Quantum dimensions and modular forms in chiral conformal theory

Volume 242, number 2

PHYSICS LETTERS B

7 June 1990

QUANTUM DIMENSIONS AND MODULAR FORMS IN CHIRAL CONFORMAL THEORY I n - G y u K O H 1,2, St6phane O U V R Y 3 and Ivan T. T O D O R O V 4

Division de Physique Th~orique s, Institut de Physique Nucl~aire, F-91406 Orsay Cedex, France Received 15 March 1990

Extended chiral field theories with an internal quantum symmetry are characterized by a modular form M(z) covariant under a subgroup F°Nof F=SL(2, Z). This subgroup is known to be the modular group of a theory with twisted boundary conditions. To each modular invariant partition function Z of the ADE classification corresponds a form M but some of the M's are images of two different Z's.

1. The far-going parallels between q u a n t u m universal enveloping ( Q U E ) algebras (or " q u a n t u m g r o u p s " ) and rational conformal theories [ 1 - 6 ] originate (at least, in p a r t ) in E. Verlinde's observation [7] that the ~ ( 2 ) ~ current algebra fusion rules [ 8 - 1 0 ] are r e p r o d u c e d by products o f SU ( 2 ) characters ch/(0k), for O k = n / ( k + 2 ) satisfying chk+ l(0k) = 0 . The value

dq(1)=cht(O~)- qt+l q _ q q- i- l - i -- [ l + 1 ] q=e in/(k+2)

( 1)

o f the character plays the role o f q u a n t u m d i m e n s i o n [ 11,1 ] o f the finite-dimensional representation l o f the q-algebra

Uq-~ Uq(Sl( 2 ) )

(2)

(l twice the spin). A t t e m p t s were m a d e to use Q U E algebras for describing h i d d e n gauge symmetries o f chiral conforl On leave from the Physics Department, Korea Advanced Institute of Science and Technology, P.O. Box 150, Chongryang, Seoul, Korea. 2 E-mail koh%[email protected] 3 Also at" LPTPE, Universit6 Pierre et Marie Curie, F-75230 Paris Cedex 05, France. 4 On leave from the Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, BG-1784 Sofia, Bulgaria. Unit6 de Recherche des Universit6s Paris XI et Paris VI associ6e au CNRS.

mal models [ 12-15 ] ~l as well as their integrable deformations [ 16,17 ]. The chiral vertex operators o f a conformal field theory ( C F T ) [ 10,4 ] are substituted in this approach by chiral fields carrying an internal " q u a n t u m " (say Uq) index. This allows, in particular, to diagonalize all b r a i d ( a n d fusion) matrices [ 15 ]. The state space o f such a theory is a direct sum o f tensor products o f finite d i m e n s i o n a l Q U E spaces a n d conformal Hilbert spaces. I f we view the chiral theory as a building block for a two-dimensional C F T then the Q U E multiplets o f conformal primary fields carry information about the corresponding local two-dimensional fields. Different m o d u l a r invariant theories (like the tetracritical Ising and three-state Potts m o d e l ) with the same central charge are represented by different chiral fields. One can also speculate that the chiral theory has an interest o f its own, or as a building block o f a string theory. The purpose o f this p a p e r is to introduce a m o d u lar form M ( z ) which carries i n f o r m a t i o n about the field content o f the theory (allowing to restore the q u a n t u m d i m e n s i o n o f the internal space associated with each field). To each m o d u l a r invariant p a r t i t i o n function Z ( r , f ) = ~ N~uZ~(r)Zu(z)

(3)

2,,u

#1 The lectures [ 15 ] review, in particular, unpublished work by D. Buchholz, I. Frenkel, G. Much and I.T. Todorov.

0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)

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PHYSICS LETTERS B

of the ADE classification [18] we make a corresponding "chiral modular form" M(r) in which the conjugate character Zu is substituted by the quantum dimension dq(/~) = lim '~It(~) ~ o ;~o(T) '

(4)

interpreted as a characteristic of of the internal space of the corresponding chiral field. It is demonstrated that the form

M(z) = ~, Na.udq(It)Za(r )

(5)

3.,It

is covariant under the modular transformation TST (and its inverse, STS), where T and S are the familiar generators of F: T: z--,r+ 1, S." z ~ -

1

-. T

(7)

The transformations T N and TST generate, in general, a subgroup of the group F°={(;

bd) eF, a , d = + l ( m o d N )

b=O(mod N) } ,

(8)

which is the modular group for a theory with twisted boundary conditions [ 19,20 ] (an "orbifold theory" in the current terminology). We conjecture that M ( z ) is in fact a modular form with respect to the whole group (8). 2. The quantum dimension ( 1 ) is a ratio of S matrix elements where S is the modular inversion (6)

[7] S'o dq( l)= s~ .

dq(l)=dq(k-l) .

(10)

Notice also that the quantum dimension is real. We are now in a position to give in table 1 the complete list of the su (2) modular forms ( 5 ) associated to the invariant partition functions [ 18 ]. The particular combinations which enter in these modular forms have been derived by direct computation. Quantum dimensions are regarded as numbers invariant under T and S whereas the Kac-Moody characters transform as

T: Z,(r+ 1) (~ . F / ( l + 2 ) =exp].zm L 4 ~ ) 2 4 1 (k_~2)]}

= k~2

Xl(Z),

k sin ( ~ (/+l)(/'+l))k+2 Xr (z) . t~o

(ll) Note also that under TST (z---,z~ ( z + 1 ) ) and T N the forms transform covariantly with multipliers qk/4 and ( - 1 ) k . As stated in the introduction, the correspondence between modular invariant partition functions and chiral modular forms are that the conjugate character in the former is substituted by the quantum dimension in the latter. However this correspondance is not one-to-one: indeed the modular forms associated to A4p_l and D2p+l are the same, due to the property of the outer automorphism (10). The same is true for the forms associated to D~o and E7, where one also uses dq(8) =2dq(2) for q~,= _ 1. Thus two different Z's are associated to the same M if they are related by an automorphism of the fusion rules [ 11 ]. The results given in table 1 can be generalized to other current algebras. For example, in the case of su (3) l the modular form

dq(e)Z~.~3)+dq(D)Z~3)+dq(D)X~ (3) (9)

The symmetry of the extended Dynkin diagram associated to the su (2)k Kac-Moody algebra gives rise 206

to the outer automorphism l ~ k - I which leaves the quantum dimension invariant,

(6)

On the other hand, each character Z~ in a rational conformal theory is multiplied by a 2-independent phase factor for z-~z+N with a suitable N. For an (2) k theory we have

TNza(Z)=(--1)kZ3-(Z), for N = 4 ( k + 2 ) .

7 June 1990

(12)

transforms under TST and T 12 with multipliers q2/3 (q = e i"/4) and 1. A last comment on table 1 relies on conformal embedding [ 21 ] that allows to derive a non-diagonal

Volume 242, number 2

PHYSICS LETTERS B

7 June 1990

Table 1 ADE su (2) modular invariant partition functions and chiral submodular forms which transform under TST and T U(N=4 (k+ 2 ) ) with multipliers qk/~(q= exp [in/(k + 2) ] ) and ( - )k. The modular forms corresponding to A4~_~and D2p+t, and to E7 and D~o,are the same. The Ztdenotes the su (2) Kac-Moody character of isospin l/2. Type

Parameter(s)

Modular invariant partition functions

Chiral submodular covariant forms

Ak+ 1

k = l , 2 ....

k ~ ]Ztl 2 I=O

I=0

k

Z d.(t)z~

k/2-2

D(k/2) + 2

k=4p, p= 1, 2,. . . . . . .

~ IZt+Xk-tl2+2lXk/zl2

I~0

I=O

even k/2--2

D(k/2)+ 2

E6

~ IZtl2+lXk/212+ ~ (XtZ~-t+C.C.)

k=4p-2, p= 2, 3, ...

even

k = 12

]Zo "~'X612-]- IX3 ~-)~7 ]2"[- IX4 +ZlO [2

1=0

l~ 1

dq(l)x~+do

Xk/2+ ~ [do(k-l)x,+d,(l)Xk-~l

I~0

odd

even

/=1

odd

[dq(O)+dq(6) ]()~o+,~)+ [dq(3)+dq(7) ](X3+Z7) + [dq(4) +dq(10) ] (X4+Z~o)

E7

k= 16

1~0"~-X16]2"]-IX4-~/~12[2"~"1t(6"~10 [2 + IX812+ [ (Z2 -[-)~14)/~ -{-C.C ]

[dq(0) +dq(16) l (Xo+X,6) + [dq(4) +dq(12) ] (X, +X,z) + [dq(6 ) +dq( 10 ) ] (X6+Z,o) + do( 8 )Zs + [dq(2) +dq(14) Ix. + d.(8) (Z~+Z,,)

Es

k = 28

[Zo-]- ~(10 "~Xl8 "~-X28 [ 2

[dq(0) +do(10) +dq(18) + dq(28) ] (Zo+X,o +X,s +X2s) + [dq(6) +dq(12) +dq(16) +dq(22) ] × (X6+Xl2 +Xt6 +X22)

"~ 1~6 "[-XI2"~16 "~2212

m o d u l a r form o f su ( 2 ) 4 from the diagonal from ( 12 ) o f su ( 3 ) i. The b r a n c h i n g rules from su ( 3 ) to su ( 2 ) are ~su(3) ---~/~0Jr'Z4 ,

where y~ k) (0, z) is the level k K a c - M o o d y character in the integrable representation 2 (twice the spin), and Z ~ . (z) is the coset character. U n d e r the m o d u l a r t r a n s f o r m a t i o n z--. - 1 / z one has 2Jk)(0, - - 1 / Z ) = ~ S(k)~'x~k)(O, z) ,

Z ~ (3) --~Z2 ,

2'

(13)

Z ~ (3) -~Z2"

Y, Sau~ ~'u'~'Z~,u,.,(z),

Z~u~(-1/r)=

(17)

,Tt'lt,t~ r

It follows that ( 1 2 ) branches to

dq(O) (Zo +Z4) + d q ( 2 ) Z 2 ,

where the coset S~Ul ~' matrix elements are (14)

(q = e ~"/6) that corresponds to the m o d u l a r form associated to D4 in table 1. 3. It is well-known that one can construct m i n i m a l conformal theories with central charge c = 1 6 / ( k + 2 ) ( k + 3 ) by taking cosets o f K a c - M o o d y algebras as

S;~'u',,' = S ( k ) ~ ' S ( 1 ) u ' S ( k + 1)71~, ~.uv = S ( k ) ~ ' S ( l )u'S(k+ 1 )*~'

(we have used the unitarity and s y m m e t r y properties

SS*= 1, S T = S ) . In anology with eq. ( 9 ) we take the q u a n t u m d i m e n s i o n o f the coset to he ~'000

dq(2/tv) - ~'au. cooo " o000

su(2)~ Xsu(2)l SU(2)k+l

(15)

p

(19)

This leads to

dq(2~tv) = S ( k ) ° S ( 1 )° S ( k + 1 ).o

The character f o r m u l a for this coset is given by

z~k)(o, Z)X~u')(O, Z)= ~ ZZu~(Z)Z~k+~)(O, Z ) ,

(18)

S(k)°S( 1 )°S(k+ 1 )~o (16)

=dq,(2)d~(/~)d,3(v) ,

(20) 207

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PHYSICS LETTERS B

with qt =e'~/(k+2),

The simplest off-diagonal (A, D) type is given by q2 =eni/3,

q3 =eni/(k+3) •

(21)

The integrable representations of level 1 are given by 0 or 1 with congruence class (number of Young tableaux) 0 or 1. In addition we have cong. class (2) + cong. class (/z) =cong. class(v) rood 2,

(22)

and the representation/.t is uniquely fixed once 2 and v are given. Thus the coset character Zany(r) can be abbreviated to Z ~ ( r ) . Also dq~(a) = 1 for all integrable representations of level 1. Thus for the coset model (20), the quantum dimensions of the primary states are given by dq(,,].p) -~ dql (/t.) de3 ( v ) ,

(23)

where only SU(2)k and SU(2)k+~ appear. We recall first that the modular invariant partition functions of the coset model (15) are given by (A, A), (D, A) and (E, A) where the diagonal forms A ofsu(2)k+~ (or SU(2)k) combine with the forms A, D, E o f s u ( 2 ) k (or SU(2)k÷l) [(D, D), (E, E) and (D, E) types of partition functions are absent ]. To construct modular forms for the coset model ( 15 ) we can follow two paths: either we repeat the strategy followed in the current algebra cases, namely substitute the conjugate character by the quantum dimension, or we directly use the results given in table 1 and by coset construction build the corresponding modular forms. Both paths obviously lead to the same result because ofeq. (23). The simplest (A, A) type is given by

dq( I)zl + dq( a)za W dq( ¢ )Z~

(24)

where/, a, E are the primary operators of the Ising model su(2)l×SU(2)~/su(2)2 with conformal dimension 0, ~, ~, respectively. Quantum dimensions follow from (23) with I = ( 0 0 ) , a = ( 0 1 ) and ¢= (02); they read dq( I) = 1, dq( 6) :,v/2 a n d dq( ¢ ) : 1. The characters in eq. (24) are precisely the conformal characters Z~(r) where (2v) = (00), (01) and (02). The multipliers under TSTand T 48 are exp (hi/ 24) and 1 respectively. F°8 is not the maximal modular subgroup for the Ising model; the form (24) transforms covariantly also under T ~6 with for multiplier exp( - 2ni/3). 208

7 June 1990

dq(l)Zl'l'dq(~-)Z~: q'-dq(~)Xa't-dq(Z)z z

(25)

in the three-state Potts model (su(2)3X su(2) l/su(2)4). The primary operators/, ~, a -+ and Z -+ have conformal dimension 0, 2, ~ and ] respectively. The modular form (25) is non-diagonal because the Z's are linear combinations of coset characters, ~1 =~00 "~-~04, Z¢ =/~10 "JI-/~14 Xa-+ =/~12 ,

ZZ± =X02 .

(26)

But, since the quantum dimensions of Zoo and/~o4 are equal, as well as the quantum dimensions of Z~o and ,~14 [ due to (10) ], the modular form can be cast as in (25), the quantum dimensions read d ~ ( I ) = l ,

dq(~) =sin(2n/5)/sin(n/5), dq(a +-) = s i n ( 3 n / 5 ) / sin ( n~ 5 ) and dq ( Z -+) = 1, respectively. 4. As discussed in the introduction, one can take modular forms in table 1 as characterizing chiral fields with an internal quantum index. Alternatively, one can entertain the idea of using modular forms as building blocks for a string theory with twisted boundary condition. To achieve modular invariant partition functions one can either take both chiral and antichiral modular forms or one can take several copies ofchiral modular forms [for example nine copies of k = 16 modular forms in su(2) ]. Two of us (I.G.K. and I.T.) would like to thank IPN for kind hospitality. One of us (I.G.K.) would like to thank KOSEF for the research contracts. One of us (I.T.) would like to thank Igor Frenkel for suggesting the idea of considering a chiral modular form involving quantum dimensions in the spring of 1989. It is also a pleasure to thank Jean-Bernard Zuber for useful discussions (in particular for pointing out the equality between M(E7 ) and M(D~o) ). References [ 1] N.Yu. Reshetikhin, Quantized universal enveloping algebras and invariants of links I, II, Leningrad preprints LOMI-34-87, E-I 7-87; A.N. Kirillov and N.Yu. Reshetikhin, Representations of the algebra Uq(Sl(2)), q-orthogonal polynomials and invariants of links, Leningrad preprint LOMI-E-9-88.

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