Nuclear Physics B356 (1991) 387-438North-Holland

QUANTUM GROUP EXTENDED CHIRAL p-MODELS

L.K. HADJIIVANOV, R.R. PAUNOV x and I.T . TODOROV*'

Institute for Nuclear Research artel Nuclear Energy, Bulgarian Academr ofSciences,BG-1784 Sofa, Bulgaria

Received 24 September 1990

The quantum symmetry group U,, of an extended chiral conformal model is determined bythe requirement that symmetry transformations commute with braid group statistics operatorsand by the relation between fusion rules and tensor product expansions of a certain class of U,,representations. For thermal minimal "p-models", involving no more than p - 1 unitary lo%`estweight representations of the Virasoro algebra Vir, U,, is the quantum universal enveloping(QUE) algebra U,,(sl(2)) with deformation parameter q satisfying q + q - ' = -7 cos r, /p (qr = -1.p =4.5,.. . ) . To each 2-dimensional local field labelled by a pair of nonnegative integers v . v(0 < v, v < p - 2) we make correspond an analytic chiral field 0,, of weight -1,, and q-spin I,, .The correlation functions of 0,, transform under an 1-dimensional unitary representation of thebraid group . As a result we reproduce the ADE classification of 2-dimensional p models interms of their extended chiral counterparts . It turns out that U,,extended chiral p-modelsalways involve non-unitary and indecomposable representations of Vir .

1 . Introduction

A rational conformal field theory* (RCFT) is characterized by single-valuedlocal 2-dimensional (2D) euclidean Green functions that can be written as finitesums of products of (multi-valued) analytic and anti-analytic ("right and leftmovers"') conformal blocks. The space of such (say, analytic /I-point blocks)carries a finite-dimensional representation of the braid group Z � [6-10] . It hasbeen noticed (first in the framework of the lattice models [11,12]) that similarrepresentations of Z � are generated by the universal R-matrix associated with anappropriate QUE algebra (see refs . [13-15] and references therein) . The analogybetween quantum groups and RCFT has been subsequently explored in a numberof papers [4,16-30] . (For another line of development see refs . [31-33].) In

* Present address : Math. Phys. Sector, International School for Advanced Studies, I-34014 Trieste,Italy .

* ' Present address: Division de Physique Théorique (Unité de Recherche des Universités Paris 11 etParis 6 Associée au CNRS), Institut de Physique Nucléaire, F-91406 Orsay Cedex, France .

* The abundant literature following the seminal work of Belavin, Polyakov and Zamolodchikov (BPZ)[1] can be traced from the reprint volume [2] and from recent reviews (see refs. [3,4] among others).The term RCFT was first introduced in ref. [5],

0550-321 .1/91/$03 .50 «') 1991 - Elsevier Science Publishers B.V . (North-Holland)

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L.K. Hadjiicwrtoi- et al. / Chiral p-models

particular, the idea has been put forward [18,21,22] that substituting chiral vertexoperators (CVOs) by multicomponent fields which transform as (irreducible) Uqtensors one can trivialize the braid group action. We demonstrate in the presentpaper that this program can indeed be carried through, but at a certain price : oneneeds to extend the physical Hilbert space of the conformal theory by includingnon-unitary (and indecomposable) representations of the Virasoro algebra Vir.A given set of CVOs corresponds, in general, to more than one local 2-dimen-

sional model, i.e . to more than one set of 2D local (Bose) fields . The operatorcontent of a local 2D theory is reflected in its (modular invariant) partitionfunction [34] . The study of such partition functions yielded the ADE classificationof minimal (and su(2) current algebra) models [35] . The resulting 2D primary fieldscan be written as finite sums of products of analytic and anti-analytic (right andleft movers') CVOs. The key to the construction of chiral primary fields withinternal quantum numbers is to replace the anti-analytic CVOs by appropriate Uqvertex operators (gVOs). This allows us to reproduce the ADE classification in theframework of extended chiral RCFT.We outline more precisely the problem and its solution in sect . 2 and then

proceed to the computation of 4-point functions, thus verifying our statements inthe simplest nontrivial examples .A preliminary account of some of our results (concerning "diagonal" or_"A-

mseries" models) is given in ref. [3é,1 . Si~. . . .iIar results can be obtained for the su(2)current algebra. A more conceptual appro c:: to that case is developed byGawedzki [37] (who relates it to unpublished work of Felder).An alternative view on the quantum symmetry of the chiral Ising model has been

developed by Mack and Schomerus (see ref. [38] pp. 388-427) who apply to it thetheory of superselection sectors exhibiting morphisms of the chiral observablealgebra that generate all positive energy representations of Vir with c from itsvacuum representation.

2 . UQ-extended chiral p-model. Statement of results

2.1 . PRIMARY CHIRAL FIELDS. RELATION BETWEEN q-SPIN AND CONFORMAL WEIGHT

One way to define a quantum field theory consists in displaying the vacuumrepresentation of a (local) "algebra of observables" and a set of primary "charged"fields (commuting at space-like separations with the observables) whose repeatedaction on the vacuum generates the entire state space // of the theory . (Theseparation of the fields into local observables, which leave the vacuum sectorinvariant, and charged fields, which intertwine the vacuum with other superselec-tion sectors, is borrowed from the algebraic approach of Haag-Kastler-Doplicher-Roberts whose present status is reflected in refs . [38-40] .)

The observable algebra of a minimal chiral model always contains (and in two ofthe three infinite series of models is generated by) the stress-energy tensor

T(z) _

LnZ-n-2

11EZ

whose Fourier-Laurent modes L� together with the central charge c span theVirasoro algebra Vir. (In all other cases the observable algebra is generated by oneor three integer-spin primary fields.)A unitary minimal model [1,41] corresponds to

where

c = c(p) = 1 - 6/p(p - 1),

p=4,5, . . . ,

(2.2)

and involves -1,(p - 2)(p - 1) unitary irreducible positive energy representations ofVir (or a subset thereof closed under the BPZ fusion rules [1]). It turns out thatevery diagonal minimal model as well as any non-diagonal model for even pcontains a (closed) submodel generated by the 2D primary fields corresponding tothe first line of the Kac table, cf. refs . [1, 42] . The chiral (right movers') projectionof this line contains p - 1 unitary lowest weights

2 PJ1�

L .K. Hadjiivanor- et al. / Chiral p-niodels

r = (p - 1)/p,

[ Ln ') L,,,]= (n - m)L,,,,n + l'-2cn(n2- 1)Sir+M.o 1,

389

,k � = (Zv+ 1)r- 1 = 4v+ 4 1 - dv -1,

(2 .3a)

v=0,1, . . .,p-2 . (2 .3b)

(This set, as well as its subset corresponding to even v, is closed under the fusionrules and hence gives rise to a quantum field theory.) The 2D primary fields of ap-model Q(z, z; v, v) are thus characterized by a pair of conformal weights av anddv [of the set (2.3)] restricted (by 2D locality) to integer "helicity":

dv -4v EZ

(v,vE10,1, . . . . p-2}) .

(2.4)

The operator content of p-models is verified to be directly related to the ADEpartition functions of su(2)k current algebra models [35] for p = k + 2*. To eachsuch a we make correspond a (v + 1)-component chiral field 0� (of conformalweight 4,) which can be represented as a polynomial of degree v of a formal

*This is an optimal procedure for the An _ , and the D21+3 (p = 41 + 4) series . The cases of the

Dz1+2 (p = 41 + 2) series and the exceptional E series are treated in subset . 4.3 .

390

L.K Hadjiiranoi , et al. / Chiral p-models

variable iil'

.z , l1

v

' /7-

Il

Il =o

Here ¢,,�, (fit being the q-analog of the "magnetic quantum number") transformscovariantly under the third q-spin component (the Cartan generator of Uq):

q/]J3

J; +Ill~l~Ill

_~l" Illq

We are using the q-deformed binomial coefficients

(2.5a)

(2.5b)

n I ,

11 ! v-)l

[11] :=

- -111

. (2.6a)[j

qq

The value of the deformation parameter q (a root of -1) is related to the label pof the model by the quadratic equation

[2] =q+q- ' = 2cosT/p ( => q=e±`-- /P) . (2.6b)

The fusion rules of the theory are given by a symmetric cubic matrix {NA ,,,,,} ofelements 0 and 1 [43] (NA,,,, = 1 iff the invariant 3-point function~OA( zl , 111)Oj z,, 11, )O,,( z Y , It3)> does not vanish . The correspondence vH 2 I� = vbetween conformal labels and U,, weights satisfies the following basic property :NAB� , majorizes the BPZ fusion rules and is majorized by the rules for addition ofsu(2) spins :

BPZ

ti inNAliv

< NAj.v < N~ ,, .

for the spin and the BPZ fusion rules the upper bounds are

nspill -min(A,~,) =-',(A +~, - JA -t,I),

1, BPZ =min(nsp in,s,i", p-2-

(2.7a)

The BPZ fusion rules depend on the algebra of observables. We reproduce herethe formulae for the unextended case (in which the vacuum sector is generated bypolynomials of the stress-energy tensor T acting on (0>) . All three N's vanishunless

v= 1A-,u( +2n,

n = 0,1, . . .,n� ;

(2.7b)

nspin) .A~

(2.7c)

A further restriction on Iv comes from the braiding properties of the "1 j-symbol"(A.13) which should (and do) coincide with those of a diagonal theory (with v = v).

It would be interesting to formalize the desired properties of the (multi-valued)correspondence v ++ 1,, in order to be able, conversely, to rederive the ADEclassification within the framework of an (extended) chiral theory. We contentourselves in this paper with verifying that the ADE correspondence does lead tobraid covariant chiral Green functions. The fact that the list is exhaustive followsfrom the results of ref. [35] because of the possibility of reconstructing the 2DGreen functions from the chiral ones (sect. 5).

2.2. BASIC PROPERTIES OF CHIRAL GREEN FUNCTIONS

The Uq transformation properties of a field

of q-spin 1 are summarized by

qJ;(p( z, it)q-'; =q-10( z, qtt)

(2.8a)

1+O(z,111 =q-'th(z ; u) .1+ +1,;~( z . it)a-J; .

(2.8b)

where .1� is the finite difference operator

O(z, qu) -O(z,q -'it)1`O(z, u) -1 u'1= [n ]u(2.9)

(for q = e -'"IP we set q 1/2 = ei-' 12p).

The Green functions of the theory

(a) Möbius invariance

LX Hadjiii-atior- et al. / Chiralp-models

391

v�1�

z�u�

are assumed to satisfy the following requirements :

= <O1 cb(z l , u,) . . .O(z", u,t)j 0> ,

il1

dgzi- 1

aziG(gz , , . . .,gz,,)

=G(z,,z2, . .i

where g E SU(1,1) (and we have suppressed the labels Pig 1j, ui)

az + ,8gz = -

_

with air -,8,8 = 1,ßz +ii

where E is the co-unit (A .3) ;

agz ) _ (ßz + â)-2 .az

(2.10)

.,z� ),

(2.1 la)

(b) Uqinvariance which is implied by eq. (2.8) and the vacuum invariancecondition (for any X E Uq)

XI O> = .E(X)10>,

(Le.

q ± J;10> = (Oi ,

J,10> = 0),

(2.l2)

392

L.K Hadjiiranoi , et al. / Chiral p-models

(c) They obey differential equations coming from appropriate null-vector condi-tions;

(d) There are factorization properties associated with small distance expansions ;(e) The monodromy matrices (generated by squares of statistic operators, to

be defined below) act on G multiplying it by phase factors . Moreover, for v, =v, = . . . = v� the n-point Green functions transforms under an 1-dimensionalunitary representation of the braid group IZ� .

(f) The normalization of the 2-point functions is

v 1E ~i,,-nZi,2_1jq-1 /2u,(- )q '/2 u2~ - ,

z,

zt,

(z,, = z, - z,) where 11)""21" is a homogeneous polynomial of "U, 1'2 of degreev satisfying

1'cI [L.,(-)(~21v- [v~[L~1(-)1..2,v -' - -. ~.,LZ,1(-)L~2,v~

[",(-)c2]0 = 1 .

n=0,1, . . .},

(2.13)

(2.14)

Its explicit form is given by eq. (A.13) and it can also be characterized by therecurrence relations (A.20) and (A.25).Requirements (c) and (d) need some specifications . We assume that the "null

vectors" of order v + 1 associated with the lowest weight states

Ov(0, u)l0) = 1 v, Il)( = 1p ; v, u)) ,

(2 .l5)

that have, according to ref. [1] the form (L"_̀,' + . . . )J v, u), vanish in

7,1' ; inparticular,

L_ 1 10) = 0,

(L2 , - rL- 2 )11, u) = 0,

(2.16a)

(L3_ 1 -4rl,_ 2 L_, +2r(2r- 1)L- 3 )12,u) =0 .

(2.16b)

The second set of null vectors associated with 1p; v, u), at level (p - 1 - v)(p - 2),gives rise to an invariant subspace in the Verma module (of lowest weight 4� fromeq. (2.3) and central charge c(p) from eq. (2.2)). These null vectors cannot be setequal to zero, since they will be found, as a consequence of our requirements, tohave non-vanishing matrix elements with some vectors in ~V. The simplest exampleof such non-zero null-vectors is encountered in the Ising model (for p = 4). We

have

but we shall find that

.

lu, vG(4)- ( - 1 ) Av

U2�n=o Z1 Z2

L.K. Hadjih-anot- et al. / Chiralp-models

L�(L? 1-3L-2)14;2,u>=0, n=1,2, . . ., (2.l7)

<0I02(ZPul)02(Z2 'y u2)(L- 1 - 3L-2 )14;2,u> *O .

(On the other hand the vectors 14; 2, u ) do satisfy eq. (2.16b).)We shall spell out the factorization properties (d) for the 4-point functions in a

diagonal theory (suppressing the index It, = v/2)

lu,

v

vG(4) =G1

Z lu l Z2u2 Z3u3 Z4U4

It satisfies two mutually dual expansions (s- and u-channel)

lu, v v lu,G(4) = Y Sv-E.c +2it

r1=0

Z1 Z2 Z3 Z4

tu, /2

v/2

v/2

u/2x J;, +(v - /1)~2

,ul 112 u3 u4

Here ~; and l/, are the U9 blocks defined in appendix A (see eqs. (A.15), (A.17)),while S� and U2� are Möbius-invariant conformai blocks satisfying the shortdistance relations

v

lu

tL/2v/2

v/2

lc/2ll�

;z3 z4 u l u 2

u3 u4

l,t. < v .

393

(2.18)

(2 .19a)

(2.19b)

for Z23 --~ 0 .( 2.20b)

lu,Z

34~+.]

SA

v vCAv , ,t,(O I01,1,( Z1) Ov( Z2)OA Z3) 1 0>')

Z 1 Z2 Z3 Z4

for Z34 -* 0 .) (2.20a)

Z2.1,,- .12,,U2�

v vCvv2i,< () 101£( Z1) 02n( Z2)Og( Z4 )I0 1

z l z2 z3 Z4

39-1

L. K. Hadjiirarrot- et al. / Chiral p-models

Here q6j z) are the chiral vertex operators of refs . [6,41 (so that

<Oi~~.( Z1)~( Z2)~A( Z3)i0> = COi ( Il

0

il ) .

v

~

h )ZÀ

~

0 )Zi0i >

cf. subsect . 2.1I . The z-dependence of the 3-point function in eq . (2.20) is known:we have

0i 0u(Z 1)0J Z,)0A(Z3) i 0>

CArc!' Z 12A _,~` -,t Zl -,A-,AZ A

21)3 -,A -,~~ ,

(2 .

CAU ,, being the "structure constants". Energy positivity implies that any n-pointcorrelation function is analytic and single-valued in the complex neighbourhood ofthe real open set

3 .1 . DUAL SETS OF CONFORMAL BLOCKS

Z1 >Z,> . . . >z�- 1 >z� > -z� -l'

3. Four-point functions involving a pair of step operators

(2.22)

If we choose an arbitrary normalization for the 4-point blocks then each term in(2.19a, b) should contain a coefficient a� , b� , respectively. The statement is thatthese coefficients can be chosen in such a way (in accordance with (2.13)) that theduality condition (2.19) holds . The braid group action on such 4-point functions(with A, = v) is given in sect . 3 . Here we note that the set of free phase factors(fourth roots of unity) in eq. (2.13) is mapped into itself under the (cyclic) braidgroup '3, (see eq. (3.33b)).For v = A the 4-point function (2.18) is determined uniquely up to an overall

phase factor, provided that 2 v <p - 1 . For 2 v =p we exhibit, in the case of theIsing model, a solution that depends on two additional parameters . For t, 0 vindependent phase factors (powers of i) appear, in general, in each term of theexpansions (2.19) .We introduce in sect . 5 . an Uq-invariant hermitean form in the tensor-product

ring of "physical representations" of U,1 . Its restriction to Uq-invariant blocks givesrise to a positive semi-definite braid-invariant inner product, allowing us toreconstruct the monodromy free 2D Green functions.

We proceed with the proof of the statements of sect . 2 . starting with the simplestcase of the 4-point function (2.18) for A = 1 . The field ¢ 1 (of q-spin 1/2) is termedfollowing ref. [11 a step operator .

L.K. tladjih-anoi- et al. / Chiral p-models

395

Skipping for the time being the {c` ;}-dependence we can write Gc4' in this case as

v vG(4) = Z-2-IG

14 11Z W

Z = Z13Z24/Z149 W=Z12Z34/Z149 Z-W=Z23 ,9 3.1

where G� is a homogeneous function of z and w of degree -2A,. The level-2null-vector condition (2.16a) implies the second order partial differential equation

2 1 1 4,, d(a_ +a�.) +Y a

-+ -a�.-

- 2"»G11(" , v =o-

(3.2)(Z W z

2w z W

The space of homogeneous solutions of eq. (3.2) is 2-dimensional. It is spanned byeither the s- or the u-channel blocks

1 v v 1S"±1 - Z232.1�

77-`fv+1(n),Z1

Z2Z3

Z4

v v

z2 z 3

Z

(3 .3a)

1

-:"z23

92n(Z4

here A,, and A 1 , are given by eq. (2.3), -q is the Möbius-invariant cross ratio

W Z12Z34 I Z14Z231-i= ,Z13 Z 24 Z13Z24

while fa and g,� obey the related hypergeometric equations

(3.4)

~ ( 1-~1)f~' + {2 - (v+1)Y-(4-(v+3)r)r~}fâ+2a,,(1-Y)f~,=0, (3.Sa)

ri( 1-71)g2rt + {2-2Y -(4 -(v+3)r)(1-r])}g?rt +2h,,(1-Y)g2rr =0,

( 3 .5b)

3y(ß

L.K Hadjin -anon et al. / Chiral p-models

and are singled out by the initial conditions

fl. - 1(0) = Ci . e~- ICI,- 1 ~~ 1

(r+ 1)r- 1f , + 1(i) ~ Cl e.Il+ iCI , 1 l e, 1 71

go(0, v) = 1,

g2(1--

1l'1 v) ,., Cr , . 2C121( 1 -71 )2r-1

for

71 --> 1 .

(3.6b)

(The C,,,,i, are the structure constants of eq. (2.21) .)If we fix, arbitrarily, the structure constants by setting

F(r)F((v+ 1)r- 1)

F2(r)F((v+2)r- 1)Cll i ) i,- 1 =

+

Ci(~)2C1t 2) 1 -F(2r- 1)F(vr)

F(2r)F(2r- 1)F(vr)

C(o)

C(o)

- F( r) F((v + 2) r - 1)It'v+1 ~.+1~ , 1 - F(2r- 1)F((v+ 1)r) '

then we find the fusion (or duality) relation

go(1 - n' v)

__

fz. -1(~1)g2(1 - 7l , v)

FO

fv+ l( 7%)

'

for

[v+2]

[2][v + 1]

[v+ 1]

where the q-numbers [n] are defined by eq. (2.6). A straightforward derivation ofeq. (3.8) is given in appendix B. We only note

°

[2][v+ 1]

sin-rr(I -r)

which remains true for the normalized F-matrix (see eq. (3 .29) below)

0, (3 .6a)

(3 .7)

(3.8)

F.--

1

0

Fo

«,,

0

,(3 .l0a)Yv 0 ß,

where

which requires

L.K Hadjdranov et al. / Chiralp-models

397

c(0)1,-,C(O)

Co) C(o)IV

v-Ivl

~

1vv+I v+I v lF' =at._

Clvt,-lCv-Ivl "

v

Clvv+lcv+Ivl,

Cvv2Cl21Yv

C(o) C(o)

(a"Pvyv

1)

3.10bvv2 121

(see eqs. (3.7) and (3.26)) since it is consistent with the involutivity property

F2 = l,

(3.11)

a? - [2] [v ]/[v + 1],

avß"Y" = 1-

(3.12)

3.2. HECKE ALGEBRA REPRESENTATION OF THE BRAID GROUP $4 FOR v = 1

Consider an arbitrary n-point conformal block as an analytic function ofz,, . . . , z� with primitive analyticity domain given by a (simply connected) complexneighbourhood of the set (2.22) (in which Izi I > Izi+ I I for i = 1, . . . , n - 1). Todefine a permuted block we have to specify a homotopy class of paths in C" withno coinciding arguments, along which the original block is to be continuedanalytically . The set of all path-dependent permutations is generated by the basicstatistic matrices Ri, i = 1, . . . , n - 1 (and their inverses) . The operator Ri = R<<"+,permutes a pair of neighbouring factors 0" (z i ) and 0"'jzi + l ), involving analyticcontinuation along a path that exchanges zi and z i + I in the positive direction sothat

Ri :

Zii+l ---> e"Zii+1 , (zip Hzi+lj

for

j *i, i + 1) .

(3.13)

Coming to the 4-point blocks we observe that the operators R, and R3 arerepresented by diagonal matrices in the s-channel basis (2.20a) (times a permuta-tion of the associated field labels). Their eigenvalues are determined by the actionof R I and R3 on the corresponding 3-point functions:

( RI"')

= ei7r(vA -du S

= (RI.")l a~~

a~~ 3 aa~ . (3.14)

The statistic matrices Ri satisfy the Yang-Baxter equation and are related to thefusion matrix (3.10) (in the case tL = 1) as follows :

R "l "R 1

1I"Rl " = R21"R1l"R"

2" = e -2"e,, F .

(3.15)

39N,

The two equations (3.15) can be used to evaluate R ;`` and R " in terms of F andthe diagonal matrices (3.14). The result for R;" is particularly simple:

We note that the fusion operator F (unlike the elementary statistic operatorsRt ) maps 4-point blocks into 4-point blocks with the same order of arguments. Thisobservation suggests the study of an extension of the monodromy (or pure braid)group by adjoining fusion transformations. Consider for the sake of definiteness4-point blocks of type (Is v Pp,). Then the extended monodromy group Gp =Gp(;, v v0 is the normalizer of a group generated by three elements,

L.K. Nrarljiiawaoa~ ei cal. / Clairal p-eauadels

subject to the relations

m ' F=Fin,,

III

=IIIP ~ Centre(Gp ) ,

III 2pp = 1 .

by normalizer we mean that

R;"= F-'R"F .

III,= (R,.,`,)2;`')`' and

F-' = 1

(nt ; = 1

F, (3 .16a)

(3 .16b)

for

(p - 1)/.L v

even) ;

(3 .16c)

gEGp =~ (R;") - 1 gR1"EGp etc . (3 .16d)

It is readily verified that for tL = v Gp coincides with the monodromy representa-tion of the braid group '0 4 of four strands on the set of (4-point) conformal blocks .(For v <p - 2 its (v + 1)-dimensional representation on the space spanned by theblocks S,� , fi = 0,1, . . . , v is unitarizable ; for 2p - 1 > v >p - 1 it admits a (2p -v - 1)-dimensional subrepresentation.)We display as an example the case u = v = 1 in which a simple Hecke algebra

representation [44) of 934 is realized . In this case the "normalized" T4 generators

bj = q - ' - ej ,

(3 . 1ôa)

bj = e . r12R] =i (Rj = Rj1 ' , .1 1, 2, 3) , (3 .17a)

satisfy the equation

(bj+q)(bj-q-')=0 for q= (3.17b)

Moreover, introducing the Temperley-Lieb projectors ej by

L.K Hadjih-anot - et al. / Chiralp-mortels

399

we find the standard relations

e

_ [2]ei ,

eiei+ ,ei =ei

for i = 1,2

(or i = 2,3) .

(3.18b)

The 2 x 2 matrices e, and e3 are equal and in the s-channel basis characterized by(3.14) they are diagonal :

In a unitary basis, in which [2]'y, = [3]ß, [see eqs. (3.8), (3.10)] the matrix e2 is alsohermitean:

We note that the defining relations for the braid group ~34 generators,

et = e3 =(0

0

,

([2] = q +q- ' = 2cos 7r/p) .

(3.19a)0 [2]

e2

[31

- ffil

([3] = I +q2 +q -2)

[]

=q - '{q-2 - 1 +b lb2 +b2 b, -q- '(b, +b2 )},

(i=1,2),

and

b,b;=b 3b,,

are a consequence of eqs . (3.18), (3 .19) and of the commutativity property e,e3 =e3e, which follows from eq. (3.19a).

Note that the condition F2 = 1 (3.16b) requires the additional relation

b,b 2 + b2b, + (q - q-')(b, + b2 ) = 1 -q 2 -q -2 ,

(3.19b)

(3 .20a)

or

e,e2 + e2e, = [2](e, + e 2 ) - [3] ,

(3.20b)

which is satisfied by the matrices (3.19) . We shall display in this simplest case themonodromy subgroup . .l/4 or 934 and shall demonstrate that its extension GPcoincides with the (Hecke algebra representation of) '84 . The group .//4 is

:I(0

L.K Hadjdranot- et al. / Chiral p-models

generated by

(q-2

0

q([3] - 1 )

(q-' -q) [3]bï =

b2 =0

q 2

(q -' - q)

[3]

q -' ([3] - 1)

b l b261_, =

1

q([3] - 1)

q-'( 1 - q-2 )

[3]

,,

[2]

q(q2 - 1 )

[3]

q -' ([3] - 1)

GP contains in addition the O(2) fusion matrix

i

(i

V-[-3] ,F mi V-

1

Ui2 = 1 ,

QlQ2Ql = U2u1(7-2 = { F} .

3.3. THE (ANTI)SELF-DUAL GREEN FUNCTIONS

m

[v+ 2] 11

(,, +1)/2`

[v + 1][21

[2]

~

(3.21a)

(3.21b)

as well as -F (3.20a) (for even p the central element b P, = -1 = b2 belongs to-1/4 ). It follows from eq. (3.16d) that GP also contains

and, similarly, b, . Hence, it coincides with T 4. The identity b, = b3 implies thatthe factor (matrix) group 184///4.,184///4 coincides with the6-element group Cam ;ofpermutations. Itstwogenerators a; _ (bi ) i=1,2aresubject totherelations

Itiseasily verified that the U. blocks

+1)/2 and 2~0 , I computed inappendixA(see eqs.(A.18)and(A.19))satisfy afusion relation similar to eq. (3 .8)

The fact that the products of the diagonal (and hence also of the off-diagonal)elements of F0 and 9- 1 coincide implies that one can make the transposed of theU.-fusion matrix to coincide (up to a sign) with the inverse of F0~ . Fixing thenormalization of Y() _ .~~ ~ ) and 21o, - //(()') in accordance with eq. (2.13) we find

L.K. Hadjiii anor et al. / Chiralp-models

401

that the desired change of normalization reduces to a couple of sign factors

or

9-o =

0

-(1

0a1( -1)v-I ( 1

0) ~tgOFO

1)v1)

-0 -1

This implies that the Green function (2.18) for ju = 1 is given by

1G(4) _- G(Z I U,

v v

z2u2 z3u3 zau4

(3.23)

G(4)= ( -1 ) vz1 2.1,Z23-2'"n -a`'(go(1 -n,v) -/o+gi)(1- ,q ,v)Z`I), (3.24b)

where f�°) and g2) are defined by eq. (B.6) (gô) = g° ) and

(3.25a)

It is clear that the change in normalization of the conformal blocks

(o) = vl)(

I)/2+n

1

' (

1)/2+n v

Remark.can be compensated by an inverse change for the U9 blocks, only products ofnormalization constants appear in the Green functions. If v + 1 <p - 2 (i.e. if theBPZ fusion rules are respected by the extended theory) we can use the resultingfreedom to make the statistic matrices of sect . 3.2, and hence the fusion matrixunitary. (As a result the relevant representations of the universal UQ R-matrixincluding the representations of the braid group that belong to the commutant ofUq will also be unitary.) This gives the following values of the structure constantsC� 11� introduced in eq . (2.21) and appearing in eq. (3.6) (assuming that they arefully symmetric) :

C lvv-1 -Clv-lv- r(1 -r)r(2r- 1)r(vr)r(2--(v+ 1) r) ' (3.26a)

(v + 1)r- 1_

2_

2Cvv2C121

2r-1

C1 vv-1C1 vv+1

F(r) r(2 - 2r)r(1 - vr)r((v + 1)r- 1)

(3 .26b)

(Eq . (3.26) agrees with the result of ref. [1] obtained by demanding "crossingsymmetry" of the 2-dimensional euclidean correlation functions, as well as themore recent systematic computations in ref. [10].)

Defining the U,, structure constants N,,,_,, in terms of the normalized 3-pointfunction

we find for the (symmetric) unitary normalization

N4

=

[211 ]![211 + 1]![21,]![21, + 1]![2I;]! [2 ~.j + 1]!tst_t; ([I1 +1,-Ij![1;+I1 -1,]![1,+4-1,]![I1 +1,+1~+ 1]!)2

(3.28)

(cf. ref; [151). The corresponding conformai and Uq fusion matrices are thensymmetric and involutive and differ by at most a sign

[v+?]~( [V+ 1]

LK âlîirljiiiimor - et al, f Chiral p-models

1, 1~

Il I, I;I'hIz

a11'

113

11 1

11,

113(3.27)

[v+2][v + 1]

[v]-

[v-+, I

(3 .29)

We observe that for v =p - 2 the matrix F is diagonal, since

[p]=0

forgp= -1 .

(3.30)

For this value of v the conformai structure constants C, vv+ , and Cv v 2 vanish,

C1 p _2 p _, = Cp _ 2 p _22 = 0 . (3.31)

However, !V,.,` , ; diverges for 211 = 1, 212 =p- 2 and 213 =p - 1

so that itsproduct with C, p _, p_ , remains finite . That is why the Green function (3.24)involves for v =p - 2 the non-unitary representation of Vir of lowest weight APwhich is forbidden by the BPZ fusion rules.

Define the U, statistic matrices N ; is the space of Uq blocks by presentingthe generator of the universal R-matrix [13-151 that interchanges the arguments(i,i + 1) as $; i + 1 91 i where 13;1+ , stands for a permutation .

Proposition .

The Green function G4 transforms for v = 1 under a 1-dimen-sional unitary representation of the braid group s~ 4 .Proof.

The Uq statistic matrices 91, and T 3 have a diagonal action on thes-channel blocks (A.15) whose eigenvalues are determined by their action on thecorresponding 3-point functions . Using the expression for the universal R-matrix

of ref. [151 (corrected in ref. [161) we find

L.K Hatljiii°anot- et al. / Chiralp-models

403

_91

h

12

= ( -1) /,+/,-1q-,(,+4+1,(1,+1)+1,( 1,+0S'

11,10 1111 ,

_gl 12

I~ ( = 913)~

t3.32a )1 0 11 .

Here J1 I I'

1, I is the statistic matrix acting on a product of U.-vertex operatorsrthat carry q-spins I, and 12 and are sandwiched between states of q-spins 1; and1f . Combining eq. (3.32) with eq. (3.14) for j, = 21,, v = 212 and A = 21 we findthat the product of diagonal statistic matrices is independent of q (or p):

R,911

R__

p

v oR .u/2

v/2

= ei(.. /axA,-,u--v2)( - 1)(i/2xiL+v-A)0 A 0 A/2

= ei( .. /2)(li+v+(2n-1)w-jil- liv) for A= Iv-tLI +2n .

(3.33a)

In particular, for 1u, = v the n-dependence disappears and we have

.A, := R,91 1 -R v

v

~[

v/2

v/2

_ v,0

2n

0

n

-i

'

( n =O,1, . . .,v) . (3 .33b)[

Combining this equation (in the special case v = 1) with the property 'F ='RO Fo = -1 [see eq. (3.23)] and with the relation (3.15) between fusion andbraiding we obtain the desired result.

Simple as it is, the representation of the braid group so obtained does notreduce to a representation of permutations since the squares of its generators arenot 1,

.ßi2 = -1,

(i= 1, 2,3) .

(3.34)

(In general, for I, =12 = v/2, .:~d2 is related to the overall fusion matrix (3.23) bythe following counterpart of eq. (3.15)

1)vgv(v/2+ 1 )e -2:.i.],,tRF = iv-'( -1)v,

(3 .35)

since t j F = ( -1)v and where we have used the relation e - 2°' i .1"q v(v /2 +') _ ( - i )v-'

for d v given by eq. (2.3) and q = e",-r( - 1 ). The result is

in accordance with eq. (3.33b). Eq. (3.34) then follows for odd v's, in particular, forv = 1 .)

L.K Hadiiiiw"zor et al. / Chiral p-models

It is instructive to give the example of the 4-point function of the step operator

in the Ising model (v = 1, p = 4), that is an elementary algebraic function

1

1

1

1

1/s

G

Z13Z24

G ( ,q ; (u})lsingZ1111 Z-)u .) Z3U3 Z4u4 Z12Z23Z34Z14

l~ '

where the s- and u-channel expansions of G l are

1

1/2)) 1 /2 _1/2 1/2

1/2

1/2GI = ( ~ (1 + (1 - rl)

(q

ul - q

i1?)(q

u 3 - q

u4)

(3.36)

-(2(1- (1 - 0)1/2))1/2(q_lu l - g11 3)(q

_IU 2- qu4)

+(q1/'u1-q-114)(q_ 1/2U,-q1/2U3)), (3 .37a)

G

('(1 + 1~1/2))1/;(q- 1/2111 -ql/Zu4)(q-I/2u2 -g1/2113)1 -2

+(2( 1 - 77 1/2 )) 1/2(q -1u 1 - qu3)(u2 - 10

+ (q -1/2u 1 - gl/2u2)(q-3/2u3- g3/2u4) -

(3.37b)

(In verifying the equality between the two expressions we use the relation [2] --= q +q -1 = 4 for the Ising model). We note that the coefficients of the two hypergeo-metric equations (3.Sa, b) coincide for v = 1 and that furthermore,

(2(1+ /2 1/21

f (1 - ~ 1/2)) 1 /2k2

= 'r21

1

('(1 + (1-r7)1/2))1/22 (

1/21-(1-77)1/2))

In general, for 11 0 12 , the covariance group of 4-point blocks (II 12 12 11 > can begiven by the normal extension Gp =Gp(1 1 12 12 11 ) of the Uq monodromy groupgenerated by

9J2 = T2I 3

= diag{q-21(1+1)+21 1(11+1)+21,(12+1)

(3.38)

IE 11 1 1 -I2 1 +n,

n =0,1 . . .}l (3 .39)

and the fusion matrix

(cf. eq. (3 .16)) . Similarly, the covariance group of G(4) isgenerated by .' ~ and `

F.

2G

z,U ,

4. Green functions involving indecomposable representations inintermediate states

4.1 . THE CASE OF FOUR SPIN-I (v =2) FIELDS IN THE ISING MODEL

Let us consider the (diagonal type) Green function

where the Möbius-invariant part G2y satisfies, as a consequence of eq. (2.16b), athird order differential equation

d3

712(1 _rl)2

d 3

+ (6(r - 1)(2r - 1)712+ (Pr - 2)(2(v + 1)r - 3)(1 - 1)2

d-2(2(4v + 5)r 2 - (6v + 23)r + 12)71(1 -- n)) d71

+4A�(1 - r)(3(2r- 1)71 - (2(v+ 1)r-3)(1 - n)) ~G2v(n,{u}) =0 .

The s- and u-channel bases of solutions of this equation are defined for generic rthrough the expansions

v v 2=Z

-2.1,

-2.177

,.

Z2U2 Z3u3 Z4U414 Z23

G2v =

d2

- (2(4r - 3)71 + (6 - (3v + 2)r)(1- ij)) q(1 -,q) diq2

1 v/2 v/2 1fv+2rt(17)`~ev/2+n= _ 1

u l

U2

U 3

U 4

L.K Httdjsi;i-arrot - et al. / Chiral p-models

-71,v)W"~u j

v/2 v/2 1

u2 U3 U4

(4.2)

(4.3a)

(4 .3b)

406

L.K Hadjiicwnot~ et al. / Chiral p-models

and the initial conditions

vr- 1i._2

=C2i.a~_2

j1Î) ~ C2~~vÎ

fv+2(~) ~ C2 . '+'n2(i'+1)rr-2_ for o --+ 0,

(4.4a)

2r- 1go(01 v) = 11

92( 1 - 71, v) ~ Cv,.2C222( 1 - n)

FO) =

for q -3, 1 .

For a particular choice of normalization (see appendix B.2.) the u-channel basis,say

is related to the s-channel one, If~°)}, by the q-rational fusion matrix

1 [v+2] [v+3][3]

[2][3][v]

[3][v + 1]

[2]'`

[v+ 2] - [v]

[2]2[v ] [v + 3][4]

[2]

[41[v]

[41[v + 11[p + 2][v - 1]

[v - 1][v]1 [v]

[v+ 1][v+2]

I'

This is precisely the inverse transposed of the U,, -fusion matrix

Iv- 1]

[2][v]1[v+ 1]

[v+2][v - 1][v + 2]

[v - 2] - [v]

1[2][v][v + 1]

[v + 2]

[2][2][v + 2][v + 3]

[2]3[ v + 3 ][2][3][4][v][v + 1]

[3][4][v + 2]

[3][4]

(det F,) = -1) .

(4.5)

(4.4b)

_ P/2 v12 '(4 .6)1

1

For v =p - 2 the matrix (4.5) (and its inverse) become ill defined. The ream isthat the solutions fp°_) 2 and fp° ) (as well as g2) and g4) for p = 4) are propor-tional to each other in that case (as displayed in appendix B.2) . We are in factdealing with indecomposable non-unitary representations of Vir in this case . TheVerma module of lowest weight 4p-2 = (p - 2xp - 3)/4 involves an invariant("null") subspace of lowest weight

In particular, for the Ising model (p = 4) the corresponding lowest weight vector isa multiple of

[cf. eq . (2.17)], a vector that has non-vanishing matrix elements in our extendedstate-space . The correct (linearly independent) s- and u-channel bases are thengiven by eq. (B.18) so that the (regular) fusion matrix F has a triangular form

F=AFO~ B - ' =

for

1

[3]

-[2]

[3]

L.K Hadjiicwnoi- et aL / Chiral p-models

( 2

_4

U>

(4.8)L_ I

3L-2)IC = 12- 9 '12 =,21 ; I

L

Qp =Ap_2 +p - 2 .

4.7

i o o

i oz

A =

0

1

4

,

s - ' =

0

1

0 0 1

0 0

0- [2][ p - 2]

[p]1

407

for [p] =0

([p-k] = [k] = -[p+k])

(4.9)

(4.10)

1 [p] [p-1]+[p+1]

[3] [2][3][ p - 2] [3][p - 1]0 1 0

[p - 3] [p - 3]1

[p - 2]-

[p - 1]

4()8

L.K Hadjüranor et al. / Chiral p-models

Eq. (4.6) suggests that the corresponding regular Uq fusion matrix should be

We have

but

R.=`A-1RO'B=

It is straightforward to vertiy that `A -1 indeed transforms (ill defined for [4] = 0)u-channel blocks into regular linearly independent ones (see eq. (A.27)).Although the 2-dimensional (Hecke algebra) representation of Z 4 for the Ising

model (q4 = -1) is a finite matrix group its 3-dimensional indecomposable repre-sentation, associated with the fusion matrix (4.11) (for [3] = 1), is an infinite group .This is also true for its subgroup G4 ( I; = 1,

i = 1, 2, 3, 4) generated by

1 0 0

1 0 1MI =

0

-1

0

,

i~ =

0

1

2-'/2

0 0 -1

0 0 -1

1 0 2im 2 =R,m2Ri~' =

0

-1

00 0 -1

a,2 = 1,

(only true for [3] = 1),

1 0 2m2 = RM,R = 0 -1 0 ,

0 0 -1

generates an infinite abelian group since

1 0

14- = MIR = 0 -1 -2-1/2 ,

0 0

1

zn 1ol .

2 = 2m,

m2 = 1,

(We note that ~ is inverted by the fusion matrix : ~-' = a~a = am .) This examplealso illustrates the fact that the element th2 (unlike M2) cannot be constructed byproducts of m, and 11 .

[3] 0 11

0 1[2] , (4 .11)-1

0 0[3]

L. K Hadjiit°anon et al. / Chiral p-models

The fusion matrices (4.9) and (4.11) are independent of [ p] . But according to theremark after eq. (A.27) of appendix A it is only for the Ising model (p = 4) that themodified u-channel blocks

1/0()>

1l~o)

j'~p/2-2

1/, I = `A -' I

~~) I = I

l/(0 )

I =I

p12-1

(4.12)2

~0)

4 ()- ([2]2/[4])1/;o)

-yp12

diagonalize the second Uq statistic matrix

1 2 .The braid-invariant Green function. -in this case (p = 4) is not uniquely given by

eqs . (4.1) and (4.3). There is a 2-parameter family of such functions

2

( a

bG

) ,22(~ {ul ) = 1: f2n(17)

t +f2 a'~1 +

[2~-

[3J

2It = O

2

_

g2tt( 1 - 17) iltt +920- q)(a1/, +bll2) .n=o

4.2 . FOUR-POINT FUNCTIONS AND FUSION MATRICES INVOLVINGA PAIR OF GHOST FIELDS

The study of higher, say 6-point, Green functions with "physical external legs"(i.e . with v; <p - 2) requires the knowledge of 4-point blocks involving non-unitaryrepresentations of Vir. The simplzst such example of the class (4.1) considered inthis section corresponds to v =p - 1.For v =p - 1 the allowed s-channel intermediate values of Uq weights are

21=p - 3, p - 1 and p + 1 . The representations with the same eigenvalues (3.32)of the statistic matrix 91 i are the partially equivalent ones corresponding to q-spinsI and p -I - 1 (since(- 1)21+1-pq(p-1)(p-1-1)-1(1+1) _ (_ 1)21+1-pgp(p-21-1)- 1

for qp = -1). For 21, = 2, 2,2 = p - 1 these are the values 21 = p - 3 and p - 1 .Choosing in this case

`B

`B(2,p- l,p - 1,2)

[2][p-2]

,

[p l0 1

(4.13)

(4.14)

410

L.K. Hadjih -anoi - et al. / Chiralp-models

we obtain from eq. (4.11) (with A given by eq. (4.10))

0

F P-12

P-2

1 =AFo,B - '=

is indeed finite for [4] --> 0.

[3] [2] 11

1_[2]

2

[2][2] 1

1

[2]

-1 - 2[31

[3]

[2][3][P - 1]

[3]

0

1

[2][P - 1][p+ 1]

(4.15)

1

[p -2]

3 1P -1] - 22[P-1]

[] [p-2]

[]

1

1

1

[3] [2][3]

[3]

t - 1= a .

(4 .16)

For the Ising model, p = 4, it is straightforward to verify, using eqs. (A.30) and(A.31) of appendix A, that the nontrivial combination `BSS;

1 3/2 3/2 1

1 3/2 3/2 1

U I

U 2

113

U 4

U 1

11 2U 3

U4

[2][5] J' (O)

1

[4] 1/23/2

U,

3/2 1

113 U4

(4.17)

[2][P - 1 ][31 -

[p+ 1]1

p -1 p -1 [P +1] [P - 1] 12 2 = `A -' z~ () 'B =

[2][P - 1]1 _

[P+ 1] [2]1 1[2][ P - 1] 10 [3][p + 1]

-[3]

L.K Hadjiü -anot - et al. / Chiral p-models

411

The general braid invariant combination of U,, and conformal blocks is . in thiscase, the diagonal one:

G23( 77') 1 11)) -

!-r 'fit + 1 /2S2n+ 1 -

/011=0

»=0

4.3 . D-E CLASSIFICATION OF NON-DIAGONAL p-MODELS

The BPZ fusion rules (defined in eqs . (2.7)) are obeyed by the quantumdimensions [v + 1 ] (which can be viewed as the ratio of dimensions of irreducibleVirasoro modules of lowest weights ,,, and 0, see ref. [43]):

This explains the success of the diagonal assignment 21,, = v and suggests a secondsolution, 2Î� =p - 2 - v, in view of the identity [v + 1] _ [ p - v - 1]. To examinethis second possibility we study the change of 91, eigenvalue (3.32) for h =1, = Iunder the involution 1 --), Ï. Setting

BPZllApt

[A + 1 ][tc + 1] = TNBPZ[v+ 1] _

[IA - ILI +2n + 1] .v

it=0

(4.18)

The cases (a) and (b) correspond to the D21+3 and D, 1+ , series of nondiagonalpartition functions in the ADE classification [35] . If primary fields of odd v arepresent then 12j =j* (For j = 1 this follows from the operator product expansion oftwo fields of q-spin 1/2, that is two step-operators or two Op - 3's.) Thus thesecond possibility, a non-diagonal primary field 02j, can only appear for a

P 1 1 1'.=

I0 nI= ( -

11)+21(1+ 1)

, (4.19a)

we find

pit

=( -1)21+1i,,Pt for 21+21=p-2 . (4.19b)

There are exactly two cases in which P; is left invariant :

p=41+ 4, (-1) 21+1 = 1, (4.20a)

p=41+ 2, (-1) 21 = 1 . (4.20b)

These two cases are also singled out by the requirement

dp-3-2j -42j+1 = (p - 3)(p/4 - j - 1) E Z . (4.21)

412

p(= 41 + 2) submodel in which Op-2 is added to the observable algebra and thecondition that its representations are single-valued excludes the appearance of oddv. It is consistent to adjoin the field Op-2 = 041 to the algebra of observables sinceits dimension (or "spin")

are only single-valued for even v, since

'd o =0,

L.K Fladjiieai:oc et al. / Chiralp-models

dp-2= (p - 3)(p -2) =l(41 - 1),

(4.22a)

is integer in this case. On the other hand, the (possibly) non-zero 3-point functions

<dp-2-J0p-2(Z)I4v, - Cp-2-vp-2vZ

dp-2 +A, - dp-2 -v _ (p - 3)v/2 = (41- 1)v/2 .

(4.22b)

The presence of an Uq scalar of weight (4.22a) implies that there are two differentprimary fields 0(2 )(z, u) for each (even) conformal label 2j: the field 0(2;) hasq-spin j, the field 02;) has q-spin (p - 2)/2 - j =21- j; the fields -02Î" have thesame q-spin 1, but have different intrinsic parity E _ ± which appears as amultiplicatively conserved quantum number. The z-independent Uq vertex opera-tor .0" plays a role of a spurion field of q-spin 21.There is a more economic way to describe the quantum symmetry of the chiral

(D4) Potts model corresponding to p = 6. The complete nondiagonal minimaltheory can be viewed as a su(3) coset space model (cf. ref. [45]):

su(3), x su(3),~c=2+2-

8-2 - 4 .su(3)2

3+2 5

The lowest weights of the model corresponding to level 2 su(3) representations are

di=4i= 15

®2 =4i= 23,

2dli - s*

Their fusion algebra is displayed in ref. [46] [see eq. (3.40)] . The quantumsymmetry of the model is described by Uq(sl(3)) with q = e-"15 and will bediscussed elsewhere. The intersection of this model with the original ("thermal")p = 6 model only involves the vacuum sector and the two conjugate sectors 2 and 2of weight

. These are closed under the fusion rules 2 x 2 = 2, 2 x 2 = 0, etc.The algebra of observables is also extended for three exceptional (E� ) models

with p = 12,18, 30. The phase factors p() (4.19) for the corresponding spurions are

all equal to I

9 _

- ls- lo = 1Po - q

We consider an example of the D21+3 series (4.20a) involving a pair of non-diag-onal primary fields of q-spin 1/2. We shall demonstrate the fusion (although notthe monodromy) invariance of the Green function

or

Ps -q_1o .6 =1,0

L.K Hadjiir-aaor et al. / Chiral p-models

2

1

4l+ 1 1/2

4l+1 1/2

2

1G

,zl ul

z2 u2

z3 u3

z4 u4

Z14~=Z~~at+t.n-41-1/(41+4)G241+1

;

with the Möbius-invariant factor given by either

I 1/Z 1/2 1G241+~ - h-fâi~ 1( 71)~3j2 u l u2 u3 u4

413

(4.24a)

(4 .24b)

G241+1

(1 _

[q- 112u ( -)gl/2u41 2(q-1/2u2 _

ql/2u3) +agi)(I _ ~l)~iol-- go(I

)

1

(4 .24c)

(This example is a priori suspect since the 3 x 3 conformal fusion matrix of type(4.5) should be matched by a 2 x 2 q-spin matrix corresponding to the spin-1/2

p = 12 : 46 = 8, Po = q-6.4 = l . (4.23E6)

P = 18 : 416= 60, Po = q-8'ls = 1 (4.23E7)

p=30: 110 = 24, A18 = 78, 42s = 189,

414

L.K Hadjiiranot , et al. / Chiral p-models

field of conformal weight 441+, =A , + 1(41 + 1) . Here

and

g;t'1 are thesolutions of eq. (4.2) for v = 41 + 1 normalized according to eqs. (B.13), (B.11) and(B.14). The U,,, blocks are given, in agreement with eq. (3.25) and appendix A by

1 1/2 1/2 1.~ 3/2 = .

3/2il, il,

11 3 114

= [3 ] -1 (il 1 -114){-g3/21,1113 -[2](111114 +ll 7 l1 ;) - q-3/2 112114

+ [3](q- ;/2,1111, + g3/22113114)} ,(4 .25a)

.t .1/; _ (11 1 - 1l4)( - q-3/2 u 1 11 ; +11 1114 + ,12 11 ;-q3/--',1,1l 4 ),

(4 .25b)

1l11 = (ll 1 - 11 4 )(q -1 11 1 - gil4)(q- 1/2it , - g 1/211 ;),

(4.25c)

llio> _ (,11 -,14)(,11,14 +,1,111

-[2] - '(q -3/2 11 1 11 2 + g3/2 11 ;,14 + q -5/2 11 1 113 + g 5/2,1 2 114 ) } .

(4.25d)

They obey the fusion relation

co1= ,.. co1~~t =

Il è~ 312-mc1,1

where R is obtained from Ro of eqs . (3.22) and (3.23) by exchanging the first andthe second column :

[2] [2][3]

(det~- =1) .

(4.26b)

On the other hand we have a duality relation between the triplets g ;';; andfai

(('1-i+2, 1 of conformal blocks. According to eq . (4.5) the corresponding ~fusion

matrix is

(The elements 13 and 23 vanish in agreement with the BPZ fusion rules.) By achange of normalization one can decouple 94 and f4t+3. We shall demonstratethat the constants a and b + in eq. (4.24) can be chosen in such a way that theresulting 2 x 2 matrix F can be related to the Ug fusion matrix (4.26) by

F=

11 0

[3]

[2] z0 a

[4]

1b - ' 0

[2][3]21

[3] 0 b+'-

[4] 1

[2]

-[3]

1

Indeed we find that the four linear equations for the three constants a and b +_ arecompatible and we have

a =1/[3] ,

b-= [2]/[4],

b + = -1/[3]2 .

(4.29)

5. Reconstruction of monodromy free two-dimensional Green functions

(4.28)

5 .1 . SEMI-DEFINITE BRAID-INVARIANT INNER PRODUCT IN THE SPACE OF Uu BLOCKS

We saw in subsect. 3.3 that for /, + v <p - 2 the 4-point Uq blocks of weights(A, v, v, A) give rise to unitarizable representations of the extended monodromygroup GP(I., v, v, A) (in particular, for tL = v, to an unitary representation of thebraid group 9-14). This suggests the existence of a braid-invariant sesquilinear formin the space of U,,-blocks that is positive-definite at least for q-spins consistentwith the BPZ fusion rules. The resulting inner product is unique up to anormalization whenever the associated representation of the braid group is irre-ducible .

L.K. Hadjiir-anot-

1

et al. / Chiralp-models

1

415

[3] [2][3]20

[2]2 1[4] [3] 0

(4.27)

[4] [4][3]1[3] [2]

416

L.K. Hatljiira®aoi~ et al. / Chiral p-models

ire shall construct an extended factorizable* sesquilinear form C, .) on thetensor product ring generated by the irreducible Uq modules ?' ;/ (21= 0,1,-,p - 2 for qP = -1). Two different irreducible modules (corresponding to q-spins11 I, ) are defined to orthogonal while the inner products of monomials in 7,,-2,are

The normalization is chosen in such a way that the norm square of the invariant2-point function (A.13) is equal to the "quantttnt dimension" of ?` ;/

O

®172t11( ® )q1 2tt2 'l . Iq-1/2u1(- )q1/2tt,]_l

)

= ( -1 )2/e2 .i/

1-rit l+#» s ..1 21it

,u

)®=e [I+in

S»i®

21

J[21

_1 ~I-~nt-nt

-raa-n: j~ / -n:lt/+n:1 + '11 ( )q

1 2M, W_ -!

I

q -n, =[21+ 1] .

11 12 13112 _ (( 1 11,

13 )

( 1,

1,

13jj(U1 u2 U3

tt1 112 113 ' u 1 112 u3

it l+mul-,nr)1 2

(5 .2)

A considerably more complicated computation yields the following norm square ofthe 3j-symbol (invariant 3-point function) (A.14) (which is non-zero if Ii + Ij - I,,- isa non-negative integer for any permutation U, j, k) or (1, 2,3))

__ [11+12- 1J![12+13_111![11+13-121![11+12+13+111

[211]![2I2 ]! [2I3 ]!

(5 .3)

Remarkably, the right-hand side vanishes whenever the BPZ fusion rules areviolated (i.e . for 1 1 + 12 + 13 >p - 1). If, on the other hand, I, + 12 + 13 <p - 2, sothat the normalized 3-point functions (3.27) are well defined, their norm square

*The form (Î, g) is factorizable if (1 1 ® 12, g 1 ® g2)

g1), Y2, 92),,, whenever f1 ®f2 and9 1 0 92 belong to the tensor product / '1 0 7 2.

has a simple form:

) 2

V l'

12

13

= ([21 1 + 1][212 + 1][213 + 1])'/2 ,

for

I, +I +1 <

-2.u1 u2 u3

2 3(

This type of expression (the square root of a product of q dimensions) extends tonormalized n-point blocks.The 4-point blocks provide the first example of a multidimensional space of UQ

invariants ; denoting by Inv 2'2,, ® 721, ® 7"2,; ® 7214 the subspace of U9 invari-ants of the tensor product ?' 21, ®7 21,® 7"113®7214 we have

dim Inv 7/2, , 0 7'212 0 7/'2,r, ® 7,-214 = 21min + 1 ,

(,min = min(I,,12)) .

(5.5)

The s- and u-channel blocks form orthogonal bases

i

5(o)

h

I2I

ul u 2

[21, + 1][212 + 1]

N411121

L.K Hadjiii-anoc- et al. / Chiralp-models

417

12

Il

1l

121 2I,~~o>u3 u4

ul u2 u3 u4

[Zh ]! [212 ]! [ZI ]![2I+ 1] ( [1, +I2 -1]![1 +1, -I2]! [I+12-I] ]! [I, +12 +I+ 1]!

ZC (o)~ = Sn,r'[2 1, + 1][2I2 + 1]lN?,,nN?I,n

(5 .4)

(5.6a)

3,,,,,[2 1, -n]! [212-n ]![n]!4

418

L.K

culjih -canoc- et cal. / Chircal p-models

Eqs. (5.2), (5.3) and (5.6) exhibit the factorization property

We note that for 1, + Ia + I =p - 1 =

1

1

-.~

(aD~

1

1,( tt

t' )ttt 1

lt ,

I~

1`1

`1-

ct ~

tt 2

ct

c'

a

and obey the orthogonality conditions

11 3

h 11

113 tt4

I the norm of -j`((" vanishes

p/2 - 1 -1,

I,

I,

p/2-1-1,

tt ; 114

(5.7)

again in accord with the BPZ fusion rule . Similarly, the finite linear combinations®, and ._j, /, (

P/2- [2][p - 2][ p]- °~p12_ 1 ) appearing in eq. (4.12) (involvingindecom

sable representations of Uq ) have zero norm:

[,]'[4] -l À/;mÎÎ~- [2J[p- Il[ pJ[u -'[p-3] -' _ [2][3J-'[pJ =o,

(5 .9b)

/2 ) _ - [2J[ p - 2J[pJ

2/2-yl = - [ P - l ] [p ] = o, ('5.10a)

[Z]2[4] -lll~~l0» 112= -[pJ[p- l][2] - ' [P-2J -' =o .

(5 . 10b)

5.2. GAUGE-INVARIANT TWO-DIMENSIONAL GREEN FUNCTIONS

We define the fields and Green functions of the anti-analytic sector by thediagonal assignment v = 2 1,, assuming that the corresponding ("left movers") U,,blocks transform under the conjugate representation of Uq, so that e.g . the 2-point

function U,, weight v is

1,1ei;

e ) =[qll2i,-l ( -)q-'12çi2 ]`

(5.l1)

We define the (manifestly gauge-invariant) Green functions of the 2-dimensionaltheory by the inner product of sect . 4. In particular, only 2-point functionscorresponding to the same q-spin (21,, = 2IF, = v) survive and we have

(010'( Z 1�i l ; v, v)U(Z" Z2 ; v, v)10

where

or

v v/2 v v/2 v

v= G

,G

= 12 12Z1 il1 Z2

Z1111 Z,11,

The unusual normalization factor is justified by its vanishing for unphysicalpropagators. We thus restore the monodromy-free euclidean 4-point Green func-tions

IL ju v v v v IL N.G

Zi

Zl

Z2

Z2

Z}Z 3

Z4

Z4

+ 1][v + 1]Z-2.1,Z-2 .1,.f-2-1-i-2-I-.14 3 14 23 ~

G~~vv( ~1 ~ ~) _

~

g2n( 1 - ~1) g2~~( 1 - ~1) ~

no = min(~u., ju,, v, v; p - 2 - N,, . . . )

0<Il<n ( )

LX Hadjiiî`rrsrc)r - et al. / Chiralp-models

G~~

NBPANBPA.fA07 VAOi)a

(5 .12)

(5 .14a)

(5 .14b)

(the normalization factors being included in fA , f� ) 92,, and g2n) . The fact that(G L , GR) is monodromy-free and self-dual (i.e . invariant under fusion) is a

4-Ni

LK

acljlàranor et ai. / Chiralp-models

consequence of the G,, invariance_of the inner product and of the correlationtween phase factors in G a and GL. The latter is obvious for diagonal invariants

and folio from the relation p' = Pt between the factors (4.19) under either ofthe conditions (4.20); it is also verified for the nondiagonal pairs (4.23) correspond-

ing to the E-series. The F-invariance follows from eq. (2.19) and the relation

(-1)"'" = (-I)j! i"

(5.15)

valid for the nondiagonal pairs in eqs. (4.21) and (4.23).We shall relate eqs. (5.13) and (5.14) to the examples of the preceding sections

displaying the hidden normalization factors . The diagonal Green function (3.24)(corresponding to p, = 1) yields, according to eqs. (5.6), eq. (5.14) with

similar expressions are obtained from eq. (3.24b) for the product of g's: go goremains unchanged while

92,92 = [v + 2][2] -2[v + I] -I g(o)g(o) (5 .16c)

The 4-point function of cr(z, z; 2,2) in the Ising model, whose chiral counter-part (4.13) is rather involved, is particularly simple because of the vanishingof the norms. Its M6bius-invariant part is given by a single term fo(rq)fo(1) _go(1 - Ogo( 1 - ii).

The present paper was preceded and stimulated by a joint effort of DetlevBuchholz, Igor Frenkel, Gerhard Mack and one of the authors (I.T.) to understandthe role of quantum groups in RCFT. We thank Jurg Fr6hlich and KrzysztofGawedzki for enlightening discussions at the final stage of this work and PaoloFurlan and Yassen Stanev for a careful reading of the manuscript.

Part of this work was done while R.P. and I .T . were visiting La Division dePhysique Théorique (Unité de Recherche des Universités Paris XI et Paris VI,associée au CNRS), Institut de Physique Nucléaire, Orsay, and while L.H. andR.P. visited the 11 Institut für Theoretische Physik, Universit5t Hamburg . Weacknowledge the hospitality of these institutions as well as financial support fromCNRS and from DFG. The work was supported in part by the Ministry of Science,Culture and Education of Bulgaria under contract No. 403 .

f'.-&q)f.- 1(ij) = [v+ 1][2]-1[v]-'f!°-'1( 7J ).f'̀.°-'1( ij), (5.16a)

fa+1(~a)fv+1(°~)_

_ [~+2][2] -1[v+ 1] -lfY°+~~(~1)Îv°+~~(7I) (5.16b)

L.K Hadjiic°araoc- et al. / CJ:iralp-models

Appendix A. The QUE algebra Uq. Finite dimensional representations .Uq blocks

Uq = Uq(sl(2)) is a Hopf algebra [13] defined as follows .(i) It is a complex associative algebra generated by four elements q :~: J3 and J +

subject to the relations

qJ.sq_J;=q-J

3q J3 = 1 ,

q2J3 _q_2J3

q _q- I

[cf. eq. (2.6)].(ü) Uq is equipped with a coproduct, i.e . an algebra homomorphism A: Uq -~

Uq ® Uq satisfying

4(q±J;) _q±J3®q± J3, ,1(J+) =qJ; ®J ++J+®q-J3 .

(A.2)

(iii) There is an algebra homomorphism E: Uq --+ ,%C, the co-unit, such that

E(q ±J3) = 1,

E(J±) = 0,

and an antihomomorphism y: Uq -3- Uq , the antipode, defined by

y(J+) = -q :4- 'J+ .y(q ±J;) =q ±J3

Each

"",2, admits a canonical basis { JIm >} such that

J + JIm~= [I+m ][I±m+1]JIm±1) .

421

qJ;J+= J+qJ3 ' ,

(A.Ia)

(A.1b)

(A.3)

(A.4)

For a satisfying eq. (2.6b) (qn = -1) Uq has p - 1 irreducible finite dimensionalrepresentations n of q-spin v/2 and positive quantum dimension

dg(7v) = [v+ 1] = tr, .~, g2J3,

v=0,1,2, . . .,p-2,

(A.5)

(q ±J3 -q± in )IIM>=0,

(A.6a)

(A.6b)

If q is a root of unity (in particular, for q given by eq. (2.6b) the ring of tensorproducts generated by the above Uq modules contains new finite-dimensionalrepresentations of quantum dimension 0 including some indecomposable Uq mod-ules .

422 L .K Hadjiiawnoa~ et al. / Chiral p-models

The following results of Fröhlich and Kerler [30] are relevant for our discussion.Proposition A.1 .(a) Let A + j, < p - 1 ; then

(b) If 0 < A, AL < p - 2, A + IA >p, then

p-2t 2p-4-A-m ®UAp./P-i ®

®

//,. ,v=2p-2-A-h.(A+g,-v even)

(A.7a)

(A.7b)

where OAS = 1 if A +,u --= p - 1 (mod 2) (®� L = 0 otherwise). Here //;, are 2p-dimensional indecomposable U,1 module with an invariant subspace isomorphic to

The quantum dimension of the factor space

is [2p - v - 1] = -[v + 1]

so that dQ(i/;,) = 0; also dq(/p- ,) = 0.

Proposition A.2 .

The indecomposable representations r/i, and the irreduciblerepresentation Z,-- l of zero quantum dimension generate an ideal in the tensorproduct ring of U,l representations.An n-point Uq invariant

11

121 �. . .

EInv/21, 0/21~® . . .®/21,1 ,

(A.8)tt, u,

ti ll

is a homogeneous polynomial in (u,, u,, . . . , u � ) of degree

satisfying (as a manifestation of J + invariance) the finite difference equation

I,q l~+ . . . +1� .

u,

+q-11 +1;+ . . . +1 � 5J2

q- IU2

q - l u,,

I, 12

qu,

u 2

I l 12. . . +q-11- --- -1-1 .9111

qu,

I�

q - l u

Irr

qu2 unl =o .

(a.io)

1

where

?;F(u,, . . .,u;, . . .,u� )

For vi = 21; satisfying

LX Ha(/jiia - aaarui- et al. / Ciziralp-inalels

423

F(u,, . . .,qui, . . .9un) -F(u,, . . .,q

(q - q -' )ui

v;=0,1, . . . . p-2,

1

1 )u,

t2

-_[q -1/211 1( -)q 1/21121 21

1 21

j(-j) 1-IFIq-17111;+r17Ll2-M

in _ -1 I +m

(A.12)

i = 1, 2, an invariant 2-point function only exists for 1, =12 = I and is given (up tonormalization) by

(A.13)

Similarly, an invariant 3-point function only exists if I; + Ij - Ik E Z+ for allpermutations (i, j, k) of (1, 2,3) and is then a multiple of the (unnormalized)3j-symbol

12u2

[q-(13+ 1)/2u1( - )q(13+1)/2u211 1 +1= -13

X [q("_ -1)/2u1(_)q(1-1,)/2u3J11+13-1,

X [q-("+ 1)/2u2(-)q("

+1)/2 u31 12 +13-11 .

(A.14)

The n-point invariants are evaluated, in principle, in terms of 3- (and 2-) pointblocks . For instance, a basis of 4-point invariants is given by the "s-channel" blocks

13

u 3

I1 12( -W-

In=

~~~q»>

-1

u l u2

13(A.15)

424

LX Hadjüvanov et al. / Chiral p-nrodels

where the mixed 3-point functions are defined as expansion coefficients of thepolynomials (A.14):

Il I, I =U1 u 2 u

Here and in what follows the superscript (0) on Uq blocks reminds us of the specialchoice (A.14) of normalization of 3-point functions. We are also using dual"u-channel" blocks

111/ΰ)

U1 U-1 U3

1

1, 13g1n

u 2 U3

1/2 v/2 v/2 1/2

r

21

] 1/2

Il

12

'in= -t

I+m

U1 U2

m(A.16)

(A.17)

We shall write down the expressions for these (unnormalized) bases in the specialcases I1 =14= 1/2 and 1, 1, = I3= v/2. In the first case the space of invariants is2-dimensional and we have

ul u2 u3 U4

_

)

(q-(" 1)/4u l - q(v+1)/4U2)(q-(v+1)/4U3 - q(v+1)/4U4)[q-IU2

U ]v-1

(A.18a)

1/2 v/2 v/2 1/2:~'2 (11)(v+ 1)/2

Ul U2 U3 U4

= [V + 11-1

` (gv-1 /2 U 1 - q1/2-vu4)(q-v12"2 _ gvl2U3)[U2( - )U3 ] v-1

+ [v ] (q(v-s)/2ulu2+q(5-v)/2u3u4_ulu4 - u2u3)[q-lu2( - )qu3] v-l} ; (A .18b)

the expression for WO can be written in general, for I1 =14=A/2

ju,/2 v/2 v/2 ju,/2~o = ~o u l u2 u3 u 4

- [q -1 /2Ul( - )q 1 /2U41~L [q-1/2U

2( - )g 112U31 v', (A . 19a)

L.K Hadjiic-anor et al. / Chiralp-models

425

while the expression for //;") will only be reproduced for r,L = 1 :

1/2 v/2 v/2 1/2~Î",

s

su l u2 u 3 u4

In writing eq. (A. 18b) we have used the relation

For I, =14 = 1 we shall restrict our attention to the case 12 = 13 = 1 . Then,

are given by

= [q -'u2( - )qu3]v-1 { [2] - 1(gv/2u2 + q-v/2u3)

[q_ 1/2u2(- )g l/2u3]v= (q-v12u2 - gv/2u3)[u2( -)U3] "- ' -

(A.20)

,%i") _ (q - lu, - qu2)(q -1 u3 - quo)

92") = q-2uiu2 + g2u2u22

X(q -3/2u, +q3/2u4 ) - U,ü4 - u2u3) . (A .19b)

u2 u3

1A=0,1,2

11 4

`~"=`%")=(g-lll,-qu2)(u1 -u2)(g-lui-gu4)(u3-u4)ß

(A.21a)

X jq - lu,u2 +qu3u4 - [ 2]- 1

(qu, +q - 1u2)(qu3 +q -lu4)) 9

(A.21b)

-[2]2[4] -1(g-l U,u2 +qu3u4)(qu1 + q-lu2)(qu3 + q-1u4)

+ [2] [3 ]- 1[4] -1 (g2u1 +q-2u2+ [ 2]2u '

u2 )( g2u2+q-2u2+ [2]2u3u4 )'

(A .21c)

We reproduce for the reader's convenience the intermediate steps in computingthe expressions for . ', and .t'2 . The starting points are the following special cases

26

L.K. Hacljinwnot~ et al. / Chiral p-ntoclcls

of eq. (A.14)

(I

1

1 )dl l

dl'

=(q -I ttl - gdl~)(tl l - tl~(Q -I tl~ -gtl),

1 1 2)II Iit-,

dl

- [ttl( °)tl~aLQ-I tl,( °)qtl,~ "

-

The three terms in the sum (A.15) in .,/', are

1 1

1 1

1 1q

~

= q-I tllll2(q -1 11 1 - Qlc~)(Q -I tl~II Iit,

- 1

1

11 3

114

l 1

1

11 l 1

1

1 1q- I

=tc I

tc,

1

-1

11 3

114)

~qu3tl4(q - 'it, - gtl,)(q- l U3 - qu4)

1 1

1 1

1

dl I dc, 0 0 113

The first two terms on the right-hand side of eq. (A.21c) come from the contribu-tions

1 1q 2

u l u 2

1 1q-2

u u2

1= [2] - '(q-I tt, -gtl2)(qUI +q-IU2)

22 )f 2

z 1 f ?2

v/2 v/2

X(q-I tt3 -gtl4 )(qtl3 +q -l tl4) .

u3

u3 u4

1=q -2UiU2

u4

The u-channel computation can be done with the same ease for the moregeneral Uq block

1n=0,1,2,

u2 u 3 u4v>2 .

- gtl4) ,

Indeed, 1/,(,°) =1/() is given by eq.the relations

- q

q

v/2 v/2

112 U3

v/2 v/2q2

U2 U3

L.K. Hadjiir -anoi- et ai. / Chiralp-models

427

19a) for 1A = 2; to compute ®/, and //, we use

v/2 v/2

1 1

1

1

112 113 -1 U, 1 U4

v/2 v/2 1 1

1

1+q-1

U2 U3 1 U1 -1 U4

_ -(U1-114)[q-1112(-)q"311,-1(UIU4+U2113),

1

0

- [2] -1 ( 1t1-U 4)[q -1 U2( - )q11

2 1 2 1

~ .-2_ [q-312U2( - )g3/2113]

t12112-2 t11 2 tl4

(v/2

v/2

21 (

1

2

1q-2

U2 U3 2 U1 -2 114

v/2 v/2

2 1 2

U2 U3 -1 U 1 1

-g- 1

v/2

v/2

2 1

1

U2 U3 1 U 1

(q-2111 + g2114)(q~,/2u, + q-`'/2U3)

9

4

[q-3/2112(-)g3l2v-2

11~] Ujll2 ,1

9

U4

-[2][4]-'[q-3/2 t1

~l2U . ] v-2(- 2U +q 2 tl )(

°12U,+

''12u_ )(11 11 +11)2(-)q.

3

g

1

4

q

g

3

1 4

428

etc. The result is

//20) - [2]2[4] -1 //1o )

v/2 v/2

112 113

L.K Hadjiic,anoe , et al. / Chiral p-models

i~

1 1= (111 -114)['7-l"2(-)q"3,J

-111114-112113

+ [2]-'(q-2111 + g2114)(qv12u, + q-v12U3)1 ,

(A.22a)

1

v/2 v/2

1

u l

112

113

U4

- [ q -3/2112( - )g 3l2113]v-2

{11;114 + U2U2 - [2]2[4] -1(q-2111 + g2114)

X(q "12112+q -v/2113)(UIU4+112113) + [2][3]-'[4]-1

X (q-4112 +g4U2 + [2]2U1l14)(gvU2 +q-vU2 + [2]2 U2U3)~ .

The above expressions are well defined for generic values of q or, if q is a rootof unity of the type (2.7), for p > 5. For p = 4, the case of the Ising model, theexpressions for and //2° ' involve divergent coefficients . In that case we cansubstitute ~~2 and //2 by

= q -2U2

112 + g2U32 2U44 - [2](g -1UIU2 + gU3U4)(U1u3 + u2u4)

+ [2][3] -1 {gll~U3+q -1 112u4+u,U 2(q -lU2+quâ)+11 3U4(q -1 1121 +qu2)

+ [4]ulu2U3U4) + [3 ] -1 (U2U2+u2U2),

(A .23a)

=U 1U 2 +U21t2 - [2](UIu4 + u2U3)(q-1111U3 + gU2u4)

+[2][3] -1 {(u,u3+U2u4)(q -1 Uil12+qu3u4) + [4]uIu2u3u4

+q-3

UiU 3 + '73112114} - [3 ] -1 (q -2 UÎ 11 2 + g 2U3U4) .

(A.23b)

We derive from here the simplest duality relations

(A.22b)

(A .24)

We observe that the terms involving [4] -1 on the left-hand side of eq. (A.23b) are alsocancelled for p > 4 and v > 2. Indeed, using eqs . (A.22) and the relation

as well as the identities

we find

L.K. Hadjiir`araot - et al. / Chiral p-models

429

[q -111 2( - )ql{;]i'- 1

= (q"12-2U2 - q2-`'12u3)[q -312u,( - )g3l21131r_2,

(A.25)

[4] = [2]([3] - 1) = [2](q 2 + q -2 ) ,

[1] + [3] + [5] = [3]2

,

(A.26)

1 v/2 v/2 1

1

v/2 v/2 1~120~

- [2] 2[4] -1ll 1 li2 u3 U4

111 u2 113 114

= [q-3/2,17(

;/2u, ] `,- 2(11 2U 2 + 1122,1l

[2](111114

u`u_ )

/2- )g

3

1

4

2

3 -

-

,

~ (q

~2111U3 +q`'

117114)

+ [2][3] -1 ((UIU3+112U4)(q -1 111U2+glt 3U4 ) +q -I,- 'it~U2+q I,+1 U2112

+ [4]U1112U3U 4 ) - [3] -1(q `-4U~Cl2+q 4-Vll2Uf .

We apply eq. (A.27) in subsect. 4.1 to the case v =p - 2 in which the q-spin 1 and2 contributions to the tensor product expansion (A.7b) (for A = t, = p - 2) belongto a single indecomposable representation r12 .Remark .

The u-channel blocks diagonalize the U,t R-matrix 91 2 , we have

1 v/2 v/2 1l~

_ (_ 1) v-~~q -n(n + 1)+v(i'12+ 1)) 1/

=0

A.28u1

u2 u 3 U4

[q -1 2 11,( - )g 1 /2u 3 ]~,

(A.27)

[cf. eq. (3.32)] . 1/1 and r/2 correspond to the same eigenvalue of N 2 iff -q 2 = q6or q4 = -1, i.e . for p = 4 (v = 2) . Thus only for the Ising model (for which 1/(') isill defined) the linear combination (A.27) is an eigenvector of 912*Knowing (A.27) we can compute J,v/2-1 by extending the second fusion

relation (A.24) to arbitrary v > 2 (owing to eq. (4.5) and the relation a � ' ='F0between the U,, fusion matrix R,~ and its RCFT counterpart F1,) . Using for v > 2the identity

_ [q-31211

3/21131 v-2 ('' -311 2

[2]tl t1

+

~-vu`)2( - )g'

g

2 -2 3

g.

3 ,

(A .29)

4311

L.K Hcaljiiiwnoi - et al. / Chiral p-models

we find the following relation for (1 v/2 v/21) U(, blocks

= [3] -'//(, - [ 2]_[4] -r~

;)) + //2")_ [q-3/,11,(

-)q3/2t,3jv 2~[3] -1(q,,-3112 - [2111 2tt 3 +q 3- "tt;)

X

1 11 22 -

l, . +

ll2

+ 11 2,12 + il 21l2 -

`' -41l 21l2 +

4-`'1l2il2(q-

1

[2]11 1

4

g

4)

1

4

3

[31 -1 (q

1

2

g

3

4)

+ [2][3] -l((11 1 113 +il21l 4)(q-1 t, 111,+gt13t,4) +q-"-1 1,1112

+q`+11121124+ [4]i1111211~t,,~) - [2]( "Iü,~ +u 2 tl3 )(q- `' /2 11 1 tt3 +q"/2t1211 4)}

- [q -/211 2( - )g3/21131~ .

2(q-"/2111 - [2]11 1 11 2 + qv/2112)

X(q -`'/2112 - [2]11 311 4 + q"/-2110) .

(A.30)

We also reproduce the expressions for

%; and

/; + 1 in the special case

v = 3 (needed in subsect . 4.2 .)

13/2u, u, «i yeat! 1 2 t,3 u4

= (q-5121,1113 +

q512"2114 -11 1114 -11 2113 ){q-21, 1112 - q

21123114

+ [3] -1 (g 3/211 3(g5/211 1 11 3 + [2]2q-5/2112114 + [2](Llll,4 + 112u3))

-q-3/2l! 2 (q-5/2ll2tl4 + [2] 2-q 5/2- 1l lll3 + [2](1l 1 1, 4 +

u2113)))}, (A .31a)

1 3/2 3/2 1

.~s/zIl i

11 2

113

il4

=q-5/21l ~u2 - q5/,̀ll~ll4 - [5] - 1

X {([3]2CjSl2i111l3 + [2][3](112113 + 111114) +[2]2q-5/21l2114)q-21111l2

- ([2]2g5/2

u1u3 + [2][3](112113 + 111114) + [3]2q-5/2u2114)g2113114~

+ [2 ][4] -1[5] -ll([3]gs/21l2 + [2] [3]ulcl2+q-5/2tt2)

x (q s/2[3]1l2 + [2][3]u311 4 +q-5/'

tl4)q_3/2

u2 - (q .5/2U2 + [2] [3]ulu2

+ [3]q_5/2tl2)lll5/211~ + [2] [3]ll3ll4 + [3]q ®5/2 11~)C) 3/2 113 1 .

(A .3Ib

L.K. Hadjiii -cn:oi - et al. / Chiral p-models

431

Appendix B. Integral representations and fusion relations for4-point functions

B.I . FOUR-POINT FUNCTIONS INVOLVING A PAIR OF STEP OPERATORS

The solutions of the s-channel equation (3.5a) specified by the initial conditions(3.6a) are proportional to the hypergeometric functions

Ctv- ,dv-,(71)=F(-2Av, 1 -r;2-(v+ 1)r; n)

T( 2 - (v + 1)r)

F(1 -r)F(1 - Pr)

Civv+, J v+,(77) _ 7iv+ >r-1F(vr, l - r ; 2A, ; ,q)

T((v+ 1)r)T(1 - r)F((v+2)r- 1)

Here we have used the standard convention CN.vA = CA,,,,as well as the shorthandnotation (2.3a):

Av=dv+4, - dv_, =(v/2+ l)r- 1,

(r=(p- 1)p),v=0,1,. . .,p-2 .

(B.2)

1,, 2 refer to the integral representations associated with the Coulomb gas picture[471

I,(a,b,c ;71) = f xUQ(u - 1)b(u -77y.du= f it-a_b-c-2(j -t)b(1 -71t)rdt

1

0

- F(b+1)T( - a - b-c- 1) F(-c, - a - b - c - 1 ;T( -a - c)

,2(a,b,c ;ri) = f u"(1 - u)''(r1 -u)`du0

(B .3a)

T(a + 1)T(c + 1)=

71 i +a+"F(-b, a + 1 ; a + c + 2 ;,q)

.

(B.3b)T(a+c+2)

432

L.K HadjUranor et al. / Chiral p-models

The parametric integrals (B.3) obey the simple crossing relation

I,(b, a, c ;1

sin rb

sin Trc

I,(a, b, c ; rl )12(b, a, c; I - ,q)

sin 7r(a + c) (sin 7r(a + b + c)

sin 7ra

I2(a, b, c; ri)

The fusion relation (3.$) is deduced from here by noting that, for

we have

a = -r,

b= -vr,

c=2A,

(-a-b-c-1=1-r) (B .5a)

sin 7r(a+b+c)

sin 7r ( 1-r)

(-1)v+Isin 7r(a+c)

-sin Tr((v+ 1)r-2)

[v+ 1]

'

sin Trb

IV]sin m-(a +c)

[v+ 1] '

and by using the "q-rational normalization"

F( r)

Cv]fv°-) i( - 1) = 1-(1-r)T(2r- 1) [v+ 1]

I,( -r, -vr,2A� ;

T(r)

T(2 - 2 r)go( 1 - 71,v) =

h2(1-r) Ic(-vr, - r,2A� ;1 -

(B .4)

(B .5b)

c

=

12( -r, - vr, 2A � ; ,1) ,

(B.6a)f~+c

T(1 -r)T(2r- 1)

(-1),+ir(r)[v]

9z>>(1 - ~1

v) -

12( - vr, - r, 2J1,, ;1 - q) .

(B.6b)T(1 - r)T(2r - 1)

L.K Hadjiia -ai :oc- et al. / Chiral p-n:odels

433

B.2 . FOUR-POINT BLOCKS OF FOUR (v = 2) CHIRAL VERTEX OPERATORS OFWEIGHT 2r - 1

An s-channel type basis of solutions of eq. (4.2) is given by the Dotsenko-FateevCoulomb gas integrals I; = I;( - 2r, - vr,2Av; 2r l q), i = 1, 2,3

h =1.du, ul 2r( u, _ 1)-vr(u1 -,0)2A,,Jfildu2 u2

2r( u2 _ 1)-vr(u2 -,1)2A`.uii>

>

I

( 1 _,I(1 _s ))2A, .

I

(1 - t)2r((1 - 71)(1 -s)+st}2A,,

ds

2(v_ I)r- I

2r.-If

dt

vr

2r0f

s

(1 -s)

o

t {1 -s(1 -t)}

r(2r)r(1 -2r)r(1 -r)T(1-( v- 1)r)f(1 - vr)r(2 - (v + 1)r)r(r)

(in deriving the last equation we have used Dixon's theorem, see e.g. ref. [48] vol.1, sect. 4.4, eq. 5);

focI2 =

du,ul 2r(u 1 - 1)-vr(u' -~7)2A,.j~du2u22r(1 - u2)-vr( 71 - u2)2AJ-uii

0

-

vr-I

I

(1 -,7(1

S)) 2A,

I

(1 - t)2A,.(1 - (1 - s)tri}2r

- 77

.I ds

vr

2r

f dt

t 2r 1 -

t vr0

s (1-s)

o

(

)

r(1 - Pr)r2(1 - 2r)r((v + 2)r - 1),.r 71~,r_ I

"I

r(2 - (v + 2)r)r(vr)

I3 - f~du,u-2r(1 _u1)-vr(~ - UI)2A,, ,.du 2 u22r(1 - u2)

-vr(~ -u2)2A, .uiâ

0

~0

I

(1 - S )2A,,

I

(1 - t)2r(1 - st)2A,,

J

ds

2r_ 1(

-

vr J

dt

t2r

1 -

st

vro s 1 71s) o

(

)

77 2(v+I)r-2

-r2(1 - r)r((v + 3)r - 1)for

71 -3- 0 .

(B.7c)[2]((v+2)r- 1)r((v+ 1) r)

These integrals obey again a manageable fusion relation,

--* 0

(B.7a)

71 --; 0,

(B.7b)

I,(b, a, c ; gl 1- ,n)

I,( a, b, c ; gl71)

12(b, a, c ; gl 1 - 77)

= F,

1,(a, b, c ; gl ri)

,

(B.8a)

13(b, a, c ; g l l - 77)

13(a, b, c ; g 177)

L. K Nacljiie ,aitot~ et al. / Chiral p-models

where the (unnormalized, 3 x 3) fusion matrix F, satisfies

s(a +c)s(a +c +g/2)

0

0F,

0

s(a +c)s(a +c +g)

00

0 s(a+c+g/2)s(a+c+gl)

with

= (Fjo, i,j = 1,2,3,

(B.8b)

s(x) = sin Tx ,

c(x) = cos Tx

(B.8c)and

F� = s(b)s(b+g/2),

F,,= -s(b)s(c),

F� =s(c)s(c+g/2),

F,,= -2c(g/2)s(a+b+c+g/2)s(b+g/2),

F� = (s(c)s(a+b+c+g/2)s(a+c+g)

-s(a +g/2)s(b)s(a +c))/s(a +c +g/2),

F,; = 2c(g/2)s(a +g/2)s(c -g/2) ,

F;,=s(a+b+c+g/2)s(a+b+c+g), F;,= s(a)s(a+b+c+g),

F;3 = s(a)s(a +g/2) .

(B .8d)

For a= -2r, b= -pr, c=2A,,=(v+2)r-2, g=2r- we find

IV- 1]

[V+ 1]

[P+3]1)v[2] [v-1] -1 z [2][2v+2]

1)v+'[2][v+3]

[v + 1]F, _

) [ ]

[v](

)

[v][v + 2]

(

)v + 2[ ][ 2]

-

[ v + 1 ][2][v]

[2]2[v ] [v + 2]

[v + 2]

The fusion matrix FOB (4.5) is obtained from F, by a diagonal transformation,

a � 0 0 a,, 0 0Fo ~ =

0

b�

0

F,

0

p,,

01,0 0 c,, 0 0 1

(B.9)

(B.10)

with q-rational elements

[V+ 1]

[v+ 2]=

011_ -[2][v]

,

where (in agreement with eq. (B.7a))

LX Hadjüi -araov et al. / Chiralp-models

[ v + 2]

_ [,V

- [v][v - 1]vF'v ( ) [4]

C" 2

[2];

we note that for v = 2 the left and right diagonal matrices are inverse to eachother and are related simply to the U,,-structure constants (3.28)

a2a2 = b, J82 = C, = 1

[3]2a2(n) =N~i� [2n + 1]

for a2(0) = a,,

a2(1) = b2 ,

a2(2) =c, .

(B.12b)

The corresponding s- and it-channel bases are given by

f<<o ' ) ( 77) _ (y/a,,)I,(-2r, - vr,2A,, ;2rlr7),

f<<o)(7l) = (Y1A,)12( - 2r, - vr,2A,, ;2r177),

fcm (

=

1 (-2r, - Pr, 2A, ;2r177),

(B.13a)

go( 1 -q, v ) = g(() ')(1 - 77, v) = ya,. 1,( - Pr, - 2r, 2A,,, 2rI 1 -

g2 0(1 - q, v) = yb,,12(-vr, -2r,2A,,,2rI1 -77),

gâ' )( 1 - q, v) = y1-3( -vr, - 2r,2A,,,2rI 1 - r~),

(B.13b)

(2r - 1 ) T( r) F(2 - 3r) [3]

-(2r- 1)[2_

]T(r)y

T2 (1 - r)T(2r- 1)T(2-2r)

T(3r- 1)T- (1 -r)

(BA la)

(B.12a)

(B.14)

For v =p - 2 (r = 1 - 1 /p) 12 and 13 have the same monodromy ; in particular

435

lan-dt,_2=2(v+ 1)r-2-(vr- 1)1,,-,,_ 2 =pr- - 1 =p-2 .

(B.15)

Moreover, these two solutions of eq. (4.2) coincide for this "boundary value" of v .Noting that A,,- ., = p - 3 (for A,, given by eq. (B.2)) we find the following small 71

-t36

behaviour of 1,

1,(-2+2/P,3-P-2/P,p-3;2-2/pl-q)

p-4+,1,

1,-

) !p-2 2 - 2/p

(-0~F(p-3+2/P)F(4-P-2/P)(P-3 1:(

k

L .K Htuljiii ,aizoi - et al. / Chiralp-models

p-3

r(k - i +n/2)X(P-2-k)r( P+k-3+p/2)

2p-6+2/p (P-2- 1/P)(P - 3 - 1/P) . . .(1 - 1/P)

(P - 3+ 1/P)(p - 4+ 1/P) . . .1/P

(P-2)sin2-7r/p'

sin -rrp(1 -r)

sin -rr(1 -r)

The nonsingular fusion matrix F (4.9) is then reproduced.

(B.16)

which is easily verified to coincide with the right hand side of eq. (13.70 forv =p-2. The three linearly independent solutions of eq. (4.2) for this value of vcan then be taken as 11 (which is a polynomial of degree 2p - 6), 1; and the limitof the analytically continued (in f-) linear combination

1lim.

{1,(-2r,(2-p)r,pr-2;2r~~r~) - I3(-Zr,(2-p)r,pr-2 ;2j' i ~l)Î,-~ ~ - 1~~ [Pr

l

(B .17)

The corresponding s- and u-channel bases fu and g2� are related to (B.13) by

fn-4

ff,-2 =

fp

fc()p-4

B fp-2,

RO)

B =1 00 1

0 0

0-.8p-2 2

1

90 ) g0)) 1 0 0

92 =A g2)) ,94

g(4)) 0 0 1

L.K. Hadjiii-anoi - et al. / Chiral p-models

437

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