Anyonic realization of the quantum affine Lie algebra Uq(AN − 1)

29 February 1996

PHYSICS

LElTERS

6

Physics Letters B 369 (1996)3 13-324

ELSEVIER

Anyonic realization of the quantum affine Lie algebra U4( &_

1)

L. Frappat a, A. Sciarrino b*‘, S. SciutoC*‘, P. Sorba” a a Ltrbomroire

de Physique

h Diparrimenfo

Theorique

ENSLAPP:

di Scienze Fisiche.

c Dipartimento

di Fisica

Chemin de Bellevue

Universirri Teorica.

di Napoli

Univertritri

BP I IO, F-74941

“Federico

di Torino,

II”,

Annecy-le-Vieux

ond I.N.RN.,

and I.N.EN..

Cedex. France

Sezione di Nap&.

Sezione di 7i,rino.

Italy

Iialy

Received 3 November 1995 Editor: L. Alvarez-Gaumk

Abstract We give a realization of quantum affine Lie algebra U,,( &_ 1) in terms of anyons defined on a two-dimensional lattice. the deformation parameter q being related to the statistical parameter v of the anyons by q = eia”. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel [I] fermionic construction of undeformed affine Lie algebra.

1. Introduction The connection between anyons and quantum groups was originally discovered in [ 21 for the case of S1,( 2) and later extended in [ 31 to S&,(n) and in [4] to the other classical deformed algebras. For a review see [ 51. Anyons are particles with any statistics [ 6,7] that interpolate between bosons and fermions of which they can be considered, in some sense, as a deformation (for reviews see for instance [ 8,9]). Anyons exist only in two dimensions because in this case the configuration space of collections of identical particles has some special topological properties allowing arbitrary statistics, which do not exist in three or more dimensions. Anyons are deeply connected to the braid group of which they are abelian representations, just like bosons and fermions are abelian representations of the permutation group. In fact, when one exchanges two identical anyons, it is not enough to compare their final configuration with the initial one, but instead it is necessary to specify also the way in which the two particles are exchanged, i.e. the way in which they braid around each other. In the present paper we will use creation and destruction anyonic operators, equipped with a prescription about their braiding; such a prescription also induces an ordering among the anyons themselves [ 21. However we remark that anyons can consistently be defined also on one dimensional chains; for simplicity, in this paper mainly we will use anyons defined on an infinite one-dimensional chain; at the end we will extend our results to anyons defined on a two-dimensional lattice. ’ Supported in part by MURST.

! URA 1436 du CNRS, associCe B I’Ecole Normale SupCrieure de Lyon et B I’Universitk de Savoie. 0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved SSDfO370-2693(95)01518-3

314

L. Frappat et al. / Physics Letters B 369 (I 996) 313-324

He_re we extend the realization of classical deformed algebras to the case of the deformed affine algebra Uq( AN-I). As the fermionic construction of undeformed affine algebras, the anyonic construction on a linear chain gives a value of the central charge equal to one. However at the end, see Section 3, we briefly discuss how to get central charges equal to an integer positive number, considering anyons in a two-dimensional lattice. In the case of deformed affine algebras two kinds of explicit realization are known and only for the case of affine SZ,( n): a construction in terms of an (infinite) combination of the Cartan-Chevalley generators of Sl,(cc) build up with bilinears of deformed bosonic or fermionic oscillators ([ lo]) or a vertex operator construction

([111). The paper is organized as follows. In Section 2, after recalling the fermionic realization of ~M__I, we present an anyonic realization by the use of anyons defined on a one-dimensional lattice Z. In Section 3 we explain how more general representations of the quantum afhne algebras can be built by means of anyons defined on a two-dimensional lattice; then we make some final remarks. The Appendix is devoted to a discussion of the ordering of anyons on a chain and on a two-dimensional lattice.

2. Anyonic contruction

of U,,(&I)

Consider the affine Lie algebra x~_t with generators denoted in the Cartan-Weyl basis by hy (Cartan generators) and ey (root generators) where i = 1,. . . , N - 1 and m E Z. The root system of AN-I is given by A = {e, - err, 1 5 p # cr 5 N}, the E,,‘s spanning the dual of the Cartan subalgebra of gl~. The generators satisfy the following commutation relations:

(2.la)

1

hy, e”, = [

Ui

ez+“,

(2.lb) if a + b , is a root , ifb=-a, otherwise ,

(2.lc)

(2.ld) where e( a, 6) = fl is the usual 2-cocycle and y is the central charge of the algebra. One can also use the Serre-Chevalley presentation, in which the algebra is described in terms of simple generators and relations (the Serre relations), the only data being the entries of the Cartan matrix (a,~) of the algebra. Let us denote the generators in the Serre-Chevalley basis by h, and e,’ where CY= 0, 1,. . . , N - 1. The commutation relations are:

h,,hp [

[

1

=O,

(2.2a)

=fu,peg,

(2.2b)

e+ h,, a* "P 1=6ap

together with the Serre relations

(2.2c)

L. Frappal et al./ Physics Letrers B 369 (1996) 313-324 I -

315

L’“/j

c

(2.3)

Y=o

The correspondence

between the Serre-Chevalley

h; = hy ,

and the Cartan-Weyl

e+=$,_,,, ,

e,T= f$+,_,.

eo+=eiN_,

e; = e;f_,,

bases is the following

(i = 1, . , N - I) :

N-l

xhy,

ho=~-

,

(2.4) .

.j=l Let US recall now the fermionic realization of i~-t in terms of creation and annihilation operators. Consider an infinite number of fermionic oscillators cP( r) , cA( r) with p = I,. . . , N and r f Z + l/2 = Z’, which satisfy the anticommutation relations cP(r).cAr)}

= {c~(r).&s)}

=0

and

{cL(r-),cAs)}

=&,&~.

(2.5)

i the number operator being defined as usual by nP(r) = ci(r)cJr). These oscillators are equipped with a normal ordered product such that : ci(r)c,(s)

:=

c;(rkAs)

ifs>O,

7

{ -c,(s)cJ(r>,

(2.6)

ifs < 0,

and therefore : n,(r)

:=

rip(r))

ifr>O, ifr
1

i n,(r)-1,

Then a fermionic

oscillator

realization

hy=C(:c/(r)ci(r+m) rGz’ e&,

= c

(2.7) of the generators

of xN-1

:-Ic!+~(~)c~+,(~+wz)

c;(r)c&

+ m) ,

Now, taking ec,_,,, and e;,?+

p # U?

1)

1 _<~,a<

(i=I,...,N-I),

(2.8a) (2.8b)

N.

with let us say, p > IJ and m > 0, one can compute

m), C cJ(r)c,(r SEZ =

is given by

C (cL(s-

- m)

1

m)c,(s -m) - &s)c,(s))

SEZ

=

C(

:n,(s-m):

.S$ZlOJll

- :n,(s)

:) + C

(: n,(s - m) : - : n,(s)

: +l)

sEIO.ml

p-1 =--

c i

k=u

+m,

hg

(2.9)

1

Note that the value of which proves that the central charge has value y = 1 in the fermionic representation. the central charge is intimately related to the definition of the normal ordered product Eq. (2.7); different definitions, like:

L. Frappat et al./Physics

316

: up(r)

b

:= n,(r),

or:n,(r):=n,(r)-1,

Letters B 369 (1996) 313-324

E z’ )

(2.lOa)

V’rEZ’,

would lead to y = 0. It follows that a fermionic isgiven by (a=O,l,...,N-

oscillator 1)

(2.10b)

realization

of the simple generators

of ~N-_I in the Serre-Chevalley

(2.11a)

9

h, = c h,(r) rEZ’ t?* e:(r), a- - c rEz’ where (i= h;(r)

(2.1 lb)

l,...,N= n;(r)

basis

1) - ni+] (r) =: Q(r)

ho(r) = no

: - : nj+] (r) : ,

- nl (r + 1) =: IN

(2.12a)

: - : nl (I + 1) : +S,_,p,

e,?(r)

= ct(r)ci+t(r),

eo+ =c;t(r)ct(r+

e,:(r)

=c,t,,(~)ci(~),

e;(r)

=ct(r+

(2.12b)

l),

(2.12c)

l)c~(r).

(2.12d)

Inserting Eq. (2.12b) into Eq. (2.11 a) and taking into account that the sum over r can be splitted into a sum of two convergent series only after normal ordering, one again checks that N-l ho = 1 +

c

: nN(r)

: -

c

: nl (r)

that is y = 1. In order to obtain an anyonic

c

hj ,

(2.13)

j=l

rEZ’

ra’

: = 1 -

realization

of ck~(A^N-I), we will replace the fermionic

anyons in the expressions of the simple generators of Uq(A~_l) in the Serre-Chevalley therefore the following anyons defined on a one-dimensional lattice (r E 74’): up(r) =

&(r)c,(r)

and similar expressions

,

oscillators

by suitable

basis. We introduce

(2.14)

lIpIN,

for the conjugated

operator a;(r),

where the disorder factor KP( r) is expressed

as (2.15)

v E IO, 2) being the statistical parameter and 0 (r, t) a suitably defined angle (see Refs. [ 2,5] ). The anyonic ordering being the natural order of the integers, in Eq. (2.15) the anyonic angle (see Appendix and Ref. [5]) can be chosen as O(r, t) = +rr/2, O(r,t) =-n-/2, By means of Eq. (2.16) t # 0 and e(O) = 0, &(r)

= 9

-4 c,,,

with 9 = exp( in-v).

if r > t, if r < t.

(2.16)

the disorder

E(t-r):n,(0:

,

factor K,,(r)

can be rewritten,

using the sign function

e(t)

= It//t

(2.17)

if

L. Frappat et al./ Physics Letters B 369 11996) 3i3-324

By a direct calculation, r > s:

one can prove that the a-type operators

a,(r)a,(s)

+ 9-‘up(s)ap(r)

n$kQ(s)

+ q n,(s)+)

a,(r)&)

+ a$-)a,(r)

a+(r)a+(s> p p

= 0, = 0,

ap(r

317

satisfy the following

+9-‘.+(S).+(r) P P + q aJ(s)a,(r>

braiding

relations

for

=o 3 = 0,

(2.18)

and = 1(

which shows that the operators

a,(rP

(2.19)

= uJ(r)2 = 0,

uP( r), uJ( s) are indeed anyonic

oscillators

that the last ENS. (2.19) do not contain any q-factor, the operators a,(r) standard anticommutation relations. Following Ref. [ 21, we also introduce twiddled anyonic oscillators: cl,(r)

= &(r)c,(r)

,

with statistical

parameter

V. Notice

, ui( s) at the same point thus satisfying

l
(2.20)

,

(2.21)

where R,Jr)

= I-I e-i&r.?):“,(r): 1#r

with the same statistical choice O(r,t)

=-n/2,

G(r,t)

= +71/2,

parameter

Y and opposite

braiding

(and ordering)

prescription,

ifr>t, ifr
=

&?(r),ap(s)

5L(r),~J(s)

11

I

?i,(r),uL(s)

$Jr).u,(s)

to the

(2.22)

and therefore to replace 9 with 9-l in Eqs. (2.17), (2.18). Let us add that anyons with different indices p, D obey the standard anticommutation of I’. s E Z’ and for all values of the anyonic parameter V. Finally, the braiding relations between u-type and &type anyons are given by

C

corresponding

relations,

for all values

=0,

for all

r,s E Z’,

(2.23)

=O,

for all

r + SE Z’,

(2.24)

and CP(r),uL(r) {

= 9

C,&

E(r-r):n,(r):

> E(r-r):n,,(f): . = 9 - C,E%’

$(r),a,(r) {

1

It is useful to remark that the following aj,(r)u,(r)

=i$tr)ii,(r)

identities

= rip(r)) ,

hold: (2.26)

and that normal ordering among anyons is defined exactly as in Eq. (2.7). Everywhere in this paper the generators Ef will be expressed in terms of the oscillators u and the generators E- in terms of the ii as, in the deformed case, the E-‘s are not simply equal to (E+)t, but also the exchange -’ must be performed. 4-4

L. Frappat et al./Physics

318

Letters B 369 (1996) 313-324

We can now build an anyonic realization of the quantum affine Lie algebra Uq(A^N_~) “anyonizing” Eqs. (2. I I a,2.11 b), i.e. replacing the fermionic oscillators cP with the up’s in Eq. (2.12c), with the 5,‘s in Eq. (2.12d), and, indifferently (see Eq. (2.26)), with the up’s or the $,‘s in the Cartan generators hi (see Eqs. (2.12a) and (2.12b)). Theorem 1. An anyonic realization of the simple generators centralchargey=l isgivenby(a=O,l,...,N-1)

Ha =

c

affine Lie algebra Uq( $AT__I) with

E,f= c E,fW ,

Ha(r),

(2.27)

r&z

IGZ where(1

of the quantum


Hi(r)

=h;(r)

ET(r)

=af(T)ai+l(r)

: - : IZ~+I(~) ~3

=: n;(r)

9

Ho(r) = ho(r) =: m(r) E,+(r) =q-+@+*f2)

El:(r)

: - : nl(r+

ajy(r)tq(r+

(2.28a)

=ii/+l(r)iii(r),

1) : +6,.,_,/2, 1))

E,-(r)

=c+r+*/*)

&r+

l)&.(r).

proo$ We must check that the realization Eqs. (2.28a,2.28b) indeed satisfy the quantum Uq(x,v_~ > in the Serre-Chevalley basis, i.e. for cy, p = 0, 1, . . . , N - 1,

Ha, Ep” 1= E,+, Ei1=&a

[

with the quantum

afine

Lie algebra

(2.29a)

,

(2.29b)

[Kxlq, 9

(2.29~)

fa,gEi

[

(2.28b)

Serre relations

(2.30)

where the notations

[xl,=

;-$,x

are the standard ones, i.e.

)

[

T 1

v

=

[ml,!

[n]4![m_n],!

’

a,p being the Cartan matrix of i,v_-] given by (N 2 3)

[ml&= Ill,*.*[ml,*

(2.31)

L. Frappar et al./Physics 2

--1

-1

2

()--I Gp

0

. . . . . .

-1 2

\

0 -1

0

319

-1

0

.*.

. ... . .. . ..

=

0

Letters B 369 (1996) 313-324

.a_... ; ... ... ... .. .. 2 -1 0

. . . -1 ... ... 0

(2.32)

2 -1 -1

2

and, for N = 2, by

llnp = Moreover

( > 2 -2 -22

(2.33)

.

in Eq. (2.29~)

and Eq. (2.30)

one has

(lW&.)/2. (1,”= q

(2.34)

(a,, a,) being the length of the a* simple root a,, and thereforeq*=qforall cu=O,l,...,N1. The proof of this theorem will be based on &he observation that the Eqs. (2.29a-2.29c) and (2.30), which define a generic deformed affine algebra Ug( d) ,reduce to U4( A) if the dot number 0 is removed, and to another finite dimensional algebra U,(A’) if the dot number 0 is kept and one or more other suitable dots are removed. In general, the relations defining U,(z) coincide with the overlap of those defining U,(d) and UC,( A’ ) ; therefore it will be enough to check that the equations defining U,(d) and (I, (A’ ) are satisfied. Thus our strategy will be the following: l step 1 ) - prove that the generators written in the anyonic representation (see Eq. (2.27) ) satisfy the relations (229a-2.29c) and (2.30) for (Y,j3 # 0. This proof is performed using the technique outlined in Section 4 of [ 51, that is going through the following steps: - step I a) - individuate a set of generators {hi, i # 0) of a low dimensional representation of Uq(A) ; _ step I b) - show that, once that the anyonic oscillators a, ZEare expressed in terms of the fermionic ones, the set {H;, E,*;i # 0) is related to the set {hi, ef; i # 0) by iterated coproduct in U,(d) and thus is a representation of U,(d). l step 2) - Repeat the same procedure for U,(d’), through the following steps: ~ step 2a) - individuate a set of generators {hb, eL*; cr E I) of a low dimensional representation of U,(d’) ;

eif;

I is a suitable subset of the extended Dynkin graph of U,(z),with 0 E I; _ step 2b) - show that, once that the anyonic oscillators a, zi are expressed in terms of the fermionic ones, the set {H,,E:; a E I} is related to the set {hb,ek%; LYE I} by iterated coproduct in Uq(A’) and thus is a representation of Uq(A’). In order to prove the present theorem, the step 1) is easily done once one realizes that, inserting Eqs. (2.14) and (2.20), the expressions (2.28a,2.28b) simplify and become

E,i(r) = e,ifr) q-kL’ where the generators

h,(r)

&(f-r)h”(f)

(

= H,(r)IF,= H,(r) and e,‘(r)

(2.35) = E:(r)jy=l, coincide

with those defined in Eqs.

(2.12a-2.12d), corresponding to the undeformed affine algebra A^N-I. of AN--I In fact, for any fixed r E Z + l/2, the set {hi(r),e”(r); i = 1,. .., N - I} is a representation of spin 0 and l/2, and thus also of Uq(AN-1) [4];thanks to Eqs. (2.27), (2.35), H,,E,fare the correct coproduct in U,,( A,+ I) . Therefore Eqs. (2.29a-2.29c), (2.30) surely holds for (Y,/3 + 0.

L. Frappat et al./Physics

320

Letters B 369

(1996)

313-324

For the step 2), we cut the extended Dynkin diagram by deleting a dot ,z # 0; for any fixed r E Z',the wherep+lIii:N-landl,e~(r),hi(r),e’(r),hj(r + l),ef(r+ I)}, _ a spin 0, l/2 representation of A N 1,and thus a representation of Uq< AN- I ) (step 2a) ) ; again, thanks to Eqs. (2.27), (2.35), {H,,E$;a # p} is the correct coproduct in Uy( AN-I).Therefore, Eqs. (2.29a-2.29c) and the quantum Serre relations (2.30) hold for any value of the indices cu,p, only excluding the couples (Y= 0, p = ,u or cy = CL,/3 = 0. However, one notes that these equations obviously hold for Ia - PI 2 2 where CY- p is taken modulo N. Therefore, for N > 4, we take, for instance, ,U = 2 and for N = 3 it is enough to repeat the argument twice, once for p = 1 and once for ,z = 2. To complete the proof, one has still to consider the case of U4( A^t>, whose Cartan matrix is reported in Eq. (2.33). In this case the previous argument fails to prove Q. (2.29~) and the quantum Serre relations (2.30), which take now the following form

(E$E:

- A(E,~)*E:E$

+A&$

(E,f)*4:

(E:)'E; - ~(E:)*E,~E; +AE$~(E:)*

(E:)~= 0,

(2.36a) (2.36b)

-E$ (E:)~=o,

where A = 9* + 9-* + 1. Such equations can however be explicitly checked by using the braiding properties of the anyonic oscillators a and ii. The central charge is equal to 1 exactly for the same reasons of the undeformed case. As the deformation of any affine Lie algebra does not change its central charge, we will not repeat this argument in the following. It is worthwhile to remark that the presence of a power of 9 in front of the anyonic oscillators in the expressions (2.28b) of Eof(r) (necessary in order to reproduce Eq. (2.35) ) reflects the non vanishing of the central charge y and is related to the definition Eq. (2.7) of normal ordering; no power of 9 in front of the anyonic oscillators in Eq. (2.28b) would be needed if normal ordering were defined according to Eq. (2.10a) or to Eq. (2.10b), corresponding to vanishing central charge. Let us remark that we could prove Theorem 1 by using the embedding u4( A^N-t ) c Uq( A,) according to a theorem due to Hayashi [ lo], that we report here in our notations:

Theorem2. Let ki, fl+(iE Z) be the simple generators of U,(A,), satisfying

I

k;, .f; [

1

fi’> fj-

=

fl2ij

f:

I -a,,

I=0

(2.37b)

9

= S;,; [kily 9

c C-1)’

1-

Qj e

[

(2.37~)

1

with the infinite dimensional

U;,j

(fyq;

(f$O,

(2.37d)

4

Cartan matrix

=

.2. . -1 .2. . -1 *. .0. . .0. . *. . i . . . .0. . -1 *.0 . -1 Then

equations: (2.37a)

k;, k,i= 0, [

the following

1.

(2.38)

L. Frappat et al./Physics

Letters B 369 (1996) 313-324

321

(2.39)

and

where LY= 0.1 , . . . , N - 1, give a representation

of the simple generators

By means of Eq. (2.35) one easily checks that the generators those defined in Eq. (2.39), once that the identification

of Uq( iN_ 1) .

H, and E,” defined in Eq. (2.27)

coincide

with

b(~ - l/2) = kn+,w , e:(f! - l/2) = f:+NL, where!EZandcu=O,l,...,

N-l,ismade.

3. More general representations

and two dimensional

anyons

In the previous section, we have built a representation of the deformed affine Lie algebras c/y( A^,__1) by means of anyons defined on an infinite linear chain; as the corresponding fermionic representation, it has central charge y = 1. Representations with vanishing central charge could be built in the same way by using alternative normal ordering prescriptions Eq. (2.10a,2.10b). Representations with y = 1 and y = 0 can be combined together by associating a representation to any horizontal line of a two-dimensional square lattice, infinite in one direction, let us say the horizontal one. So by M copies of one-dimensional representations with central charge equal to 1 one can get representations with the value of the central charge equal to M. Note that by combining one representation with central charge equal to M with a finite number of (one-dimensional) representations with vanishing value of the central charge one obtains an inequivalent representation with the same value of the central charge. We do not discuss here the problem of the irreducibility of these representations, The extensions to two dimensional lattice infinite in both directions can also be done, but it requires some care in the definition in order to avoid convergence problems. We will show now that the use of anyons defined on a two-dimensional lattice naturally gives the coproduct of representations with the correct powers of the deformation parameter 9. Each site of the two-dimensional square lattice is labelled by a vector x = (xl, x2); the first component x1 E Z’ is the coordinate of a site on the line x2 E Z. As it is shown in the Appendix in Eq. (A.8), in a suitable gauge, the angle O(x,y) which enters into the definition of two-dimensional anyons can be chosen in such a way that H(x,y)

=

+77-p i -5-p

9

ifx2>y2.

3

if

(3.1)

x2 -c y2 ,

while when x and y lie on the same horizontal line, that is x2 = y;?, the definitions of Section 2 hold. Two-dimensional anyons still satisfy the braiding and anticommutations relations expressed in the general form in Eqs. (2.19-2.25). ones, and making the corresponding Replacing in Section 2 one-dimensional anyons with two-dimensional replacements in the sums over the sites, we easily get that H, = xa,(x) x

t

E: = c

E:(x)

(3.2)

t

x

where the sum over the vector of the lattice has to be read as the sum on the (infinite) over the (finite) line x2, that is H,=xH,(linexz), x2

E,f=xqa x2

f C,,, E(vz-xz)H,(tine ~2)

E$(line

x2),

line xi and the sum

(3.3)

L. Frappat et al./Physics

322 where

Letters B 369 (1996) 313-324

and Ha (line x2), E,‘( line x2) are the simple generators of the U, (2) defined in the These equations show that the coproduct rules are fulfilled and the set {H,, E:} gives a

qa = q(an,an)/2

previous

section.

representation of the algebra U,(fi) with central charge equal to the sum of the central charges associated to each line of the two-dimensional lattice. It is important to remark that, for sake of simplicity, we have proved all the theorems of Section 2 by exploiting the representation of anyons in terms of fermions, the disorder operators giving the correct coproducts; however, this is not necessary and purely “anyonic” proofs of our Theorems can be given: it is easy to get convinced that the braiding and anticommutations relations of Eqs. (2.19-2.25) are the only relations necessary to prove that the simple generators built by means of anyons satisfy the commutations relations and the Serre equations defining the deformed affine algebras. We emphasize that anyons, defined by the braiding and anticommutations relations of Eqs. (2.19-2.25), have nothing to do with q-oscillators, that were even used to build representations of deformed algebras. In the previous sections we have discussed the case of 141 = I. The case of q real can also be discussed and we refer to [2] for the definition of anyons for generic q. In conclusions we h%ve presented a method to get representations (in general reducible) of the affine untwisted quantum algebra Lr,(A,v_l) with positive integer value of the central charge. The role of the definition of the ordering of the anyons and of the normal ordering is essential in the above construction. One can naturally ask if this construction can also be generalized to the case of the other affine untwisted quantum algebras (B, C, D series) and to the twisted affine algebras, or if a construction of a deformed Virasoro algebra can be obtained by means of anyons, or if anyons can also be used to realize affine deformed Lie superalgebras. On the same line, one could think that anyons are natural objects to use in statistical mechanical models in which quantum groups arise. We hope to address these issues in some future work.

Appendix A A crucial role in the construction of anyons in terms of fermions coupled to a Chern-Simons field, which endow them with a magnetic flux, is given by the angle 0 (x, y) under which the point x is seen from the point y. As the angle 0 (x, y) appears in the exponent of the disorder operator K(x) (2.15) multiplied by iv, v being not integer, it must be thoroughly defined without any ambiguity (see [ 21 and references quoted therein). On a two-dimensional lattice, we fix a base point B and associate to any point x a path from B to x (see Fig. A. 1). The angle 0 (x, y) is defined as O(x,y)

= B-+&,

(A.11

where 00 is a constant and Bs is the angle under which the oriented path Bx is seen from a point y* that will be eventually sent to y inside one of the four cells which have y as vertex. The point y* is shifted from the point y by an arbitrarily small vector in order to define unambiguously the angle Bs even if the path Bx passes through the site y. The simplest choice is to fix B as the infinity point of the positive n-axis and to associate to each point x the horizontal straight line coming from B (see Fig. A.2). Choosing y* = y + E(U, + uY) with E -+ O+ and u,, uY unit vectors of the axis x and y, one easily gets for two points on the same line (x2 = ~2) BFx=O,

if XI > YI ,

BTx=-IT,

if XI -c YI ,

and therefore,

(A.2)

fixing 00 = 7r/2,

323

L. Frappat et al. /Physics Letters B 369 (1996) 313-324

Fig. A.2.

W~x,y) = reproducing

tap

C

7

XI > YI ,

if

-7712

CA.3)

if XI < YI .

9

Eq. (2.16).

For two generic points x,y on the two-dimensional lattice, the angle Bs can get any value in the interval [ -T, T). However, in the braiding relations, and therefore in all calculations of this paper, the only quantities coming into play are the differences 0(x, y) - O(y, x). It is easy to check that, for any couple x. y. the definition of Fig. A.2 gives x2 > YZ XI >Yl

+r, (“>(x,y) -- 8(y,x)

(A.4) x2 < Y2

-r,

The delinition

if x2 = y2 ,

= if x2 = y2 ,

XI
of the angle 0 (x, y) induces an ordering

x+-Y

if

O(x,y)

- O(y,x)

= +7r,

X-iY

if

O(x,y)

-O(y,x)

= -7r.

This ordering x+-y

coincides

w

among the points of the lattice ( 21. We say

(A.5)

with the natural one for points on a horizontal

XI > YI

for

x2 =

line: (A.6)

YZ .

As it is possible [5 ] to deform continuously the functions the quantities O( x, y) - O( y, x) ( we will choose

O(x,y)

x=y,

with the only restriction

of keeping

fixed

CA.7)

and therefore O(x,y) =

{

w/2 -r/2

3

7

ifx2>y2, if x2 < y2.

CA.8)

References 1 I 1 A.J. Feingold and I.B. Frenkel, Adv. Math. 56 (1985)

117.

12 I A. lmda

and S. Sciuto. Nucl. Phys. B 401 (1993) 613. 131 R. Caraeciolo and M.A.R.-Monteiro, Phys. Lett. 3088 ( 1993) 58.

14I

M. Frau. M.A.

R.-Monteiro

and S. Sciuto, J. Phys.

A 27 ( 1994) 801.

324

L. Frappar et al./Physics

Leners B 369 (1996) 313-324

IS 1 M. Frau, A. Lerda and S. Sciuto, Anyons and deformed Lie algebras, DFIT 33/94-DFf-US 2/94, to appear in Proc. of the Int. School of Physics “E. Fermi”, Course CXXVII Quantum Groups and Their Application in Physics. 16) J.M. Leinaas and J. Myrheim. Nuovo Cim. 378 ( 1977) 1. 17 ) E Wilczek, Phys. Rev. L&t. 48 ( 1982) 114 and Phys. Rev. L&t. 49 ( 1982) 957. 18 1 E Wilczek, in Fractional Statistics and Anyon Superconductivity edited by E Wilczek (World Scientific Publishing Co., Singapore 1990). I 9 I A. Lerda, Anyons: Quantum Mechanics of Particles with Fractional Statistics ( Springer-Verlag. Berlin, Germany 1992) j 101 T. Hayashi, Commun. Math. Phys. 127 ( 1990) 129. I I I 1 I.B. Frenkel and N. Jing, Proc. Nat]. Acad. Sci. USA 8.5 ( 1988) 9373.