Fusion rules and (sub-)modular invariant partition functions in non-unitary theories

Volume 215, number 4

PHYSICS LETTERS B

29 December 1988

FUSION RULES AND (SUB-)MODULAR INVARIANT PARTITION FUNCTIONS IN NON-UNITARY THEORIES I.G. K O H ~ and P. S O R B A 2

Fermi National AcceleratorLaboratory, P.O. Box 500, Batavia, IL 60510. USA Received 12 September 1988

Fusion rules in non-unitary Kac-Moody and minimal conformal theories are obtained from their modular properties. New modular invariant partition functions in the non-unitary Kac-Moody theories are classified. From the discrete symmetries of the system we obtain new sub-modular invariant partition functions in the Kac-Moody and non-unitary conformal theories.

1. Introduction

Near the critical temperature statistical systems in two dimensions exhibit scaling behaviour. At the critical point they are described by conformal field theories [ 1 ]. Actually the conformal theories predict off-critical properties. The critical exponents ~ of the heat capacity ( C o c t - " ) , magnetization (MoctP), susceptibility ( Z ~ t - ; ' ) , correlators ( ( a ( R ) a ( 0 ) ) o c t - ' ~ ) , coherence length (~oct -V) and magnetization ( M o t h ~) (t and h are the reduced temperature and magnetic field) are described by the spin and energy operator's conformal dimensions h~ and h, as "=

1-2h, h~ 1-h----T' fl= l - h , '

7=

1-2ho r/=4h~ 1-h-----~' ,

1

u= 2(l-h,) - - ,

h~

a= 1-h~ --.

(1)

The numerical values of the critical exponents (o~, fl, 7, r/, v, a) in the Ising and three-state Ports models are known as (0, ½, 7, ¼, 4 ) and (½, ~, ~3, rs, 4 5 ~, ~4) respectively, and agree with the conformal dimensions (h~, h,) = ( ~ , ½ ) a n d ( ~~, 3) 2 in Ising and three-state Potts models, respectively. More generally correspondences between two-dimensional statistical systems and the unitary conformal theories are widely studied. Relationships between the non-unitary conformal theories and statistical systems have been less well studied. However, there is at least one example in the statistical mechanics for which a minimal (i.e. with a finite number of primary fields) non-unitary theory shows up. It is the so-called Lee-Yang singularity [ 3 ]. The properties of a system can be computed from the distribution of the zeroes of the grand partition function where these zeroes are brought by the complex magnetic field (conjugate to the spin operator). The density of zeroes exhibits a divergence near the critical complex magnetic field, and this led to identify [ 4 ] it with the c = - ~ h,, = - ½ nonunitary theory. From the mathematical point of view, the minimal (both unitary and non-unitary) representations of ref. [ 1 ] o f the Virasoro algebra are the modular covariant representations: see ref. [5 ] which contains in addition a study of the A [ ~~ unitary and non-unitary modular covariant representations. Motivated by these aspects, we investigate properties of the non-unitary representations of K a c - M o o d y A [~ and minimal conformal theories in this letter. In a minimal representation one can actually take subsets of Department of Physics, Korea Advanced Institute of Science and Technology, P.O. Box 150, Chongryang, Seoul, Korea. 2 Laboratoire d'Annecy-le-Vieuxde Physique des Particules, Chemin de Bellevue, BP 909, F-74019 Annecy-le-VieuxCedex, France. 2 See ref. [ 2 ] for a review. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

723

Volume 215, number 4

PHYSICS LETTERSB

29 December 1988

operators whose choices are specified by the modular invariant properties of the partition functions. The dynamics of these theories are encoded into the operator product expansions. Using Verlinde's approach [6 ] we compute the fusion rules from the modular properties. New non-unitary modular invariant partition functions of A I ~) are found and are presented in an A - D - E classification similar to the one obtained in ref. [ 7 ] for unitary A I ~) representations. Some of these theories enjoy additional symmetries which are also reflected in the fusion rules. We illustrate this with the well-known three-state Potts model. In the presence of such symmetries the fields at the opposite sides of the parallelogram can be different up to the action of these discrete symmetries (twisted boundary conditions). The partition function of the resulting theories are now invariant under a subgroup of the modular group, and we construct such submodular invariant partition functions.

2. Fusion rules

(A) Kac-Moody system The integrable representations of the SU (2) affine K a c - M o o d y algebra at level me N are denoted as 0, with n = 0, 1, ..., m. Here n is twice the SU (2) spin. Their fusion rules are known [ 8,6]: min(n+n',2m--n--n' )

O,, ®~0,' =

Z n" = In--n'

O,".

(2)

I . n + n ' - - n " = 0 rood 2

More generally, one can consider a rational level m = t / u with relatively prime te~_ and ueNN satisfying 2 u + t - 2 >i 0. The modular invariant or admissible representations [ 5 ] are labeled by two positive integers k and n such that

(~/.:,,;O<~k<~u-l,

O<~n<~2u+t-2.

(3)

The unitary integrable representation case is recovered by putting u = 1, te N. The characters for the admissible representation (3) are given by [ 5 ]

Zk:,,( z, z) =

0,.... (r, z/u)-Ot,_,~(r, z / u ) Oh2(,,f,z)__O__h2(,'C,Z. )

(4) '

with 0b,a(r, z) =

exp[ 2niza(j2 +jz) ] ,

(5)

be=u[+_(n+l)-k(m+2)].

(6)

jeY+b/2a

and

a=u2(m+2),

Due to the following properties of the theta functions: Ol,.a( T , Z )

= 0_/,,a(T

,

- - Z ) =Ob+ 2a,a( T, Z ) -~-O_b+ 2a,a('t, -- Z ) ,

(7)

the characters for 1 ~
.

(8)

The character is dominated in the large-Im ~ limit by exp [2niT( - ~ c + h ) ] , where 724

(9)

Volume 215, number 4

c=3m/(m+2),

PHYSICS LETTERS B

h={[(n+l)-k(m+2)12-1}/4(m+2).

29 December 1988

(lO)

The h value becomes negative for k ( m + 2 ) - 2 < n < k ( rn + 2 ). The character (4) may contain negative coefficients in the power expansion of exp (2nir). For u = 1, h, in eq. (10) is reduced to the Casimir eigenvalue of the spin ½n representation normalized with respect to 1 ( m + 2 ) as

h=½n(½n+l)/(m+2).

(11)

The general modular transformation is [ 5 ] S
(12)

and satisfies SS*= 1. The fusion rules Oi ~ ) ) = E Aok~k k

are computed from eq. (12) with

A~ik= ,.E

So----~'

(13)

where index i denotes a representation (k: n) and i = 0 is the index of the identity representation. Note that A is symmetric in i and j, and one obtains minln+n',2(2u+t--2)--n--n'} ¢/,': n @ 0 / , " : n' =

E

n"=

In--n' I,n+n'--n"=0 rood 2

( -- ) [ (k+k')/u]

Ok+k' mod u:n" •

(14)

where [ ( k + k' ) / u ] denotes integer division without remainder (e.g. [ 4 / 3 ] = 1 ). When [ (k + k' ) / u ] = 1, the n' on the right-hand side has to be replaced by n' ~ (2u + t - 2 ) - n'. Eq. (14) is of course reduced to eq. (2) for the integrable representation u = 1 and te~q. Let us illustrate eq. (14) in the case m = ½ with (t, u) = ( 1, 2). The fusion rules then read ~1:3~1:2 =--~0:2

,

~ 1 : 0 ~ 0 : 3 = "JI-~ 1 : 3 .

(15)

The associativity E A A m= ~iik~kl k

~ AitkA~/"

(16)

k

of the fusion rules in eq. (14) has been confirmed, even in the presence of negative fusion rules. A computation of modular invariant partition functions for rational level m = t/u leads to an A - D - E classification generalizing the one obtained in ref. [7] for unitary AI ~) representations. Note that the unitary cases are re-obtained from eq. ( 17 ) by restricting t to be positive integer and u to be one, the sum over k = 0 ..... u - 1 then disappearing. 725

Volume 215, number 4

PHYSICS LETTERS B

29 December 1988

u-- 12u+t--2

A2u+t-I'

Z

D21,+2:

4p=2u+t-2>~4:

D2p+l:

4p-2=2u+t-2>~6:

E6:

2u+t-2=

E7:

2u+t-2=16

IXk:n j2 ,

2

k=O

H=O

u--

2p -- 2

1

~

k=0 neven=0 u-

1

( Izk:. -~k:4p-n 12+ 2 IZ~:2p 12) ,

// 4p-- 2

~'. ~

Z

k=0 \ n c v c n = 0

2p-- 3 JZ/,-:n J 2 + [Z/,':2p-I

12+

()(,k:n)~:4p--n "1"-C . C ) ) ,

nodd= 1

lt-- 1

10

Y. ( I,~/,':O "4-,~k: 6 12-1- IZl,:3 -Fz/,-:712q- {Zk:4-I-Xk:xOI 2 ) ,

k=0

tl-- I

k=0

{ IZk:o "+-Zk:16 12+ IZk:4 +Zk: 1212-1- Izk:6 -I-Z/,-:,o 12+ Ix,:8 12

+ [ (Zk:2 +Zk: ~4)ZL, + c . c . ] } , it-

Es:

2u+t-2=28

]

( IX/,.:O "+'Zk: 10 "+Z/,.: 18 +Z/.:2g I 2 + IZ/,.:6 "arZk: 12 +Z/,: 16 -/-,~/,.:22 J 2 ) . k=0

(17)

It is worthwhile to remark that to a given algebra of A - D - E type are now associated a infinite (discrete) set o f models. For example, to the 04 algebra correspond theories such that

(t,u)=(-4,5),

( - 8 , 7), ( - 1 6 , 11), ( - 2 0 , 13),....

We not that two different allowed models do not involve the same number o f primary fields, the k index in the partition function taking the values k = 0 , 1, ..., u - 1. We have not yet checked whether eq. (17) gives the complete list o f modular invariant partition functions relative to the admissible representations in A~ ~) .

(B) Minimal conformal system Let us remind that a minimal conformal theory [ 1 ] is characterized by a pair (p, p' ) of two relatively prime positive integers. The central charge then reads

c= 1 - 6 ( p - p ' )2/pp, ,

(18)

and the primary fields are labelled as

(r,s)=(p'-r,p-s)

withl<~r<~p'-l,l<~s<~p-1.

(19)

The case IP - P ' I = 1 corresponds to the unitary series [9]. The fusion rule for a general pair (p, p' ) can be deduced from the modular transformation properties of characters, and reads minlr+r' - 1.2p" - I - r - r " , ~,.., (~Dr',s' =

~

r"= Ir--r'l +1

min'~s+s' - 1,2p- 1--s--s' } E s"=Js--s'L+l

~r",s" ,

(20)

with the constraint r+r' - r " =s+s' - s " = 1 m o d 2. The relation between eq. (20) and eq. (6.7) in ref. [ 1 ] is the following: one applies eq. (6.7) o f ref. [ 1 ] for (r, s) ® (r', s' ), (r, s) ® (p' - r', p - s' ), (p' - r, p - s) ® (r', s' ), (p' - r, p - s) ® (p' - r', p - s' ) and take c o m m o n terms in the product after making use of the identity (19). 726

Volume 215, number 4

PHYSICS LETTERSB

29 December 1988

3. Symmetries Minimal theories cannot have a continuous symmetry since an h = 1, h-= 1 field, which would be associated to a conserved current, never shows up, but they do admit discrete symmetries. The well known examples of such symmetries are the Z2 symmetry in the Ising model and the Z3 or more precisely $3 symmetry in the three-state Potts models [ 10 ]. Actually these models belong to the class of the parafermionic theories [ 11 ]. Note that such a discrete symmetry can help to break the degeneracy in a theory where more than one primary field ~.,~(z, ~) with the same conformal weights h~, h, are present [ 12]. As an example consider the (D4, A4 ) model (i.e. the three-state Ports model) with partition function Z = IZo-t-Z312+

(21)

IZ2/5 "~-)~7/512+21Z,/,~ 12+2[Z2/312 ,

where the subscripts denote the conformal weights of the corresponding primary field. We will denote the primary fields associated to the linear combination 3f characters (Zo+Z3), (Z2/5+Z7/s) by 0, and 02, and by 0~ and 0~ those associated to Z~/,s and )~2/3The modular transformation matrix relative to the six vectors (0,, ¢~', 0~-, 02, 04+ and 0~- ) is

s=~

2

-sin ½n sin~n sin~n sin ~n 2 sin ~n '

sin ~n -~osin'sn - 0 9 2 s i n 31n - s i n 3 ni ~ sin 2n

_sin ½n

092 sin ~n

sin {n -to2sinln - 0 9 s i n u'c - sin ~n ' 2 ~2 sin ~n 09 sin ~n

sin -~n -sin½n - s i n s n' - s i n ~ ,n sin 2 sin 2n

sin ~1 ~osin2n o~2 sin2n 2 sin~n 09 sin ,

sin 5n' o92sin{n co sin 2n 2 sin ~n co2 sin , n

(22)

092 sin ½n co sin ½n _

The S matrix satisfies

SS*=I,

(23)

SS=C,

where C is 1 0 0 0 0 1 C= 0 0 0 0 _ 0 0

0 1 0 0 0 0

0 0 0 1 0 0

0 0 0 0 0 1

0 0 0 0 1 0 _

The Z3 group leaves invariant 0, and 02, and provides a phase-(eigenvalue) oJ = exp (i2n/3 ) to 0~- and 0 ~ , and o~2 to ~ - and ~4. Such an action can be seen as an automorphism on the fusion rules [ 13 ] of the model. Indeed one obtains the following fusion rules from eq. (22):

0,'01=0,, 0,'02=02, 02"02=0,+02, 0,'0~=0~, 02-0~=0~+0~, 0,'0~=0~, 02'0~=0~, 03.03 =0,+0_,, 0;~0~-=0;0~=02, 04+0~-=0,, 0J-.0J-=0~-+0a-, 0J-0:=0;, 0~-0~=0; +

--

0 ; . 0 ; =0~- + 0 + ,

0~-.0~-=0~-,

0 ~ - . 0 4 = ¢ ~ -.

(24)

Fusion rule A231 in 0 J ' 0 f = 0 , is computed from eq. (13) as A23'= ( S S ) 2 3 = C 2 3 = l, and this partially "explains" why SS= C in eq. (23). The fusion rules in eq. (20) exhibit the following $3 symmetries: 727

Volume 215, number 4

PHYSICS LETTERSB

and ~O~J-,~oJO~-

Z3: 0J- - ' o90J-

0~1

\o~20;/

\0:/

\o~20:

, Z2:

0J- --' ~Y

0~-/

29 December 1988

and

\0J-/

~-1--,~\04/

.

(25,26)

\0 +

Note that eqs. (25 ) and (26) correspond to a cyclic group of order 3 and 2, respectively. Products of transformations (25) and (26) also leave the fusion rule (24) invariant.

4. Submodular invariant partition functions If a system is invariant under a group G, one may impose G-twisted boundary conditions on its conformal fields [ 10,14,15,8 ]. In the case where G = Z2 (respectively Z3) the associated partition functions are then invariant under a subgroup of the modular group [ 14 ], namely Fo (2) (respectively Fo ( 3 ) ) generated by

T':T--,T+2 i n F o ( 2 ) ,

T':T--,T+3 i n F o ( 3 ) ,

(27)

and S' which is constructed from the S and T of the full modular group as

r ~ z + 1--,- 1 / ( z + 1)--,1- 1 / ( r + 1).

(28)

The submodular Fo (2) invariant partition functions for the minimal (unitary or non-unitary) conformal theories associated to the (At,,_ ~, Ap_ ~) series are p'--I p--I

E

r=l

p'--I p--I

E Z,.,Z*t,-s = E

s=l

r=l

E Zr.,XT,.-......

s=l

(29)

When t h e p a n d p ' are restricted to IP-P'I = 1 (unitary series), eq. (29) include eq. (9) in ref. [ 14]. The spin of the operators ( = difference in conformal dimensions of the chiral and anti-chiral sectors) is a multiple of ½. The submodular Fo (3) invariant partition functions have been obtained for (p, p' ) = (p, 6) conformal theories in the series (D4, At,_ j ), and read [p/2 ]

[ (Z,., +Zj.p_,)Z~.++c.c+ Iz3.+l 2] ,

(30)

i=l

where [p/2 ] denotes the integer division without remainder. As mentioned before, the p should be co-prime with respect to 6. Operators in such theories have spin which is a multiple of ~. The p = 5 and 7 cases agree with the unitary Fo (3) invariant partition function given in refs. [ 14,15,11 ] and the p = 5 case is related to the threestate Potts model with Z3 symmetry considered in eq. (25). The simplest non-unitary Fo (3) invariant partition function corresponds to the theory (p, p' ) = ( 11, 6) with central charge c = 6 and reads

(x,., +x,.,o)X3., + (X,.z +x,.9)x3.2 +...+ (x,.5 +x,.6)x~.5 +c.c+ (Ix3., 12+.-+ Ix3.512) •

(31)

We end up this section with the Fo ( 3 ) invariant partition function associated to the level m = 4 representations ofAl '~ affine Kac-Moody algebras: (Xo +X4)X* +X2 (Xo +X4 )* + X3X],

( 32 )

5. Conclusion Some properties of the K a c - M o o d y Al l) as well as minimal Virasoro theories have been generalized to the 728

Volume 215, number 4

PHYSICS LETTERS B

29 December 1988

non-unitary case. Fusion rules have been constructed for the non-unitary representations o f affine K a c - M o o d y theories and their m o d u l a r invariant partition functions are constructed. S u b m o d u l a r invariant partition functions with twisted b o u n d a r y conditions have been worked out for the non-unitary conformal a n d K a c - M o o d y theories. The above results can be considered as a first step in a general study o f the m i n i m a l non-unitary theories. In particular, one m a y wonder whether our classification o f non-unitary A[ ~) m o d u l a r invariant partition functions is complete. F u r t h e r m o r e one should compute the fusion coefficients in the o p e r a t o r product expansions which are typically ratios o f g a m m a functions [ 1,16 ], while the fusion coefficients in eq. (13) is normalized to the integer. F r o m the given list o f the s u b - m o d u l a r invariant partition functions, which deserves to be completed, it looks likely that the s y m m e t r y group o f a m i n i m a l conformal theory is isomorphic to the invariance group o f the A D - E D y n k i n d i a g r a m associated to the model. Pasquier's a p p r o a c h [ 17 ] o f the A - D - E classification [ 7 ] suggests this conjecture. Note that, if Z2 and $3 are the s y m m e t r y group o f the Ising model and three-state Ports m o d e l respectively, the four-state Potts m o d e l involves the D~ j ~ K a c - M o o d y algebra [ 18 ], the s y m m e t r y o f the D y n k i n d i a g r a m o f which is $4. One m a y wonder whether the five-state Potts model could be related to the hyperbolic algebra [ 19 ] D H whose D y n k i n diagram, o b t a i n e d from the diagram o f D~ ~) by a d d i t i o n s o f a sixth dot connected to the central dot by one line, a d m i t s an $5 symmetry.

Acknowledgement It is pleasure to thank J.-B. Z u b e r for helpful conversations and B. Julia for sending us ref. [ 5 ] which plays a crucial role in this work. We are grateful to W.A. Bardeen and the F e r m i l a b Theory G r o u p for their k i n d hospitality. One o f us ( I . G . K . ) is i n d e b t e d to K O S E F for a research grant.

References [ 1] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [2] E.g., L.P. Kadanoff, in: Phase transitions and critical phenomena, Vol. 5A, eds. C. Domb and M.S. Green (Academic Press, New York, 1976). [ 3 ] C.N. Yang and T.D. Lee, Phys. Rev. 87 ( 1952 ) 404; T.D. Lee and C.N. Yang, Phys. Rev. 87 (1952) 410. [4] J.L. Cardy, Phys. Rev. Lett. 54 ( 1985 ) 1354. [5] V. Kac and M. Wakimoto, Proc. Natl. Acad. Sci. USA 85 (1988) 4956. [6] E. Verlinde, Nucl. Phys. B 300 (1988) 360. [7] A. Cappeli, C. ltzykson and J.-B. Zuber, Nucl. Phys. B 280[FS18] (1987) 445; Commun. Math. Phys. 113 (1987) 1. [ 8 ] D. Gepner and E. Witten, Nucl. Phys. B 278 (1986) 493. [9] D. Friedan, Z. Qiu and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575; Commun. Math. Phys. 107 (1986) 535. [10] J. Gardy, Nucl. Phys. B 275[FS17] (1986) 200. [ 11 ] A.B. Zamolodchikov and V.A. Fateev, Sov. Phys. JETP 82 ( 1985 ) 215; D. Gepnerand Z. Qiu, Nucl. Phys. B 285 [FS19] (1987) 423. [ 12 ] R. Brustein, S. Yankielowicz and J.B. Zuber, preprint SPhT/88-086, TAUP- 1647-88. [13] R. Dijkgraaf and E. Verlinde, preprint THU-88/25, in: Proc. Annecy meeting on Conformal field theories (North-Holland, Amsterdam, 1988), to be published. [ 14] J.-B. Zuber, Phys. Lett. B 176 (1986) 127. [ 15] J.-B. Zuber, in: Proc. Espoo Conf. (World Scientific, Singapore, 1986). [ 16] VI.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240[FS12] (1984) 312; I.G. Koh and B.C. Park, preprint KAIST-88/10 (1988). [17] V. Pasquier, J. Phys. A 20 (1987) L1229; Nucl. Phys. B 285[FS13] (1987) 162. [18] S.K. Yang, Nucl. Phys. B 285[FS19] (1987) 183, 639. [ 19 ] V.G. Kac, Infinite-dimensional Lie algebras ( Birkh~iuser, Basel, 1983 ); C. Saglioglu, CERN preprint CERN-TH 4854/87. 729