjugate representation ofλ.

In this paper a uni?ed approach which we call orbifold-like method,is presented and shown to lead to all the known nondiagonal theroies.The method is easily applied to high

rank groups at arbitrary level.The organization of the paper is as follows:In Sec.2,we brie?y review some characteristic features of Kac-Moody algebra,such as their unitary

highest weight representations and modular transformation properties of their characters. Then we present our approach.We start with some known theory whose partition function is Z(G)and mod it out by a discrete groupΓ.We will takeΓto be a cyclic group Z N.

It is not necessarily a subgroup of the centre of?g.In order for the modding to gives rise to a modular invariant combination with rational coe?cients,certain relations must be satis?ed,which will be explicated.In Sec.3,we apply our method to su(n)models with n=2,3,4,5,6and as an example of a non-simply laced a?ne Lie algebra,to g2models, and generate all the known nondiagonal theories and some new exceptional ones.In Sec.

4,we conclude with some remarks.In Appendix A,we gather some formulas and relations that are used in the body of the paper.

2.The Orbifold-like method

Observing that all modular invariant partition functions of su(n)models at level one are obtained by modding out the group SU(n)by subgroups of its centre15,the authors in Ref. 29for the?rst time found that not only the D series of su(2)WZW models are obtained with modding out the diagonal theories by the Z2centre,but also all their exceptional partition functions can be found by modding out the A or D series by a Z3which is not obviously a subgroup of the centre.

In this paper we are going to generalize that work to WZW models with any associated a?ne Lie algebra at arbitrary level.We have called our approach orbifold-like method, because the?nite group which one uses in modding may not be the symmetry of a target manifold.Befor going to the details of our approach,we collect in the following subsection, some facts about untwisted Kac-Moody algebras and set up our notations.

2.1.Preliminaris and Notations

A WZW model is denoted by?g k where its a?ne symmetry algebra is the untwisted Kac–Moody algebra?g associated with a compact Lie algebra g,and the positive integer number k is the level of the Kac-Moody algebra(see Refs.30,31for details).The primary?elds of the model can be labelled by the highest weight representations of horizontal Lie algebra g.In the basis of fundamental weightsωi of g these HW representations are expressed byλ=Σr i=1λiωi whereλi’s are positive integers(Dynkin labels)and r is the rank of g. Imposing unitarity condition,restricts the numbre of HW representations which appear in a theory at a given level.These representations which are usually called integrable representations satisfy the relation2

(see e.g.Ref.32).The character of an integrable HW representation transforms under the action of the generators of the modular group S:(τ→?1/τ)and T:(τ→τ+1)as χλ(?1/τ)=C {λ′∈B h} {ω∈W(G)}ε(ω)e(2πi

(k+ˇh)r/2 voll.cell of Q?

|Γ| h1,h2∈Γ[h1,h2]=0(h1,h2),(2.4)

where|Γ|is the order ofΓ.ForΓ=Z N which is the interesting group for us,eq.(2.4) reduces to:

Z(G/Z N)=

α=1(1,hα),(2.6)

and acting properly on it by the generators of the modular group,S and T,one can obtain the full partition function Z(G/Z N).9In Ref.35the following formula is drived for Z(G/Z N)when Z N is a subgroup of the centre of G and N is prime:

Z(G/Z N)= N α=1 TαS+1 Z1(G/Z N) ?Z(G).(2.7)

Concerning the method mentioned above we make two crucial observations.First,the orbifold method can be similarly applied to the case where G is not the covering group, and even when Z N is not a subgroup of the centre of G,but is a symmetry of the classical theory.However,in those cases in order for the modding to be meaningful,i.e.,the sum of the terms in the bracket of eq.(2.7)and therefor the whole expression be modular invariant with real coe?cients,the following relation must be satis?ed

T N SZ1(G/Z N)=SZ1(G/Z N).(2.8) Secondly,for the case when N is not prime,the expression(2.7)does not completely describe an orbifold partition function and in order to generate the full partition function Z(G/Z N),some additional terms must be included into the bracket.But then,extra constraints beyond(2.8)have to be satis?ed for modular ivariance(see eq.(A.3)).We call the appropriate moddings for a given theory which satisfy these constraints,allowed moddings.We have collected in Appendix A,a list of formulas for the cases of interest to us.

Our strategy in?nding the nondiagonal WZW theories with the a?ne symmetry algegra?g k is as follows.First we start with a diagonal theory and mod it out by some group Z N.The untwisted part of the partition function Z1,is realized by representing the action of Z N on the characters of HW representations in the left?moving sector as

p·χλ(τ)=e2πi

combinations of characters that have been found in the process of the previous moddings, a new modular invariant partition function will be found.See for example,the case of modding D8by Z9in su(3)models on page15.

Case III)Some of the terms in the bracket have negative coe?cients but after sub-tracting from it some known physical partitions or/and some modular invariant combina-tions of characters,at most a new modular invariant combination will be obtained,which is not a partition function.See for example,the case of modding D24by Z2in su(3) models on page16.

3.Applications

In this section we apply our method to su(n)WZW models with n=2,3,4,5,6,and as an example of a nonsimply-laced a?ne Lie algebra to g2WZW models.

3.1. su(2)WZW models

For su(2)models besides the usual diagonal series A h= {λ∈B h}|χλ|2at each level, where

B h= λ=mω 1≤m

is the fundamental domain andωis the fundamental weight of su(2),and a nondiago-nal D series at even levels,three exceptional modular invariant partition functions have been found at levels k=10,16,28;and it has been shown that this set completes the classi?cation of su(2)WZW models.10In what follows we will review the results of Ref. 29,where it was shown that the exceptional partition functions can be obtained by our orbifold method,and present some further calculations.The action of the Z N group on the characters of the left?moving HW representations of su(2)is de?ned due to eq.(2.9) by

p·χm=e2πi

|W| {λ∈W B h;m odd}|χλ|2,(3.4)

using the identityχω(λ)=?(ω)χλ,where W B h is the Weyl re?ection of the fundamental domain B h.Then we rewrite(3.4)in the following form

Z1(A h/Z2)=

h Q consists of the following HW

representations:

2h 1

h λ,ω′(λ′)?ω′′(λ′′)

χλ′ˉχλ′′.(3.7)

The sum overλis easily done using eq.(3.6),and it appears that the sum over one of the two Weyl groups can be factored out and give an overall factor equal to the order of Weyl group.Finally the sum over the other Weyl group must be done.Substituting(3.7)in eq.

(2.7),we easily do the sum and?nd a nondiagonal partition function at every even level given by

D h≡Z(A h/Z2)=h?1

m odd=1 χm 2+h/2?2 m odd=1(χmˉχh?m+c.c.)+2 χh/2 2(3.8a)

for h=2mod4,and

D h≡Z(A h/Z2)=h?1

m odd=1 χm 2+h/2?2 m even=2(χmˉχh?m+c.c.)+ χh/2 2(3.8b)

for h=0mod4.these are exactly the D series given in Ref.10.

E-Series

One expects to?nd possibly,exceptional partition functions at levels k=4,10,28 according to the following conformal embeddings:20

su(2)k=4? su(3)k=1, su(2)k=10? (B2)k=1, su(2)k=28? (g2)k=1.(3.9)

1.At level k=4(h=6),we start with A6and mod it out by Z3,and check that the eq.(

2.8)is satis?ed;then we do the sum in the bracket of eq.(2.7)with N=3,and ?nally get

α=1

TαSZ1+Z1 =4A6?D6.(3.10)

We continue modding by allowed Z N’s up to N=48,but nothing more than A6and D6 appears.For example,in the case N=6we obtain

α=1Tα+

α=1TαST2+2 α=1TαST3 SZ1+Z1 =4A6+D6.(3.11)

Then we start with D6,mod it out by allowed moddings but also nothing more is found. For example,in modding by Z3we?nd

α=1

TαSZ1+Z1 =2D6.(3.12) This is not surprising,since it can easily be seen that D6given by

D6= χ1+χ5 2+2 χ3 2,(3.13)

exactly corresponding to conformal embedding su(2)k=4? su(3)k=1.

2.At level k=10(h=12),we start with A12and mod it out by Z6which is allowed. After doing the sum in the bracket of eq.(A.5),we encounter the case I of Subsec. 2.2., which after subtracting the known partition functions A12and D12each with mutiplicity 2,we obtain the exceptional partition function E6,which in our notation is described by E12

α=1Tα+

α=1TαST2+2 α=1TαST3 SZ1+Z1 =2A12+2D12+2E12,(3.14)

where

E12= χ1+χ7 2+ χ4+χ8 2+ χ5+χ11 2.(3.15) In Ref.29,E12was obtained by modding A12or D12by Z3,but there,we encounter the case II of Subsec. 2.2.We also get E12in modding D12by Z6;the result is exactly the same as eq.(3.14).

3.At level k=16(h=18),we start with A18and mod it out by Z6which is allowed. After doing the sum in the bracket of eq.(A.5),we encounter the case I of Subsec. 2.2., which after subtraction A18of mutiplicity3,the exceptional partition function E18is obtained:

α=1Tα+

α=1TαST2+2 α=1TαST3 SZ1+Z1 =3A18+3E18,(3.16)

where

E18= χ1+χ17 2+ χ5+χ13 2+ χ7+χ11 2+ χ9 2+ χ9(

We also?nd E18in modding D18by the allowed modding like Z3:

α=1

TαSZ1+Z1 =D18+2E18.(3.18)

4.At level k=28(h=30),we start with D30and mod it out by Z3which is allowed. After doing the sum in the bracket of eq.(2.7),we encounter the case I of Subsec. 2.2, which after subtraction D12of mutiplicity2,leads to the exceptional partition function E30:

α=1

TαSZ1+Z1 =2D30+E30,(3.19)

where

E30= χ1+χ11+χ19+χ29 2+ χ7+χ13+χ17+χ23 2.(3.20) So in this subsection we have generated,by orbifold method,not only the partition functions which correspond to a conformal embedding like E12and E30,22but also the one which follows from a nontrivial automorphism of the fusion rules of the extended algebra i.e.E18.24It is interesting to notice that all the nondiagonal partition functions of su(2) are obtained from moddings by Z N’s,with N a divisor of2h and h=k+2.

3.2. su(3)WZW models

For su(3)models besides the usual diagonal series A h= {λ∈B h}|χλ|2at each level,where

B h= λ 2≤2 i=1m i

andλ=Σ2i=1m iωi,and a nondiagonal D h series at each level;four exceptional modular invariant partition functions have been found at levels k=5,9,21.12,13Recently,it was shown that this set completes the classi?cation of su(3)WZW models.11In what follows we obtain all of these by the orbifold method.We de?ne the action of the Z N group on the characters of the left?moving HW representations of su(3)by

p·χ(m

1,m2)

=e2πi

The partition function Z(G/Z3)can be calculated using eq.(2.7)with N=3.Following the same recipe mentioned in section3.1,?rst we write the untwisted part(3.23)in the following form

Z1(A h/Z3)=

= λ=mω1+m′α1 1≤m≤3h+2;?1≤m′≤h?2 .(3.25)

h Q

so that,just as in eq.(3.5)we can rewrite eq.(3.24)in the form,

Z1(A h/Z3)=

*There is a minus sign error in Ref.17.De?ningσ(λ)the same as in our case,the two terms in the exponential of eq.(6.4)of Ref.17must both have negative signs,in order for Mλ,λ′to commute with the operator T.

Then,we start with D6,mod it out by allowed moddings but also nothing more is found. For example,in modding by Z6we?nd

α=1Tα+

α=1TαST2+2 α=1TαST3 SZ1+Z1 =9

(χ1,3+χ4,3)+(χ2,3+χ6,1)

2 5D8?A c.c.8+E8 (3.35)

α=1

Tα+ST2 SZ1+Z1 =1

3.At level k=9(h=12),we start with D12and mod it out by Z9and do the sum according to the bracket of eq.(A.7)with N=9,?nally we encounter the case II of Subsec. 2.2.,which after subtraction D12of multiplicity12,we obtain an exceptional partition function,denoted by E(1)12with an overal multiplicity?3:

α=1

Tα+ST3+ST6 SZ1+Z1 =12D12?3E(1)12,(3.37) where

E(1)12= χ1,1+χ1,10+χ10,1+χ2,5+χ5,2+χ5,5 2+2 χ3,3+χ3,6+χ6,3 2.(3.38) This partition function corresponds to conformal embedding su(3)k=9? (e6)k=1.13In modding D12by Z2,after doing the sum in the bracket of eq.(2.7)with N=2,one encounters case II,however in this case the trace of another modular invariant can easily be seen.Actually subtracting D12,its charge conjugation counterpart D c.c12and E(1)12with multiplicities5/2,?1/2,and1/2respectively,we obtain another exceptional partition function which we denote by E(2)12

α=1

TαSZ1+Z1 =1

(χ2,2+χ2,8+χ8,2)+c.c. ,(3.40)

which does not correspond with a conformal embedding;and was found using a nontrivial automorphism of the fusion rules of the extended algebra.24We continued modding up to Z72,but no other exceptional theory appears at this level.

4.At level k=21(h=24),starting with D24and modding by Z2leads to the case III of Subsec. 2.2.,which after subtracting D24of multiplicity2,yields a modular invariant combination with some of its coe?cients negative integers,which we call M24

α=1

TαSZ1+Z1 =2D24+M24,(3.41) with

M24= χ[1,1]+χ[2,11] 2+ χ[5,5]+χ[7,7] 2+ χ[1,7]+χ[8,5] 2+ χ[7,1]+χ[5,8] 2 + χ[3,3]+χ[6,9] 2+ χ[1,4]?χ[4,7] 2+ χ[4,1]?χ[7,4] 2+2 χ[3,9] 2

+2 χ[9,3] 2? χ[2,2]+χ[2,8]+χ[8,2]?χ[4,10] 2+3 χ[4,4]?χ[8,8] 2 ,

(3.42)

where

χ[m

1,m2]

≡χ(m

,m2)

+χ(m

,h?m1?m2)

+χ(h?m

?m2,m1)

.(3.43)

Modding by Z3gives D12itself,and modding by Z4,Z8,Z9,Z18,Z24,Z36give rise to three extra modular invariant combinations,which we do not mention here their explicit form.Finally,modding by Z72and doing the sum in the bracket of eq.(A.16),we encounter the case II of Subsec. 2.2.,which after Subtracting D24and M24and their charge conjugation counterparts D c.c24,M c.c24with multiplicities12and3respectively,we are left with an exceptional partition function which is denoted by E24:

α=1Tα+

α=1TαST2+8 α=1TαST3+9 α=1TαST4+2 α=1TαST6

α=1TαST8+8 α=1TαST9+ST12+8 α=1TαST15+2 α=1TαST18

+ST24+

α=1TαST30+ST36+ST48+ST60 SZ1+Z1

=12D24+12D c.c24+3M24+3M c.c24+9E24,(3.44) with

E24= χ[1,1]+χ[5,5]+χ[2,11]+χ[7,7] 2+ χ[1,7]+χ[7,1]+χ[5,8]+χ[8,5] 2.(3.45)

This theory corresponds to conformal embedding su(3)k=21? (e7)k=1.13 So with our method we reproduce not only exceptional partition functions correspond-ing to a certain conformal embedding like E8,E(1)12,and E24,but also the one which can not be obtained by a conformal embedding i.e.E(2)12.Note that at each level the allowed modding Z N has N a divisor of3h,where h=k+3.

3.3. su(4)WZW models

In addition to the diagonal series A h= {λ∈B h}|χλ|2,where

B h= λ 3≤3 i=1m i

andλ=Σ3i=1m iωi,there exist two D series corresponding to the two subgroups of the centre of SU(4).Furthermore,up to now three exceptional partition functions have been found in levels k=4,6,8.In the following we obtain all of these by orbifold approach.We de?ne the action of the Z N on the characters of left?moving HW representations of su(4) due to eq.(2.9)by

p·χ(m

1,m2,m3)

=e2πi

and p is the generator of Z N .Thus,the untwisted part of a partition function,consists of left ?moving HW representations λwhich satisfy:?λ=20mod N .

D -Series

We follow the same recipe of calculation that was described in Subsec. 3.2.,but without going into the details,and ?nd the general form of D h series at each level.Starting with A h ,?rst we mod it out by a subgroup Z 2of the centre and obtain at every level a nondiagonal partition function which we denote by D (2)h ,

D (2)h ≡Z (A h /Z 2)= {λ|Σ3i =1im i =0mod 2}|χλ|2+ {λ|Σ3i =1im i =kmod 2}

χλˉχμ(λ),(3.48)where μ(m 1,m 2,m 3)=(m 3,h ?Σ3i =1m i ,m 1).Then we mod out the A h series by Z 4according to eq.(A.4)and ?nd at every even level a nondiagonal partition function D (4)h ,which has the form

D (4)h ≡Z (A h /Z 4)= {λ|Σ3i =1im i =2mod 4}|χλ|2+

{λ|Σ3i =1

im i =2+k 2mod 4}χλˉχσ3(λ),

(3.49)

where σ(m 1,m 2,m 3)=(m 2,m 3,h ?Σ3i =1m i ).These results agree with the ones obtained in Ref.17modulo the comment in the footnote of page 13.

E -Series

It is expected that there are nondiagonal partition functions at levels k =2,4,6,8,due to the following conformal embeddings:20

su (4)k =2? su (6)k =1,

su (4)k =4? (B 7)k =1

su (4)k =6? su (10)k =1, (D 3)k =8? (D 10)k =1.(3.50)1.At level k =2(h =6),we have only D (2)6,and D (4)6=A c.c.6.First,we start with

A 6and mod it out by all allowed Z N ’s up to N =24.Nothing other than A 6and D (2)6and their charge cojugation counterparts is found.For example,in the cases N =3,6,8

we obtain

α=1T α

SZ 1+Z 1 =4A 6?2D (2)6(3.51)

6 α=1T α+3 α=1T αST 2+

2 α=1T αST

3 SZ 1+Z 1 =4A 6+4D (2)6

(3.52)

8 α=1T α

+2 α=1T αST 2+ST 4 SZ 1+Z 1 =2A 6+2A c.c.6+D (2)6,(3.53)

respectively.Then we start with D(2)6and do the allowed moddings.Again,nothing more is found.For example,in modding by Z3we?nd

α=1

TαSZ1+Z1 =2D(2)6.(3.54) This is not surprising,since it can easily be seen that D(2)6

D(2)6= χ(1,1,1)+χ(1,3,1) 2+ χ(1,1,3)+χ(3,1,1) 2+2 χ(1,2,1) 2+2 χ(2,1,2) 2,(3.55) exactly corresponding to conformal embedding su(4)k=2? su(6)k=1.

2.At level k=4(h=8),there exist two D(2)8and D(4)8partition functions.We choose to start with D(2)8.Modding by Z2gives D(2)8itself,and modding by Z4and Z8 give a combination of D(2)8and D(4)8:

α=1

Tα+ST2 SZ1+Z1 =2D(2)8+2D(4)8(3.56)

α=1Tα+

α=1TαST2+ST4 SZ1+Z1 =4D(2)8+4D(4)8.(3.57)

The next modding which satis?es the condition(A.3)is Z16.After doing the sum in the bracket of eq.(A.10)we encounter the case I of Subsec.2.2.,which after subtracting the D(2)8and D(4)8each with multiplicity4,an exceptional partition function is found which we denote by E8:

α=1Tα+

α=1TαST2+ST4+ST8+ST12 SZ1+Z1 =4D(2)8+4D(4)8+4E8,(3.58)

where

E8= χ(1,1,1)+χ(1,5,1)+χ(1,2,3)+χ(3,2,1) 2

+ χ(1,1,5)+χ(5,1,1)+χ(2,1,2)+χ(2,3,2) 2+4 χ(2,2,2) 2.(3.59)

It can easily be shown that this partition function corresponds to conformal embedding su(4)k=4? (B7)k=1.The above exceptional partition function was obtained in the context of?xed-point resolution of Ref.19.We then repeat the above moddings starting with D(4)8 and?nd exactly the same results as with D(2)8.

3.At level k=6(h=10),there exist D(2)10and D(4)10.Starting with D(2)10the following results are obtained.Modding by Z2and Z4gives D(2)10itself,but modding by Z8and doing the sum in the bracket of eq.(A.6)we encounter the case III of Subsec.2.2.,which

after subtraction D(2)10of multiplicity6leads to a modular invariant combination,which we call M10

α=1Tα+

α=1TαST2+ST4 SZ1+Z1 =6D10?M10(3.60)

where

M10= (χ(1,1,1)+χ(1,7,1))?(χ(2,1,6)+χ(6,1,2))?(χ(3,1,3)+χ(3,3,3)) 2

+ (χ(1,1,7)+χ(7,1,1))?(χ(1,2,1)+χ(1,6,1))?(χ(1,3,3)+χ(3,3,1)) 2

+3 χ(2,2,4)+χ(4,2,2) 2+3 χ(2,2,4)+χ(4,2,2) 2.(3.61) In modding by Z5after doing the sum in the bracket of eq.(2.7)with N=5,we encounter

the case II of Subsec.2.2.,which by subtracting D(2)10,and its charge conjugation D(2)c.c.

10,

and M10by multiplicities?3/5,4/5,and8/5respectively,an exceptional partition function is found which we call E10

α=1

TαSZ1+Z1 =1

5 2D(2)10+4D(2)c.c.10+3M10+2E10

(3.64)

4.At level k =8(h =12),there exist D (2)12and D (4)12.We choose D (4)12and obtain

the following results.Modding by Z 2and Z 4gives D (4)12itself,but in modding by Z 8we

ecounter the case II of Subsec. 2.2.,which after subtraction D (4)12of multiplicity 12,an

exceptional partition function is found,which we call E (1)12

8 α=1T α

+2 α=1T αST 2+ST 4 SZ 1+Z 1 =12D (4)12?2E (1)12,(3.65)

where *

E (1)12= χ(1,1,1)+χ(1,1,9)+χ(1,9,1)+χ(9,1,1)+χ(2,3,2)+χ(2,5,2)+χ(3,2,5)+χ(5,2,3)

2+ χ(1,3,1)+χ(1,7,1)+χ(3,1,7)+χ(7,1,3)+χ(1,4,3)+χ(3,4,1)+χ(4,1,4)+χ(4,3,4) 2+2 χ(2,2,4)+χ(2,4,4)+χ(4,2,2)+χ(4,4,2) 2.

(3.66)It can easily be seen that E (1)12just corresponds to conformal embedding (D 3)k =8?

(D 10)k =1with the following branching rules:ch 1=χ(1,1,1)+χ(1,1,9)+χ(1,9,1)+χ(9,1,1)+χ(2,3,2)+χ(2,5,2)+χ(3,2,5)+χ(5,2,3)

ch 2=χ(1,3,1)+χ(1,7,1)+χ(3,1,7)+χ(7,1,3)+χ(1,4,3)+χ(3,4,1)+χ(4,1,4)+χ(4,3,4)

ch 3=ch 4=χ(2,2,4)+χ(2,4,4)+χ(4,2,2)+χ(4,4,2),(3.67)

where chi i ’s are the characters of the integrable representations of ( (D 10)k =1and

χ(m 1,m 2,m 3)’s are those of SU (4)k =8.

Then,modding by Z 3and doing the sum in eq.(2.7)with N =3leads to the case I of

Subsec.2.2.,which after subtraction D 12,D c.c.12,and E (1)12each of multiplicity 1/3another

exceptional partition function is found,which we call E (2)12

4 α=1T α+ST 2

SZ 1+Z 1 =1χ(3,3,3)+(χ(1,5,1)+χ(5,1,5))

*We are not aware of the explicit form of this partition function in the literature.

This exceptional partition function which doesn’t correspond to a conformal emedding, was recently found by a computational method which essentially looks for the eigenvectors of matrix S(the generator of the modular group)with eigenvalues equal to one,25,28and can be shown to be a consequence of an automorphism of the fusion rules of the extended algebra.28We continue the modding by allowed groups up to Z96,but no new partition function is found.For example in modding by Z24according to eq.(A.12),we get

α=1Tα+

α=1TαST2+8 α=1TαST3+3 α=1TαST4+2 α=1TαST6

α=1TαST8+ST12 SZ1+Z1 =4 D12+D c.c12+E(1)12+4E(2)12 .(3.70)

So again,as in the case of su(3)models,with the orbifold method not only exceptional partition functions which correspond to conformal embeddings like E8,E10,and E(1)12are found,but also the a partition function which does not corresponds to a conformal em-bedding i.e.E(2)12,is generated.

In this subsection we have been able to obtain explicitly all the exceptional partition functions which correspond to a conformal embedding and moreover the one(E(2)12)which follows from an automorphism of the fusion rules of the extended algebra.Note that at each level the allowed modding Z N has N a divisor of4h,where h=k+4.

3.4. su(5)WZW models

For su(5)models besides the usual diagonal series A h= {λ∈B h}|χλ|2,at each level, where

B h= λ 4≤4 i=1m i

andλ=Σ4i=1m iωi,and one nondiagonal D series at each level,up to now some exceptional partition functions have been found which we are going to obtain by our method.We de?ne the action of the Z N on the characters of HW representations of su(5)due to eq.(2.9)by

p·χ(m

1,m2,m3,m4)

=e2πi

We start with A h series,following the same recipe mentioned in Subsec.3.2.,mod it out by Z5using the eq.(2.7)with N=5.Doing the sum in the bracket,we obtain the general form of D h series:

D h≡Z(A h/Z5)= {λ|Σ4i=1im i=0mod5}|χλ|2+ {λ|Σ4i=1im i=3kmod5}χλˉχσ(λ)

+ {λ|Σ4i=1im i=kmod5}χλˉχσ2(λ)+ {λ|Σ4i=1im i=4kmod5}χλˉχσ3(λ)

+ {λ|Σ4i=1im i=2kmod5}χλˉχσ4(λ)(3.73)

whereσ(m1,m2,m3,m4)=(m2,m3,m4,h?Σ4i=1m i).These results agree with the ones obtained in Ref.17,modulo the comment mentioned in the footnote of page13.

E-Series

One expects to?nd,possibly,the exceptional series at levels k=3,5,7according to the following conformal embeddings:20

su(5)k=3? su(10)k=1, su(5)k=5? (D12))k=1, su(5)k=7? su(15)k=1.(3.74)

We will limit ourselves only to the?rst two cases in this work.

1.At level k=3(h=8),we start with A8,doing the allowed moddings and obtain the following results.Modding by Z2gives A8itself,but modding by Z4we encounter the case II of section

2.2.,which after subtracting A8and D c.c8with multiplicities5and?1 respectively,an exceptional partition function is found which we call E8

Tα+ST2 SZ1+Z1 =5A8?D c.c.8+E8(3.75)

α=1

with

E8= χ(1,1,1,1)+χ(1,2,2,1) 2+ χ(1,1,1,4)+χ(2,2,1,2) 2

χ(1,1,2,1)+χ(1,3,1,2) 2+ χ(1,1,3,1)+χ(3,1,1,2) 2

χ(1,1,4,1)+χ(2,1,2,1) 2+ χ(1,2,1,3)+χ(3,1,2,1) 2

χ(1,2,1,2)+χ(1,4,1,1) 2+ χ(1,3,1,1)+χ(2,1,1,3) 2

χ(1,2,1,1)+χ(2,1,3,1) 2+ χ(2,1,2,2)+χ(4,1,1,1) 2.(3.76)

We have checked that E8exactly corresponds to conformal embedding su(5)k=3? su(10)k=1.We carried out all the allowed moddings up to Z40and no other exceptional partition function was found.Then,we start with D8and mod it out by allowed moddings up to Z40,but again no more exceptional partition function is found.For example,modding

by Z16after doing the sum in the bracket of eq.(A.10),we encounter the case I of Subsec.

2.2.,which after subtraction D8and A c.c8each of multiplicity4,results in E c.c.

α=1Tα+

α=1TαST2+ST4+ST8+ST12 SZ1+Z1 =4D8+4A c.c.8+2E c.c.8.(3.77)

2.At level k=5(h=10),we start with D10which has the form

D10= χ1 2+ χ2 2+ χ3 2+ χ4 2+ χ5 2+5 χ6 2,(3.78) here following the Ref.19,we have used the following abbreviations

χ1=χ(1,1,1,1)+χ(1,1,1,6)+χ(1,1,6,1)+χ(1,1,6,1)+χ(1,6,1,1)

χ2=χ(1,2,1,3)+χ(2,1,3,3)+χ(1,3,3,1)+χ(3,3,1,2)+χ(3,1,2,1)

χ3=χ(1,1,2,4)+χ(1,2,4,2)+χ(2,4,2,1)+χ(4,2,1,1)+χ(2,1,1,2)

χ4=χ(1,1,3,2)+χ(1,3,2,3)+χ(3,2,3,1)+χ(2,3,1,1)+χ(3,1,1,3)

χ5=χ(1,2,2,1)+χ(2,2,1,4)+χ(2,1,4,1)+χ(1,4,1,2)+χ(4,1,2,2)

χ6=χ(2,2,2,2),(3.79) and do the allowed moddings.Then the following results are obtained.Modding by Z2 gives D10itself,but modding by Z25according to eq.(A.13),after doing the sum in the bracket,we encounter the case I which after subtraction D10of multiplicity10,results in

a modular invariant partition function,which we denote by E(1)10:

α=1

Tα+ST5+ST10+ST15+ST20 SZ1+Z1 =10D10+10E(1)10,(3.80)

where

E(1)10= χ1+χ2 2+2 χ3 2+10 χ6 2.(3.81) Next,we mod out D10by Z4and encounter case II of Subsec.2.2.,which after sub-traction D10and E(1)10of mutiplicities7/2and?3/2respetively,leads to another modular invariant partition function denoted by E(2)10:

α=1

Tα+ST2 SZ1+Z1 =1

χ6+c.c. .(3.83) What is interesting here is that starting with E(1)10and modding by Z4gives rise to the case II,which after subtracting E(1)10from the sum in the bracket of eq.(A.4)with

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人教版英语选修六Unit5 the power of nature(An exciting Job)

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an exciting job 翻译

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人教版选修Unit 5 The power of nature阅读课教学设计 Learning aims Knowledge aims: Master the useful words and expressions related to the text. Ability aims: Learn about some disasters that are caused by natural forces, how people feel in dangerous situations. Emotion aims: Learn the ways in which humans protect themselves from natural disasters. Step1 Leading-in 1.Where would you like to spend your holiday? 2.What about a volcano? 3.What about doing such dangerous work as part of your job ? https://www.doczj.com/doc/9014675230.html, every part of a volcano. Ash cloud/volcanic ash/ Crater/ Lava /Magma chamber 【设计意图】通过图片激发学生兴趣，引出本单元的话题，要求学生通过讨论, 了解火山基本信息，引出火山文化背景，为后面的阅读做铺垫。利用头脑风暴法收集学生对课文内容的预测并板书下来。文内容进行预测，培养学生预测阅读内容的能力。同时通过预测激起进一步探究 Step2. Skimming What’s the main topic of the article?

新目标英语七年级下课文原文unit1-6

Unit1 SectionA 1a Canada, France ,Japan ,the United States ,Australia, Singapore ,the United Kingdom,China 1b Boy1:Where is you pen pal from,Mike? Boy2:He's from Canada. Boy1:Really?My pen pal's from Australia.How about you,Lily?Where's your pen pal from? Girl1:She's from Japan.Where is Tony's pen pal from? Gril2:I think she's from Singapore. 2b 2c Conversation1 A: Where's you pen pal from, John? B: He's from Japan, A:Oh,really?Where does he live? B: Tokyo. Conversation2 A: Where's your pen pal from, Jodie? B: She's from France. A: Oh, she lives in Pairs. Conversation3 A: Andrew, where's your pen pal from? B: She's from France. A: Uh-huh. Where does she live? B: Oh, She lives in Paris.