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Proof of the completeness of the classification of rational conformal theories with c=1
Volume 217, number 4
PHYSICS LETTERS B
2 February 1989
PROOF OF THE COMPLETENESS OF THE CLASSIFICATION OF RATIONAL CONFORMAL THEORIES WITH c= 1 ~ Elias B. K I R I T S I S J
California Instituteof Technology, Pasadena, CA 91125, USA Received 17 October 1988
By using the Serre-Stark theorem we prove that the list of rational conformal theories with c= 1, given by Ginsparg, is in fact complete.
It has been a m a j o r goal o f conformal field theory ( C F T ) to classify all the universality classes o f critical behavior in two dimensions. Stated from the point o f view o f string theory, this would correspond to a classification o f the classical ground states o f the string. There has been a lot o f progress in recent years in u n d e r s t a n d i n g the tools to achieve such a classification, and some partial results have emerged. Despite the fact that a complete classification seems out o f reach at present, some partial classifications m a y shed light on which ingredients are i m p o r t a n t and which are not. Ginsparg, in a nice p a p e r [ 1 ], gave a list o f c = 1 theories. There are two continuous families in this list corresponding to the torus a n d Z2 orbifold models. In ref. [2] it was shown that there are no marginal d e f o r m a t i o n s that lead outside these two lines. In ref. [1 ] three disconnected models have been constructed, that could not be reached from the rest via marginal deformations. There was a widespread suspicion that his list was complete. In fact, in ref. [3 ] it was shown that u n d e r the a s s u m p t i o n that the building blocks are toroidal p a r t i t i o n functions, then this list is complete. In this p a p e r we will s u p p l e m e n t this classification by a rigorous p r o o f that the subset o f rational C F T ' s at c = 1 is complete. Work supported in part by the US Department of Energy under Contract DEAC-03-8 I-ER40050. Address after September 1, 1988: Physics Department, University of California, Berkeley, CA 94720, USA. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
In o r d e r to set the notation, we will write the partition functions o f these models. F o r the torus models, which are p a r a m e t r i z e d by the radius R o f the target space, the partition function is ~1
Z(R)= 1
Z
qP~/ZClV~/2
(1)
where PL = m/R + nR/2, PR = m / R - nR/2. The partition function ( 1 ) is m o d u l a r invariant, and is also invariant under the duality transformation R--,2/R. Define Z~=Z(Nx/2 ). Then the orbifold partition function at radius R is given by Z(R)orb = ~ [ Z ( R ) + 2 Z 2 - Z I ] .
(2)
The disconnected models which are in one-to-one correspondence with the three nontrivial discrete subgroups o f S U ( 2 ) have the following partition functions:
ZT=k(2Z3+Z2-Z,), Zo = ½ (Z4 + Z 3 + Z 2 - Z , Z I ~- ½ ( Z 5 q - Z 3 ' } - Z 2 - Z
), 1) .
(3)
The rational theories belonging to this list are the ones which contain partition functions with R being the square root o f a rational number. The partition function o f a C F T with central charge c behaves as r - d o e as [4] ~ From now on Z will denote partition functions whereas Z denotes the set of integers. 427
Volume 2 17, number 4
PHYSICS LETTERS B
Z ( r , q)-~ {exp [ 2 n i ( r - g) l} -c/24 X [ 1 + O (exp (2 niTe ) exp ( - 2zcie~-) ) ]
(4)
with e, ~-> 0. In a rational conformal theory the p a r t i t i o n function can be constructed as a finite sum o f holom o r p h i c blocks, i.e. Z = Z j f ( r ) g j ( g ) , The f , g~ furnish a finite-dimensional unitary projective representation of the m o d u l a r group. In m o d e r n representation theory finite-dimensional unitary representations are constructed using the inducing procedure. O f course the general p r o b l e m of classifying the representations o f the m o d u l a r group is an extremely h a r d p r o b l e m ~2 but fortunately partial results are available. Thus the study o f induced representations leads naturally to the study o f m o d u l a r forms for normal subgroups of the m o d u l a r group. In fact, in ref. [ 5 ] it was shown that in a rational conformal field theory c and the critical d i m e n s i o n s are rational numbers. This in particular implies that there exists a positive integer N so that an infinite subgroup o f the m o d u l a r group, A ( N ) , is represented trivially. Otherwise stated, the "generalized c h a r a c t e r s ' f , are m o d u l a r forms o f A ( N ) . The n o r m a l subgroup A ( N ) o f the m o d u l a r group F is defined as follows. Let T be the element o f F representing the translation r ~ r + 1. Then,
2 February 1989
u n b o u n d e d denominators ~4. Thus, in our case one has to look for m o d u l a r forms on F ( N ) . Then any representation of an arbitrary congruence subgroup can be induced by one on F ( N ) . At c = 1 the m o d u l a r forms that are a p p r o p r i a t e are called by m a t h e m a t i cians 1-singular. Such 1-singular forms on F o ( N ) (with characters) have been classified by Serre and Stark [6 ]. Let us define: £,,(r)=
~ q,,(,+b/2o):, tt~
g,,~,(r)= ~
(-1)'q
~("+b/2a)2 ,
(5)
with a, be ~ 2, a > 0 and as usual q = exp ( 2 ~ i r ) . f , j , ( r ) satisfied the following periodicity properties in b: L , h ( r ) = £ , - b ( ' c ) =£.2,+t,('c) = L . 2 a - b ( z ) •
(6)
Thus for each aE 77the "fundamental d o m a i n " of be 77 is0~
~
- i ' t 2a~ 1 exp(iTckb/a) k=0
×f,,k ( r )
if be77,
× ga,~-(r)
if be77+½,
(7)
A ( N ) = [UTNU -' , V U e F ] . This subgroup for N = 2 , 3, 4, 5 coincides with the standard principal congruence subgroups F (N) o f F ~3 and therefore is o f finite index. F o r N>~ 6 the index o f A ( N ) in F is infinite. F o r c < 2 the finite-dimensional representations of the m o d u l a r group are generated by m o d u l a r forms on finite index subgroups o f F. There are two classes o f finite index subgroups o f F, the congruence subgroups a n d the non-congruence subgroups. The b e h a v i o r o f a m o d u l a r form on a subgroup F ' d e p e n d s crucially on F ' being a congruence subgroup or not. In fact it turns out that m o d u l a r forms on non-congruence subgroups are not relevant for C F T since their F o u r i e r coefficients have ~2 An important reason for this is that the modular group is one of the so called Type-If groups. For none of Type-II groups we have a complete classification of their irreducible representations. ~3 On congruence subgroups we are using the notation of ref. [ 6 ]. 428
g~,b(--1/z) = ~ a- i r 2z ~k=o -I ~ exp[izc(2k+l)b/a] X£,k+w2(r)
ifb~77,
xg~,k+~/z(r)
ifb~Z+½ ,
(7')
F,j,( r + 1 ) =exp(ircb2/2a)£.t,(z)
ifa-beZ,
=exp(i~zb:/2a) g~,l,(r)
if a-be77+½ ,
(8)
and similarly for g~,b. Thus F~,~,-£,b/~1 where ~/is the D e d e k i n d ~/-function are m o d u l a r forms for some F ( N ) and if a-be77 then N = [4a,24] ~5. Thus at this point we can invoke the S e r r e - S t a r k
~4 Some general results related to the discussion above will be presented elsewhere, ~5 By [a,b] we mean the lowest common multiple of a, b.
V o l u m e 217, n u m b e r 4
PHYSICS LETTERS B
theorem [ 6 ], which states that any 1-singular m o d u : lar form on some F ( N ) is a finite linear combination of the functions F,,,b (r). Accordingly the general partition function for a rational c = 1 CFT can be written up to overall normalization as z(r,~)
= y~ N~,t,F~,b(r)F~,t;(r) ~,~ - •
(9)
2 F e b r u a r y 1989
sible invariants. Not all of them are independent, though.
L e m m a 2. Let a = a ' p 2 where p is a prime greater than one. Then for every divisor 6 o f a' we have z~,, = z ~ .
(a,b) a.F
There are three basic constraints on (9). The first is modular invariance and the second is that all multiplicities should be non-negative integers when one normalizes the partition so that the unit operator appears once. The third constraint is that the theory has to have a consistent operator algebra. We will return to this point later on. The first step will be to construct all possible modular invariants of the form (9). We call a in Fo,b(r) the " l e v e l " % From (7) it is obvious that b~Z and from (8) that a~2V +. Since the action of S a n d Tdoes not change the level o f the modular form F,,b(r) we can classify the invariants at a given level separately.
L e m m a 1. Let a e Z +. Then for any divisor 6 o f a there exists an invariant Z~. The invariant generated by ~ = a / 6 is the same. The form o f the invariant is given by d - I ,~-1 K=0 2 = 0
+ F~,,a+~.~+~a(r ) f f ,,~+,~,r_~( ¢) ] .
(10)
Proof. The line of arguments follows the discussion in ref. [ 7 ]. To save space we will just sketch the proof. Invariance under T implies that b 2 _ b' 2 = 0 rood (4a) if the term Fa,b(r)Fa,b,(¢) is to appear in the partition function. Choosing 6 such that g/a, S = a / 6 we see that ( b - b ' ) ( b + b ' ) = 0 r o o d ( 4 6 5 ) , and the solution is b = 6 m + ~n, b' = 6 m - J n , m, n~Y_. One then can show by using the arguments in ref. [ 7 ] that these solutions for every divisor 6 o f a generate the commutant o f T. We can then check that the partition function invariant under T is also invariant under S by using (7). Restricting b, b' in the domain 0 ~
Proof. The proof is obvious once we establish the following identity: N--I
y, GN2 N(2ak+/,)(r) = G , t , ( r )
,
( 11)
k=0
which follows trivially from the definition (5) by appropriately splitting the summation modulo N.
Theorem. Let a e Z +. Let ~ be a divisor of a, 3=a/(~ such that the greatest c o m m o n divisor (6, ~) = 1. For all such pairs 6/cithere exists an independent invariant Z~ given by (10). Moreover, this list of invariants is complete. Proof. Let 6 be a divisor of a such that (6, ~ ) > 1 than by lemma 2 this invariant can be reduced to an invariant at a lower level a' = a / ( & if)2. At a given level, invariants for different pairs ( & 6 ) and (6', 6' ) are distinct by inspection. Furthermore it is easy to show that relation ( 11 ) is the only relation between forms at different levels. Completeness follows from lemma 1. It is obvious that the invariant Z~ is equal to the toroidal partition function Z ( R ) (see ( 1 ) ) at R = ,xf~2t~ Duality is the statement that Z~ = Z~. Thus we have proven that an arbitrary modular invariant partition function is a finite linear combination o f the invariants Z~ (toroidal partition functions). The final step is to impose the constant of nonnegative integer multiplicities. This question has been answered in ref. [3]. In order to make the line o f thought clear we will present the statements without proofs. Details for the proofs will appear in ref. [ 8 ]. We will distinguish two cases. The first case is when no chiral (1,0) operator appears in the partition function. In this case there is no U ( 1 ) symmetry and we have to write partition functions in terms o f Virasoro characters. 429
Volume 2 17. number
4
PHYSICS
The general partition function can be written as a finite sum Z=C~~,c,Z(R,), c~!,c,=l, in order to have a unique (0,O) operator.
Lemma 3. The multiplicity of (s’,O) operators, for Z, and is 1 SFH+, s<5 is given by 1+2[s/N] otherwise. In particular Z, contains three (1,O) operators and any other Z( R,) only one. The constraint that there are no ( 1,O) operators the partition function forces c, = - 1.
in
Lemma 4. For all i, 2c,ch. This is so because we can show that in the opposite case there will always be fractional multiplicities in Z. Lemma 5. Except for c, = - f all other c, have to be zero or positive. This is true because we can show that if one of them is negative there will be at least one state that appears with a minus sign in 2. Theorem. Any partition
function Z(7, T) not containing any ( 1,O) operators has to be either of the form (2) or of the form (3). This follows for lemmas 3, 4, 5 and the constraint that operators of dimension (?,O ), s< 5 have positive integer multiplicities. The only case left to consider is the case that there exist ( 1,O) operators in the theory, so that states are in representations of a chiral U( 1) algebra. If the number of operators ( 1,O) is
430
LETTERS
B
2 February
1989
N, > 0 then c, (the coefficient of Z) has to be N, - f _ Lemma 4 still remains true for the same reason, and lemma 5 extends trivially in the statement that c, > 0 Vi. From the above it is obvious that there are only two possible solutions, either the torus models Z=Z(R)or2=f[Z(R)+Z(R’)].Themodelswith partition function 2 do not have a consistent operator algebra. In particular they do not satisfy the requirement of additively conserved U( 1) changes. Thus they do not correspond to CFT models. Thus our claim is proven. There are no other rational CFT’s with c= 1 except those in ref. [ 11. I would like to thank R. Dijkgraaf nating discussions, and P. Ginsparg manuscript.
for very illumifor reading the
References [I] P. Ginsparg, Nucl. Phys. B 295 [FS21] (1988) 153. [2] R. Dijkgraaf, E. Verlinde and H. Verlinde, Commun. Math. Phys. 115 (1988) 649. [ 31 R. Dijkgraaf, E. Verlinde and H. Verlinde, Utrecht preprint THU-87/30. [4] J. Cardy, Nucl. Phys. B 270 [FSl6] (1986) 186. [ 5 ] G. Anderson and G. Moore, preprint IASSNS-HEP-87/69. [6] J.P. Serre and H.M. Stark, Lecture Notes in Mathematics, Vol. 627 (Springer, Berlin, 1977) p.27. [7] A. Cappelli, C. ltzykson and J.-B. Zuber, Commun. Math. Phys. 113 (1987) 1. [8] E. Kiritsis, Caltech preprint CALT-68-1510, nol to be published.
PHYSICS LETTERS B
2 February 1989
PROOF OF THE COMPLETENESS OF THE CLASSIFICATION OF RATIONAL CONFORMAL THEORIES WITH c= 1 ~ Elias B. K I R I T S I S J
California Instituteof Technology, Pasadena, CA 91125, USA Received 17 October 1988
By using the Serre-Stark theorem we prove that the list of rational conformal theories with c= 1, given by Ginsparg, is in fact complete.
It has been a m a j o r goal o f conformal field theory ( C F T ) to classify all the universality classes o f critical behavior in two dimensions. Stated from the point o f view o f string theory, this would correspond to a classification o f the classical ground states o f the string. There has been a lot o f progress in recent years in u n d e r s t a n d i n g the tools to achieve such a classification, and some partial results have emerged. Despite the fact that a complete classification seems out o f reach at present, some partial classifications m a y shed light on which ingredients are i m p o r t a n t and which are not. Ginsparg, in a nice p a p e r [ 1 ], gave a list o f c = 1 theories. There are two continuous families in this list corresponding to the torus a n d Z2 orbifold models. In ref. [2] it was shown that there are no marginal d e f o r m a t i o n s that lead outside these two lines. In ref. [1 ] three disconnected models have been constructed, that could not be reached from the rest via marginal deformations. There was a widespread suspicion that his list was complete. In fact, in ref. [3 ] it was shown that u n d e r the a s s u m p t i o n that the building blocks are toroidal p a r t i t i o n functions, then this list is complete. In this p a p e r we will s u p p l e m e n t this classification by a rigorous p r o o f that the subset o f rational C F T ' s at c = 1 is complete. Work supported in part by the US Department of Energy under Contract DEAC-03-8 I-ER40050. Address after September 1, 1988: Physics Department, University of California, Berkeley, CA 94720, USA. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
In o r d e r to set the notation, we will write the partition functions o f these models. F o r the torus models, which are p a r a m e t r i z e d by the radius R o f the target space, the partition function is ~1
Z(R)= 1
Z
qP~/ZClV~/2
(1)
where PL = m/R + nR/2, PR = m / R - nR/2. The partition function ( 1 ) is m o d u l a r invariant, and is also invariant under the duality transformation R--,2/R. Define Z~=Z(Nx/2 ). Then the orbifold partition function at radius R is given by Z(R)orb = ~ [ Z ( R ) + 2 Z 2 - Z I ] .
(2)
The disconnected models which are in one-to-one correspondence with the three nontrivial discrete subgroups o f S U ( 2 ) have the following partition functions:
ZT=k(2Z3+Z2-Z,), Zo = ½ (Z4 + Z 3 + Z 2 - Z , Z I ~- ½ ( Z 5 q - Z 3 ' } - Z 2 - Z
), 1) .
(3)
The rational theories belonging to this list are the ones which contain partition functions with R being the square root o f a rational number. The partition function o f a C F T with central charge c behaves as r - d o e as [4] ~ From now on Z will denote partition functions whereas Z denotes the set of integers. 427
Volume 2 17, number 4
PHYSICS LETTERS B
Z ( r , q)-~ {exp [ 2 n i ( r - g) l} -c/24 X [ 1 + O (exp (2 niTe ) exp ( - 2zcie~-) ) ]
(4)
with e, ~-> 0. In a rational conformal theory the p a r t i t i o n function can be constructed as a finite sum o f holom o r p h i c blocks, i.e. Z = Z j f ( r ) g j ( g ) , The f , g~ furnish a finite-dimensional unitary projective representation of the m o d u l a r group. In m o d e r n representation theory finite-dimensional unitary representations are constructed using the inducing procedure. O f course the general p r o b l e m of classifying the representations o f the m o d u l a r group is an extremely h a r d p r o b l e m ~2 but fortunately partial results are available. Thus the study o f induced representations leads naturally to the study o f m o d u l a r forms for normal subgroups of the m o d u l a r group. In fact, in ref. [ 5 ] it was shown that in a rational conformal field theory c and the critical d i m e n s i o n s are rational numbers. This in particular implies that there exists a positive integer N so that an infinite subgroup o f the m o d u l a r group, A ( N ) , is represented trivially. Otherwise stated, the "generalized c h a r a c t e r s ' f , are m o d u l a r forms o f A ( N ) . The n o r m a l subgroup A ( N ) o f the m o d u l a r group F is defined as follows. Let T be the element o f F representing the translation r ~ r + 1. Then,
2 February 1989
u n b o u n d e d denominators ~4. Thus, in our case one has to look for m o d u l a r forms on F ( N ) . Then any representation of an arbitrary congruence subgroup can be induced by one on F ( N ) . At c = 1 the m o d u l a r forms that are a p p r o p r i a t e are called by m a t h e m a t i cians 1-singular. Such 1-singular forms on F o ( N ) (with characters) have been classified by Serre and Stark [6 ]. Let us define: £,,(r)=
~ q,,(,+b/2o):, tt~
g,,~,(r)= ~
(-1)'q
~("+b/2a)2 ,
(5)
with a, be ~ 2, a > 0 and as usual q = exp ( 2 ~ i r ) . f , j , ( r ) satisfied the following periodicity properties in b: L , h ( r ) = £ , - b ( ' c ) =£.2,+t,('c) = L . 2 a - b ( z ) •
(6)
Thus for each aE 77the "fundamental d o m a i n " of be 77 is0~
~
- i ' t 2a~ 1 exp(iTckb/a) k=0
×f,,k ( r )
if be77,
× ga,~-(r)
if be77+½,
(7)
A ( N ) = [UTNU -' , V U e F ] . This subgroup for N = 2 , 3, 4, 5 coincides with the standard principal congruence subgroups F (N) o f F ~3 and therefore is o f finite index. F o r N>~ 6 the index o f A ( N ) in F is infinite. F o r c < 2 the finite-dimensional representations of the m o d u l a r group are generated by m o d u l a r forms on finite index subgroups o f F. There are two classes o f finite index subgroups o f F, the congruence subgroups a n d the non-congruence subgroups. The b e h a v i o r o f a m o d u l a r form on a subgroup F ' d e p e n d s crucially on F ' being a congruence subgroup or not. In fact it turns out that m o d u l a r forms on non-congruence subgroups are not relevant for C F T since their F o u r i e r coefficients have ~2 An important reason for this is that the modular group is one of the so called Type-If groups. For none of Type-II groups we have a complete classification of their irreducible representations. ~3 On congruence subgroups we are using the notation of ref. [ 6 ]. 428
g~,b(--1/z) = ~ a- i r 2z ~k=o -I ~ exp[izc(2k+l)b/a] X£,k+w2(r)
ifb~77,
xg~,k+~/z(r)
ifb~Z+½ ,
(7')
F,j,( r + 1 ) =exp(ircb2/2a)£.t,(z)
ifa-beZ,
=exp(i~zb:/2a) g~,l,(r)
if a-be77+½ ,
(8)
and similarly for g~,b. Thus F~,~,-£,b/~1 where ~/is the D e d e k i n d ~/-function are m o d u l a r forms for some F ( N ) and if a-be77 then N = [4a,24] ~5. Thus at this point we can invoke the S e r r e - S t a r k
~4 Some general results related to the discussion above will be presented elsewhere, ~5 By [a,b] we mean the lowest common multiple of a, b.
V o l u m e 217, n u m b e r 4
PHYSICS LETTERS B
theorem [ 6 ], which states that any 1-singular m o d u : lar form on some F ( N ) is a finite linear combination of the functions F,,,b (r). Accordingly the general partition function for a rational c = 1 CFT can be written up to overall normalization as z(r,~)
= y~ N~,t,F~,b(r)F~,t;(r) ~,~ - •
(9)
2 F e b r u a r y 1989
sible invariants. Not all of them are independent, though.
L e m m a 2. Let a = a ' p 2 where p is a prime greater than one. Then for every divisor 6 o f a' we have z~,, = z ~ .
(a,b) a.F
There are three basic constraints on (9). The first is modular invariance and the second is that all multiplicities should be non-negative integers when one normalizes the partition so that the unit operator appears once. The third constraint is that the theory has to have a consistent operator algebra. We will return to this point later on. The first step will be to construct all possible modular invariants of the form (9). We call a in Fo,b(r) the " l e v e l " % From (7) it is obvious that b~Z and from (8) that a~2V +. Since the action of S a n d Tdoes not change the level o f the modular form F,,b(r) we can classify the invariants at a given level separately.
L e m m a 1. Let a e Z +. Then for any divisor 6 o f a there exists an invariant Z~. The invariant generated by ~ = a / 6 is the same. The form o f the invariant is given by d - I ,~-1 K=0 2 = 0
+ F~,,a+~.~+~a(r ) f f ,,~+,~,r_~( ¢) ] .
(10)
Proof. The line of arguments follows the discussion in ref. [ 7 ]. To save space we will just sketch the proof. Invariance under T implies that b 2 _ b' 2 = 0 rood (4a) if the term Fa,b(r)Fa,b,(¢) is to appear in the partition function. Choosing 6 such that g/a, S = a / 6 we see that ( b - b ' ) ( b + b ' ) = 0 r o o d ( 4 6 5 ) , and the solution is b = 6 m + ~n, b' = 6 m - J n , m, n~Y_. One then can show by using the arguments in ref. [ 7 ] that these solutions for every divisor 6 o f a generate the commutant o f T. We can then check that the partition function invariant under T is also invariant under S by using (7). Restricting b, b' in the domain 0 ~
Proof. The proof is obvious once we establish the following identity: N--I
y, GN2 N(2ak+/,)(r) = G , t , ( r )
,
( 11)
k=0
which follows trivially from the definition (5) by appropriately splitting the summation modulo N.
Theorem. Let a e Z +. Let ~ be a divisor of a, 3=a/(~ such that the greatest c o m m o n divisor (6, ~) = 1. For all such pairs 6/cithere exists an independent invariant Z~ given by (10). Moreover, this list of invariants is complete. Proof. Let 6 be a divisor of a such that (6, ~ ) > 1 than by lemma 2 this invariant can be reduced to an invariant at a lower level a' = a / ( & if)2. At a given level, invariants for different pairs ( & 6 ) and (6', 6' ) are distinct by inspection. Furthermore it is easy to show that relation ( 11 ) is the only relation between forms at different levels. Completeness follows from lemma 1. It is obvious that the invariant Z~ is equal to the toroidal partition function Z ( R ) (see ( 1 ) ) at R = ,xf~2t~ Duality is the statement that Z~ = Z~. Thus we have proven that an arbitrary modular invariant partition function is a finite linear combination o f the invariants Z~ (toroidal partition functions). The final step is to impose the constant of nonnegative integer multiplicities. This question has been answered in ref. [3]. In order to make the line o f thought clear we will present the statements without proofs. Details for the proofs will appear in ref. [ 8 ]. We will distinguish two cases. The first case is when no chiral (1,0) operator appears in the partition function. In this case there is no U ( 1 ) symmetry and we have to write partition functions in terms o f Virasoro characters. 429
Volume 2 17. number
4
PHYSICS
The general partition function can be written as a finite sum Z=C~~,c,Z(R,), c~!,c,=l, in order to have a unique (0,O) operator.
Lemma 3. The multiplicity of (s’,O) operators, for Z, and is 1 SFH+, s<5 is given by 1+2[s/N] otherwise. In particular Z, contains three (1,O) operators and any other Z( R,) only one. The constraint that there are no ( 1,O) operators the partition function forces c, = - 1.
in
Lemma 4. For all i, 2c,ch. This is so because we can show that in the opposite case there will always be fractional multiplicities in Z. Lemma 5. Except for c, = - f all other c, have to be zero or positive. This is true because we can show that if one of them is negative there will be at least one state that appears with a minus sign in 2. Theorem. Any partition
function Z(7, T) not containing any ( 1,O) operators has to be either of the form (2) or of the form (3). This follows for lemmas 3, 4, 5 and the constraint that operators of dimension (?,O ), s< 5 have positive integer multiplicities. The only case left to consider is the case that there exist ( 1,O) operators in the theory, so that states are in representations of a chiral U( 1) algebra. If the number of operators ( 1,O) is
430
LETTERS
B
2 February
1989
N, > 0 then c, (the coefficient of Z) has to be N, - f _ Lemma 4 still remains true for the same reason, and lemma 5 extends trivially in the statement that c, > 0 Vi. From the above it is obvious that there are only two possible solutions, either the torus models Z=Z(R)or2=f[Z(R)+Z(R’)].Themodelswith partition function 2 do not have a consistent operator algebra. In particular they do not satisfy the requirement of additively conserved U( 1) changes. Thus they do not correspond to CFT models. Thus our claim is proven. There are no other rational CFT’s with c= 1 except those in ref. [ 11. I would like to thank R. Dijkgraaf nating discussions, and P. Ginsparg manuscript.
for very illumifor reading the
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