ADE spectra in conformal field theory

13 May 1999

Physics Letters B 454 Ž1999. 59–69

ADE spectra in conformal field theory A.G. Bytsko

a,1

, A. Fring

b,2

a

b

StekloÕ Mathematics Institute, Fontanka 27, St. Petersburg 191011, Russia Institut fur ¨ Theoretische Physik, Freie UniÕersitat ¨ Berlin, Arnimallee 14, D-14195 Berlin, Germany Received 30 December 1998; received in revised form 10 February 1999 Editor: P.V. Landshoff

Abstract We demonstrate that certain Virasoro characters Žand their linear combinations. in minimal and non-minimal conformal models which admit factorized forms are manifestly related to the ADE series. This permits to extract quasi-particle spectra of a Lie algebraic nature which resembles the features of Toda field theory. These spectra possibly admit a construction in terms of the Wn-generators. In the course of our analysis we establish interrelations between the factorized characters related to the parafermionic models, the compactified boson and the minimal models. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction It is well known, that a large class of off-critical integrable models is related to affine Toda field theories w1x or RSOS-statistical models w2x, which possess a rich underlying Lie algebraic structure. Since these models can be regarded as perturbed conformal field theories, it is suggestive to recover the underlying Lie algebraic structure also in the conformal limit. Of primary interest is to identify the conformal counterparts of the off-critical particle spectrum. One way to achieve this is to analyze the quasi-particle spectrum, which results from certain expressions of the Virasoro characters x Ž q . or their linear combinations. Hitherto this analysis was mainly performed w3x for formulae of the form

x Ž q. sq

const

ql

t

A lqBP l

Ý Ž q. Ž q. l ... l l 1

.

Ž 1.

r

Here r is the rank of the related Lie algebra g, the matrix A coincides with the inverse of the Cartan matrix, B l Ž1 y q k ., and there may be certain restrictions on the characterizes the super-selection sector, Ž q . l :s Ł ks1 summation over l. Following the prescription of w3x one can always obtain a quasi-particle spectrum once a

1 2

E-mail: [email protected] E-mail: [email protected]

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 3 0 0 - 7

A.G. Bytsko, A. Fringr Physics Letters B 454 (1999) 59–69

60

character admits a representation in the form of Eq. Ž1.. It should be noted that such spectra can not be obtained form the standard form of the Virasoro characters Ž10.. In the following we will demonstrate that one also recovers Lie algebraic structures in certain Virasoro characters or their linear combinations which admit the factorized form q const

 14

y 1

y

q

 x 1 ; . . . ; x N 4 y  xX1 ; . . . ; xXM 4 y ,

Ž 2.

where we adopt the notations of w4x `

n

xqk y  x4 " . ,  x 1 ; . . . ; x n 4 "y :s Ł  x a 4 "y . y :s Ł Ž 1 " q

ks0

as1

In many cases Žsee w4,5x for details. expressions of the type Ž2. can be rewritten in the form Ž1., but now A is entirely absent or, at most, is a diagonal matrix. There are no restrictions on the summation over l, and we allow terms of the form Ž q y . l in the denominator Žwhich may be regarded as an anionic feature w5x.. Unlike the conventional form for the Virasoro characters Ž10., formulae Ž1. and Ž2. allow to extract the leading order behaviour in the limit q ™ 1y by means of a saddle point analysis, see e.g. w3,5x. For a slightly generalized version of Ž1., in the sense that all Ž q . l are replaced by Ž q y . l , this analysis leads to r

z iy s Ł Ž 1 y z j .

Ž A i jqA ji .

,

ceff s

js1

r

6 yp

2

Ý LŽ zi . .

Ž 3.

is1

This means solving the former set of equations for the unknown quantities z i , we may compute the effective central charge thereafter by means of the latter equation in terms of Rogers dilogarithm LŽ x .. Recall that the effective central charge is defined as ceff s c y 24 hX , where hX is the lowest conformal weight occurring in the model. There exist inequivalent solutions to Eqs. Ž3. leading to the same effective central charge corresponding either to the form Ž1. or Ž2.. When treating these equations as formal series, such computations give a first hint on possible candidates for characters. Alternatively, with regard to factorization, we can exploit the essential fact that the blocks  x 4y" are closely related to the so-called quantum dilogarithm and we can easily compute their contributions to the effective q central charge. As explained in Ref. w4x, each block Ž x 4yy. " 1 and Ž  x 4 y . " 1 in expressions of type Ž2. contributes D ceff s .

1 y

,

and

Dc eff s "

1 2y

,

Ž 4.

respectively. In the course of our argument, i.e. when we consider the difference of the Virasoro characters, we will also need the notion of the secondary effective central charge c˜ s 1 y 24 hXX ,

Ž 5.

XX

where h is the next to lowest conformal weight occurring in the model.

2. ADE structure Let g be a Lie algebra of rank r and h be its Coxeter number. We define the following function related to g

J g Ž x ,q . s

q const

,

y

 x1 ; . . . ; x r 4 h 2

q1

Ž 6.

A.G. Bytsko, A. Fringr Physics Letters B 454 (1999) 59–69

61

Table 1 Effective central charges for minimal affine Toda field theories g

AŽ1. n

c eff

2n nq3

DnŽ1.

E6Ž1.

E7Ž1.

E8Ž1.

AŽ2. 2n

1

6 7

7 10

1 2

2n 2 n q3

with x obeying the condition x a q x rq1ya s hr2 q 1 ,

a s 1, . . . ,r ,

h 4

Ž 7.

1 2

and for odd r we put x s q . Our aim is to find conformal models such that their characters or possibly linear combinations coincide with Ž6. for appropriately chosen q const and x. Such conformal models have quasi-particle spectra, generated by Ž6. for the related sectors, with the number of different particle species equal to the rank r. The question arises for which conformal models can we expect Ž6. to be a character? Exploiting Ž4., we readily find the corresponding effective central charge rq 1 2

ceff Ž g . s

2r2

2r s hq2

dim g q r

.

Ž 8.

On the other hand, the analysis of the ultra-violet limit of the thermodynamic Bethe ansatz w6x for the ADE related minimal scattering matrices of affine Toda field theory leads to the following effective central charges Žsee Table 1.. Thus, we see that, upon substitution of the related Lie algebraic quantities 3 of the simply laced algebras Žsee e.g. w7x., Eq. Ž8. recovers all the effective central charges in Table 1. Furthermore it turns out that for g from this table corresponding to minimal models or c s 1 models we are always able to identify several J g Ž x,q . with single Virasoro characters or specific linear combinations of them. In addition, there exist characters which exhibit even stronger Lie algebraic features. They are given by Ž6. with the values of x a chosen as follows Žwhich is a particular case of Ž7.. 2 x a y 1 s ea ,

a s 1, . . . ,r ,

Ž 9.

where  e a4 stands for the set of the exponents of the Lie algebra g. We denote this particular character as J g Ž q .. 2.1. Minimal models The minimal models w8x are parameterized by a pair Ž s,t . of co-prime positive integers and the corresponding 2 Ž ntyms . 2y Ž syt . 2 s,t central charge is cŽ s,t . s 1 y 6Ž syt . . Labeling the highest weights as h n, , with the ms st

4 st

restrictions 1 F n F s y 1 and 1 F m F t y 1, the usual form of the characters of irreducible highest weight representations reads w9x ` s,t xns,t , m Ž q . s hn , m

2

Ý

q st k Ž q k Ž n tym s . y q k Ž n tqm s . qn m . .

Ž 10 .

ksy` s, t

cŽ s , t . 24

y

Here we abbreviated the ubiquitous factor hn,s,tm :s q h n, m y r  1 4 1 by an analogy with the eta-function. The Žsecondary. effective central charge is easy to find Žsee e.g. w4x. 6 24 ceff Ž s,t . s 1 y , c˜ Ž s,t . s 1 y . Ž 11 . st st 3

For a twisted affine Lie algebra of type X NŽ k . one introduces h – the Coxeter number and hŽ k . s kh. We should use h in Ž8. for AŽ2. 2 n.

A.G. Bytsko, A. Fringr Physics Letters B 454 (1999) 59–69

62

Thus, the values of ceff which are less than 1 can be matched as follows Ž1. ceff Ž AŽ1. 1 . s ceff Ž E8 . s ceff Ž 3,4 . ,

Ž 12 .

ceff Ž AŽ1. 2 . s ceff Ž 5,6 . s ceff Ž 3,10 . s ceff Ž 2,15 . ,

Ž 13 .

ceff Ž E6Ž1. . s ceff Ž 6,7 . s ceff Ž 3,14 . s ceff Ž 2,21 . ,

Ž 14 .

ceff Ž E7Ž1. . s ceff Ž 4,5 . ,

Ž 15 .

ceff Ž AŽ2. 2 n . s ceff Ž 2,2 n q 3 . .

Ž 16 .

We see, that the matching is, in general, not unique. For Ž16. it depends on n – the first non-unique representations occur for n s 6 and n s 9 and coincide with Ž13. and Ž14., respectively. Therefore we might Ž2. Ž1. Ž2. have for instance relations between AŽ1. 2 ; A12 and E6 ; A18 . Some of these apparent ambiguities are easily explained as the consequence of a symmetry property of the characters. For instance we observe that Eq. Ž10. 3,10 Ž . possesses the symmetry: xaan,s,tmŽ q . s xn,s,aamt Ž q ., for instance x 2,6,5mŽ q . s x 1,2 m q . Of course Eqs. Ž12. – Ž16. are only to be understood as a first hint on a possibility for characters in the corresponding models to be of the form J g Ž x,q .. In order to make the identifications more precise, we have to resort to more stringent properties of the characters. We shall be using previously obtained results w4,10x on representation of characters of minimal models in the form Ž2.. In Ref. w10x it was proven, that for M s 0 and x i / x j for i / j in Ž2. the only possible factorizable single characters are y

xn2,nm,t Ž q . s hn2,nm,t  nm;nt ;nt y nm4 n t ,

Ž 17 .

y

y

xn3,nm,t Ž q . s hn3,nm,t  2 nt ;nm;2 nt y nm4 2 n t  2 nt y 2 nm;2 nt q 2 nm4 4 n t .

Ž 18 .

Combining now characters in a linear way, the property to be factorizable remains still exceptional. It was argued in Ref. w4x that s,t xns,t , m Ž q . " x n ,tym Ž q .

Ž 19 .

are the only combinations of characters in the same model which have a chance to acquire the form Ž2. with reasonably small N and M. The limit q ™ 1y of the upper and lower signs in Ž19. is governed by ceff and c, ˜ respectively, which can be seen form their properties with respect to the S-modular transformation w4x. The following factorizable combinations of type Ž19. where found in Ref. w4x

xn3,nm,t

Ž q.

" x 23nn,,tm

Ž q.

s hn3,nm,t

xn4,nm,t Ž q . y x 34nn,,tm Ž q . s hn4,nm,t

xn4,nm,t

Ž q.

q x 34nn,,tm

Ž q.

s hn4,nm,t

 nm;nt y nm4

½

y nt

y

nt

½ 5 ½

nt nt ; y nm;nm 2 2

 nm;nt ;nt y nm4

nt

2

2

nt y 2 nm nt q 2 nm ; 4 4

"

5

nt

,

Ž 20 .

2

y

5

y nt

y

nt

,

Ž 21 .

2

½

nt 2

y nm;

nt 2

q nm;

nt 2 y

q

5

6 n ,t 6 n ,t xn6,nm,t Ž q . y x 5n , m Ž q . s hn , m  nt ;nm;nt y nm4 n t  nt y 2 nm;nt q 2 nm4 2 n t .

,

Ž 22 .

nt

Ž 23 .

A.G. Bytsko, A. Fringr Physics Letters B 454 (1999) 59–69

63

Table 2 Representation of characters in the form of Ž6. and for differences of the type Ž23. in the form Ž24.. The replacement of the blocks  x 4y typed in bold by  x 4q yields the corresponding differences of characters y

g

sectors

 x 1; . . . ; x r 4 y

E8

3,4 x 1,1 3,4 x 1,2 3,4 x 1,3 3,4 x 1,2 4,5 x 2,1 4,5 x 2,2 5,6 5,6 x 1,2 q x 1,4 5,6 5,6 x 2,2 q x 2,4 5,6 5,6 x 1,1 y x 1,5 5,6 5,6 x 2,1 y x 2,5 6,7 6,7 x 2,1 q x4,1 6,7 6,7 x 2,2 q x4,2 6,7 6,7 x 2,3 q x4,3 6,7 6,7 x 1,1 y x 1,6 6,7 6,7 x 1,2 y x 1,5 6,7 6,7 x 1,3 y x 1,4 3,4 x 1,2 2,7 x 1,1 2,7 x 1,2 2,7 x 1,3

y 2;3;4;5;11;12;13;14416 y 1;3;5;7;9;11;13;15416 y 1;4;6;7;9;10;12;15416 14y 2 y 1;3;4;5;6;7;9410 y 1;2;3;5;7;8;9410 y 1; 32 45r2 y  12 ;245r2 y 2;8410 y 4;6410 5 1; 2 ;3;4; 92 ;64y 7 1; 32 ;2;5; 112 ;64y 7  12 ;2;3;4;5; 132 4y 7 y 2;3;4;10;11;12414 y 1;4;6;8;10;13414 y 2;5;6;8;9;12414 1;344y 2;3;4;54y 7 1;3;4;64y 7 y 1;2;5;647

A1 E7 A2

E6

G2 F4

Remarkably, as we demonstrate in Table 2, all identifications presented in Ž12. – Ž16. can be realized in terms of characters with the help of Ž17. – Ž23.. In particular, we identify J g Ž q . with the following characters Ž1.

3,4 J A 1 Ž q . s x 1,2 Ž q. ,

Ž2.

nq3 J A 2 n Ž q . s x 1,2,2nq1 Ž q. ,

Ž1.

5,6 5,6 3,10 3,10 J A 2 Ž q . s x 1,2 Ž q . q x 1,4 Ž q . s x 1,2 Ž q . q x 1,8 Ž q. , Ž1.

6,7 6,7 3,14 3,14 J E 6 Ž q . s x 2,1 Ž q . q x 2,6 Ž q . s x 1,2 Ž q . q x 1,12 Ž q. , Ž1.

4,5 J E 7 Ž q . s x 2,1 Ž q. ,

Ž1.

3,4 J E 8 Ž q . s x 1,3 Ž q. .

Since these identifications hint on the connection with massive models, i.e. affine Toda field theories, it is somewhat surprising that also the non-simply laced algebras G 2 and F4 occur in Table 2 4 . No connection is known between G 2- and F4-affine Toda models and Ž3,4. and Ž2,7. minimal models, respectively. At present it seems to be just an intriguing coincidence. It is interesting to notice that, as seen from Table 2, the differences of type Ž23. can also be of the form q const y

 x1 ; . . . ; x r 4 b

.

Ž 24 .

5,6 Ž . 6,7 Ž . 6,7 Ž . 5,6 Ž . For instance, x 1,1 q y x 1,5 q corresponds to b s 2 h q 4, x a s 2Ž e a q 1., and x 1,2 q y x 1,5 q corresponds to b s h q 2, x a s e a q 1, where h and  e a4 are the Coxeter number and exponents of A 2 and E6 , respectively.

4

3,4 Ž . 2,7Ž . We thank W. Eholzer for pointing out to us that our formulae in Ref. w5x may include J G 2 Ž q . s x 1,2 q and J F4 Ž q . s x 1,2 q as well.

A.G. Bytsko, A. Fringr Physics Letters B 454 (1999) 59–69

64

Eq. Ž6. does not exhaust all manifest Lie algebraic functions in which combinations of characters of the type Ž19. can be represented. For instance, the following representation

J g Ž w,q . s q const

½

hq2 8

q

5

hq 2 4

Ž 25 .

y

 w1 ; . . . ;wry1 4 h

q1

2

with w obeying the condition wa q wrya s hr2 q 1 also occurs Žsee Table 3.. 2.2. Compactified free Boson In general the Fock space of a free boson may simply be constructed from a Heisenberg algebra and the corresponding Virasoro central charge equals c s 1. The character of the Heisenberg module ŽUˆŽ1.-Kac–Moody. y is simply the inverse of the h-function, qy1 r24r  1 4 1 . When compactifying the boson on a circle of rational square radius R s 2 srt one can associate highest weight representations of a UˆŽ1. k-Kac–Moody algebra to this theory. The UˆŽ1. k-algebra has an integer level, which is k s st. The corresponding characters read w7x

'

`

xˆ mk Ž q . s hˆ mk

2

qm l

k

1 24

qkl

Ý

y

q

s hˆ mk  2 k 4 2 k  k y m;k q m4 2 k,

Ž 26 .

lsy` 2

y

where we denoted hˆ mk :s q h m y r  1 4 1 . The highest weight may take on the values h km s m4 k with m s 0,1, . . . ,k y 1. As Table 1 indicates, the Dn-affine Toda models are related to compactified bosons. In order to recover the Dn-structure at the conformal level, we shall find realizations of J D n Ž q . in terms of Ž26.. Similar to the case of minimal models, it will be helpful to study factorization of Žcombinations of. the Kac–Moody characters. 1 First, choosing the constant in Ž6. as h nn r2 y 24 we identify for even n y

J

Dn

Ž q.

s hnnr2

ˆ

 n4 n n

½ 5 2

n y s hn r2

ˆ

 n4

y n

n

½ 5 2

q n 2

y

s hˆ nnr2  2 n4 2 n

n

q

2

n

½ 5

s xˆ nnr2 Ž q . .

Ž 27 .

n

Formally this expression also holds for odd n, albeit in this case the right hand side may not be interpreted as the character related to a compactified boson. Therefore we need another way to construct J D n Ž q . in case n is n Ž q . which are analogues of Ž19.. In odd. For this purpose we consider the combinations xˆ mn Ž q . " xˆ nym y particular, the q ™ 1 limit of these sums and differences is governed by ceff s c s 1 and c. ˜ According to Ž5. and the possible values for the highest weights we have c˜ s 1 y 6rn.

Table 3 Representation of characters in the form of Ž25.. For the arguments in bold the same convention applies as in Table 2, including the numerator y

g

sectors

 w1; . . . ;wry1 4 y

A1 E7

3,4 3,4 x 1,1 q x 1,3 4,5 4,5 x 1,1 q x 1,4 4,5 4,5 x 1,2 q x 1,3

1 y  32 ;2; 72 ; 132 ;8; 172 410 y  12 ;4; 92 ; 112 ;6; 192 410

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65

Exploiting the identity Ž2.31. obtained in Ref. w4x, we find Žfor 0 - m - nr2. n xˆ mn Ž q . " xˆ nym Ž q . s hˆ mn

y

n

½ 5½ n

2

2

n y 4

m n m ; q 2 4 2

"

5

n

.

Ž 28 .

2

The counting, based on Ž4., gives the expected values of c and c. ˜ Notice that for the upper sign the r.h.s. can be identified as the product side of Ž26.: n r4 xˆ mn Ž q . q xˆ nym Ž q . s xˆ mn r2 Ž q. .

Ž 29 . DnŽ

Comparison with Ž27. then yields a formula for J

q . valid for both odd and even n

J D n Ž q . s xˆ n4 n Ž q . q xˆ 34nn Ž q . .

Ž 30 .

Finally, it is interesting to observe that, employing Ž17. – Ž20., we may express Ž26. and Ž28. entirely in terms of the minimal Virasoro characters:

xˆ mn Ž q . s x 1,3,mn Ž q .

x 1,2,3n n Ž q . x 1,2,mn Ž q .

,

Ž 31 .

n xˆ mn Ž q . " xˆ nym Ž q . s Ž x 1,3,mn Ž q . " x 23,, mn Ž q . .

x 1,2,3n n Ž q . x 1,2,mn Ž q .

.

Ž 32.

Here n s 6 l " 1, l g N if we really regard all components on the r.h.s. as characters of irreducible Virasoro representations. This restriction can be omitted if we regard Ž17. – Ž20. just as formal series. With regard to the central charge Eqs. Ž31.-Ž32. imply ceff Ž DnŽ1. . s ceff Ž 3,n . q ceff Ž 2,3n . y ceff Ž 2,n . .

Ž 33 .

Thus, the connection with minimal models is more subtle than one would expect at first sight from a simple matching of the central charges, e.g. ceff Ž DnŽ1. . s 2 ceff Ž3,4.. 2.3. Parafermions Ž w x. to the The AŽ1. n -series of affine Toda theories is known to be related in the ultra-violet limit see e.g. 6 Znq 1-parafermions w11x. The corresponding central charge, cŽ k . s 2Ž k y 1.rŽ k q 2. and characters may be obtained from the SUˆŽ2. ky 1rUˆŽ1.-coset, where k s n q 1. Introducing the quantity D j,k m s jŽ j q 1.rŽ k q 2. y m2rk the characters of the highest weight representation, which appear as branching functions in the coset, acquire the form w12x

x˜ j,km Ž q . s

h˜ j,k m

 14

` rqs Ý Ž y1. q r sŽ kq1.q

y 1 r , ss0

r Ž rq1 .

s Ž sq1 . q

2

2

= Ž qr Ž jqm . qs Ž jym . y qr Ž kq1yjym . qs Ž kq1yjqm .qkq1y2 j . , c k k where hj,km :s q D j, m y 24 r 141y. The labels are restricted as yj F m F k y j, particular the x 0,k mŽ q . are the characters of the parafermionic currents Ž .

˜

˜

Ž 34 . 0 F j F kr2 and Ž j y m. g Z. In cmk . The characters possess the

symmetries

x˜ j,km Ž q . s x˜ j,ky m Ž q . s x˜ kkr2yj, k r2ym Ž q . .

Ž 35 .

From our observations made above for characters of the ADE related conformal models one may expect that expressions Ž34. exhibit A n-type structures Že.g. possess n quasi particles and moreover acquire the form of the type J A n Ž q .. in some of the cases when they admit a factorized form. We shall now discuss this issue in detail for several of the lowest ranks.

A.G. Bytsko, A. Fringr Physics Letters B 454 (1999) 59–69

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As we have seen above, factorization of linear combinations of characters occurs usually only for the specific k y Ž . type of combinations. Now the analogue of Ž19. is x˜ j,kmŽ q . " x˜ j,2 limit of kym q . One expects that the q ™ 1 XX k these sums and differences is governed by ceff s cŽ k . and c, ˜ respectively. Since h s D1,1 , Eq. Ž5. yields c˜ s cŽ k .Ž k y 6.rk. This is confirmed by all the examples given below. For the parafermionic formulae Ž34. we do not have such powerful analytical tools Žanalogues to the factorization formulae Ž17.-Ž23. . at hand as in the case of the minimal models. Therefore, as a first step, we resort to an analysis with Mathematica. Typically we expand the characters up to q 100 . A1: In this case there are only three distinct Žup to the symmetries Ž35.. characters and they can be matched with those of the Ž3,4. minimal model: 2 3,4 2 3,4 x˜ 0,0 Ž q . s x 1,1 Ž q . , x˜ 0,1 Ž q . s x 1,3 Ž q. , Ž 36 .

x˜ 21

,

1

3,4 Ž q . s x 1,2 Ž q . s J A1 Ž q . .

Ž 37 .

2 2

Thus, all the characters in this case are factorizable and moreover J A 1 Ž q . is present among them. A 2: There are four distinct characters in this case and they can be matched with those of the Ž5,6. minimal model: 3 5,6 5,6 3 5,6 x˜ 0,0 Ž q . s x 1,1 Ž q . q x 1,5 Ž q . , x˜ 0,1 Ž q . s x 1,3 Ž q. , Ž 38 .

x˜31

,

1

5,6 5,6 5,6 Ž q . s x 2,3 Ž q . , x˜31 3 Ž q . s x 2,1 Ž q . q x 2,5 Ž q. . , 2 2

2 2 3 Ž . x 0,1 q

Ž 39 .

Only ˜ and x˜ 3, Ž q . are factorizable Žsee subsection II.A.. A 3: Since c s 1, it is suggestive to try to relate the characters to those of the compactified bosons. This turns out to be possible for all the characters Žthus factorizability is guaranteed.: 4 4 4 x˜ 0,1 Ž q . s xˆ612 Ž q . , x˜ 0,0 Ž q . q x˜ 0,2 Ž q . s xˆ 03 Ž q . , Ž 40. 1 2

1 2

4 x˜ 1,0 Ž q . s xˆ 23 Ž q . ,

x41 1 , 2 2

˜

Ž q.

4 x˜ 1,1 Ž q . s xˆ 13 Ž q . ,

s x 14

x41 3 , 2 2

ˆ Ž q. ,

˜

Ž q.

Ž 41. Ž 42 .

s x 34

ˆ Ž q. .

Furthermore, some of linear combinations can be expressed in terms of the characters of the Ž3,4. minimal model Žnotice that c s 1, c˜ s y1r2 for A 3 and c s 1r2, c˜ s y1 for the Ž3,4. minimal model.: 4 4 3,4 x˜ 0,0 Ž q . y x˜ 0,2 Ž q . s Ž x 1,2 Ž q. .

x41 1 , 2 2

˜

Ž q.

" x41 3 , 2 2

˜

Ž q. s Ž

3,4 x 1,1

y1

Ž q.

,

3,4 . x 1,3

Ž 43. Ž q. .

y1

.

Ž 44.

A 4: No characters or linear combinations factorize. A 5: Several characters and combinations are factorizable and can be expressed via those of the Ž3,4. minimal model and Dn , for instance 3,4 x˜6 , Ž q . s x 1,2 Ž q . Ž xˆ624 Ž q . y xˆ 1824 Ž q . . , 3 2

1 2

3,4 x˜6 , Ž q . s x 1,2 Ž q . Ž xˆ 824 Ž q . y xˆ 1624 Ž q . . , 1 2

3 2

6 x 0,1 6 x 1,1

6 3,4 3,4 Ž q . s Ž x 1,1 Ž q . " x 1,3 Ž q . .Ž xˆ 924 Ž q . . xˆ 1524 Ž q . . , ˜ Ž q . " x˜ 0,2 6 3,4 3,4 Ž q . s Ž x 1,1 Ž q . " x 1,3 Ž q . .Ž xˆ 324 Ž q . . xˆ 2124 Ž q . . . ˜ Ž q . " x˜ 1,2 Also we notice that x˜ 6, Ž q . y x˜ 6, Ž q . s q 5r96 . Such an identity can occur only in this parafermionic model since it requires c˜ s 0. 1 2

1 2

1 2

5 2

A 6: – no combinations factorize and the only factorizable single characters are y

x 1,7 m

˜

Ž q.

7 s h1, m

˜

y

 3 4 3  m;7 y m;7 4 7 m s 1,2,3. y y ,  1 4 1  3m;21 y 3m4 21

Ž 45 .

A.G. Bytsko, A. Fringr Physics Letters B 454 (1999) 59–69

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Summarizing these data, we see that, apart from the A1 case, none of the factorizable Žcombinations of. characters provided by Eq. Ž34. for A n can be identified as J A n Ž x,q .. However, it is plausible to speculate that in general the J A n Ž x,q . might be identifiable as characters of other conformal models having the central charge 2 nrŽ n q 3.. This conjecture is supported by the A 2 case Žsee Table 2. and A 3 case, in which we can identify

J A 3 Ž q . s xˆ 312 Ž q . q xˆ 912 Ž q . .

Ž 46 .

To conclude the discussion on factorizable parafermionic characters, we notice an intriguing fact – some characters in the A 7 case exhibit an E7 structure Žcf. Tables 2 and 3.: 2

8 8 4 ,5 x˜ 0,1 Ž q . q x˜ 0,3 Ž q . s Ž x 2,1 Ž q. . , 2

8 8 4,5 x˜ 1,1 Ž q . q x˜ 1,3 Ž q . s Ž x 2,2 Ž q. . , 2

8 8 8 4,5 4,5 x˜ 0,0 Ž q . " 2 x˜ 0,2 Ž q . q x˜ 0,4 Ž q . s Ž x 1,1 Ž q . " x 1,4 Ž q. . , 2

8 8 8 4,5 4,5 x˜ 1,0 Ž q . " 2 x˜ 1,2 Ž q . q x˜ 1,4 Ž q . s Ž x 1,2 Ž q . " x 1,3 Ž q. . .

This is the first case in which we have to combine three characters in order to obtain a factorized form. A more detailed account on the factorization of A n-related characters will be presented elsewhere.

3. One particle states The functions  x 4yq and  x14 can be written as double series in q with coefficients being P Ž n,m. Žor Q Ž n,m.. – the number of partitions of an integer n G 0 into m distinct Žor smaller than m q 1. non-negative integers Žsee e.g. w13,4x.. Applying this fact to a character of the type Ž2. with x i / x j , we obtain it in the form of a series x Ž q . s Ý`ks 0 m k q k , where the level k admits the partitioning, k s Ý a Ý i a pai a , into parts of a specific form Že.g. Ž47. and Ž48. below.. The interpretation of the pai a as momenta of massless particles gives rise to a quasi-particle picture Ždeveloped originally for characters of the form Ž1. in Ref. w3x., where a character is regarded as a partition function, x Ž q . s Ý k m k eyb E k . Here q s ey2 p b Õ r L , with Õ being the speed of sound, and L – the size of the system. A quasi-particle spectrum constructed in this way is in one-to-one correspondence to the corresponding irreducible representations of the Virasoro algebra or some modules related to linear combinations. It is crucial to stress that this procedure is not applicable to the standard representation of the characters Ži.e. of the type Ž10.. and is a very specific feature of the representations Ž1. and Ž2.. Note that the modules which are of the form Ž1. do in general Žif they do, they give rise to Rogers–Ramanujan type identities w4x. not factorize, such that the spectra related to Ž2. do not only differ in nature from the ones obtainable from Ž1., but are also related to different sectors. As just explained, a quasi-particle representation can be constructed for any factorizable character of the type Ž2. provided that x i / x j . For instance, the characters Ž27. related to a compactified boson admit a representation with Ž2 k q 1. particles. However, since we are particularly interested in spectra with Lie algebraic features, it is most natural to perform the quasi-particle analysis for the characters which admit the form J g Ž x,q .. In this way we obtain the following fermionic spectrum Žif we employ the series involving P Ž n,m.. in the units of 2prL y y

pai Ž m . s x a q

ž

h 4

q 12 Ž 1 y m a . q

/

ž

h 2

q 1 Nai ,

/

Ž 47 .

A.G. Bytsko, A. Fringr Physics Letters B 454 (1999) 59–69

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Table 4 5,6 Ž . 5,6 Ž . Bosonic spectrum for x 1,2 q q x 1,4 q . k denotes the level and m k its degeneracy k

mk

p1i s1q 5i2 , p 2i s 32 q 5i2

1 2

0 1 1 1 1 2 2 3

–< p10 : < p 20 : < p10 , p10 : < p10 , p 20 : < p10 , p10 , p10 :, < p 20 , p 20 : < p10 , p10 , p 20 :, < p11 : < p10 , p 10 , p 10 , p 10 :, < p 10 , p 20 , p 20 :, < p 12 :

1 3 2

2 5 2

3 7 2

4

or bosonic spectrum Žif we use the series with Q Ž n,m.. pai s x a q

ž

h 2

q 1 Nai .

/

Ž 48 .

Here x,m and N parameterize the possible states. In Eq. Ž47. the numbers Nai are distinct positive integers such a N i s N , whereas in Eq. Ž48. they are arbitrary non-negative integers. Notice that for the combination that Ý mis1 a a of characters the levels may be half integer graded, such that also the momenta take on half integer values in this case. A sample spectrum is presented in Table 4 which illustrates how the available momenta of the form Ž48. are to be assembled in order to represent a state at a particular level. Naturally the questions arise if we can interpret these spectra more deeply and if we can possibly find alternative representations for the related modules. First of all we should give a meaning to the particular combinations which occur in our analysis. In Ref. w4x we provided several possibilities. In particular the 5,6 Ž . 5,6 Ž . combination x 1,2 q q x 1,4 q is of interest in the context of boundary conformal field theories, since this combination of characters coincides with the partition function ZA, F for the critical 3-state Potts model with boundaries w14x Ž F denotes the free boundary condition.. It is intriguing that this combination possesses a 5,6 Ž . 5,6 Ž . manifestly Lie algebraic quasi-particle spectrum. The combination x 2,2 q q x 2,4 q , which coincides with ZB C, F in the same model possesses a slightly weaker relation to A 2 . To answer the question concerning possible representations, we recall the fact that the fields corresponding to the highest weight states satisfy the quantum equation of motion of Toda field theory w15x. It is therefore very suggestive to try to identify the presented spectra in terms of the W-algebras w16x. For J g Ž q . we can make this more manifest. Changing the units of the momenta to prL, we obtain from Ž48. pai s e a q 1 q Ž h q 2 . Nai .

Ž 49 .

Here e a belongs to the exponents of the Lie algebra. Since the generators of the W-algebras Wsq1 are graded by the exponents plus one w17x, we may associate the following generators to this quasi-particle spectrum i

pai ; Wa Ž WaWrya . Na .

Ž 50 .

In particular, the critical 3-state Potts model with boundaries would be related to the W3-algebra. We leave it for the future to investigate this conjecture in more detail.

Acknowledgements We would like to thank W. Eholzer for useful discussions. A.B. is grateful to the members of the Institut fur ¨ Theoretische Physik, FU-Berlin for hospitality. A.F. is grateful to the Deutsche Forschungsgemeinschaft ŽSfb288. for partial support.

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