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On the quantum affine superalgebra Uq(q̂(2|2)) at level one
ELSEVIER
Nuclear Physics B 500 [PM] (1997) 547-564
On the quantum affine superalgebra Uq( at level one
(2 2) )
R. G a d e Max Planck Institute for Physics of Complex Systems, 01187 Dresden, Germany Received 21 March 1997; accepted 17 June 1997
Abstract
Motivated by applications to two-dimensional disordered s~stems this study focuses on evaluation modules ~ , V~* of the quantum current superalgebra Uq(gl(212 ) ) obtained from deformations of infinite-dimensional gl(212) representations. A free field realization of Uq(~(212) ) associated to the appropriate non-standard root system is given. The intertwining conditions for vertex operators related to the evaluation modules Vz and V~* are explored to provide bosonized expressions for the coefficients in their formal series expansions. Finally, the existence of one vertex operator is conjectured. (~) t997 Elsevier Science B.V. Keywords: Quantum affine superalgebras; Two-dimensional disordered systems
1. Introduction
Recently, integrable systems with an underlying quantum affine superalgebra have been studied in context with supersymmetric models of strongly correlated electrons. Ground state properties and excitations of various models obtained from R-matrices associated to finite-dimensional representations of quantum superalgebras are analyzed by means of the Bethe ansatz method (for references, see Ref. [ 1 ] ). Further applications of (quantum) affine superalgebras to condensed matter physics include models for the description of two-dimensional disordered systems. Fixed point theories based on 2 ( N I N ) - K a c - M o o d y superalgebras govern the critical behaviour of two-dimensional I Present address: Institut fur Physik, UniversiRit Augsburg, Memminger Strasse 6, D-86135 Augsburg, Germany. E-mail: [email protected] 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0550-321 3 (97)00407-0
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electronic systems in the presence of random abelian [3,2] or non-abelian [4,5] vector gauge fields. Systems of this type have been proposed to model a particular universality class of integer quantum Hall transitions [2] or to describe dirty d-wave superconductors [4]. A detailed investigation of the g ) ( l l l ) current algebra can be found in [6]. In a more recent attempt to provide an analytically tractable model for the generic integer quantum Hall transition, a supersymmetric vertex model constructed from two g1(212) modules V and V* has been suggested [7]. Containing su(l, 1) representations from the lower and upper discrete series as submodules, V and V* are infinite-dimensional. Another affine superalgebra relevant to the description of disordered systems such as the random Dirac theory is o-~fi(2Nl2N) (see Ref. [8] and references therein). In view of these applications an investigation of models related to the corresponding quantum affine superalgebras seems promising. In general, knowledge about the representation theory of quantum affine superalgebras is still incomplete. However, the irreducible integrable highest weight representations of quantum atfine superalgebras have been classified and shown to be deformations of their classical counterparts [9]. Based on the representation theory of quantum affine algebras and the related vertex operators a method to analyse various classes of solvable lattice models in the thermodynamic limit has been developed [ 10]. The approach permits to derive integral formulas for correlation functions making use of free field representations of the vertex operators and highest weight modules of the algebra. A step towards extending the method to quantum affine superalgebras has been made in [11[. Starting from a free boson realization of Uq(sl(MIN)), M ~ N, at level one operators obeying the same commutation relations as vertex operators associated to N + M-dimensional Uq(sI(MIN)) modules are constructed. For Uq(sl(211 )), based on an analysis of the Fock spaces the existence of the corresponding vertex operators is conjectured. Subsequently a free field realization of Uq(sl(2[ 1)) for arbitrary levels has been presented [ 12]. In this paper an investigation of Uq(fl(2]2)) is started. To be able to construct the universal R-matrix Uq(gl(212)) has to be considered rather than Uq(sl(2]2)) [13]. As stressed in [ 1], a Lie superalgebra has many inequivalent systems of simple roots which yield different Hopf algebras upon deformation. Motivated by the physical applications intended in [7], all simple roots are chosen odd in contrast to the standard simple root system used by the authors of [ 11 ]. In the classical limit q --~ 1 the currents corresponding to the even roots generate a ( i , 1) ® t~(2). So far the current algebra k~(1, 1) has rarely been studied. Representations of ~ ( 1 , 1) can be build up on the base of the unitary representations of su(1, 1) [14]. Thus part of the Uq(gl(212)) modules contain deformations of the ~k(1,1) representations related to the discrete series 79~+~ or D ~-~ or to the continuous series C as submodules. If vertex operators associated to the modules V and V* exist, the homomorphisms are expected to involve affine representations of this type. A free field realization of Uq(~(212)) with the particular choice of simple roots is provided making use of the procedure presented in [ 15] for Uo(s)(2) ) at arbitrary level. Examples for weight modules are given, a more complete account of level one representations is relegated to a separate publication. The lack of A
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a general criterion on the existence of vertex operators for quantum affine superalgebrascomplicates further analysis considerably. A classification of vertex operators for Uq(gl(212) ) is far beyond the scope of this paper. As exemplified in a number of cases (see references in Section 4), bosonized expressions for vertex operators are obtained from their commutators with Drinfeld's generators of the quantum affine algebra as dictated by the intertwining condition. This procedure is carried over to Uq(gl(2f2)) to construct operators subject to the same commutators as vertex operators associated to infinite-dimensional Uq(gl(212)) modules if the latter contain an dement annihilated by the Chevalley generators fi, i = 1,2, 3. If instead the infinite-dimensional module is generated from an element v such that eiu = 0 for i = 1,2, 3, another set of Drinfeld generators is required. It is related to the standard set by the antipode. As bosonization follows from the Drinfeld realization, this implies the introduction of two free field realizations of the quantum current superalgebra. Finally, the existence of a vertex operator related to the module V* is conjectured. The paper is organized as follows. In Section 2, the definitions of Uq(})(212)) and Uq(~(212)) are given in terms of Chevalley's and Drinfeld's generators together with the relation between the latter. Evaluation modules obtained from deformations of the gl(212) modules of Ref. [7] are explained in Section 3. The free field realization is considered in Section 4. Section 5 addresses the issue of vertex operators. Section 6 contains a summary. The basis of the modules Vq and Vq obtained from V and V* by deformation is described in Appendix A.
2. The quantum affine superalgebra Uq(/~(212)) For a given (affine) Lie superalgebra several inequivalent systems of simple roots exist [ 16]. The quantum superalgebras obtained for different systems of simple roots are related by twistings [ 17]. Motivated by the physical applications mentioned in the introduction, the choice adopted in this paper differs from the standard set considered in most of the literature on the subject. Denote the set of simple roots by //{a0, a l , a2, as, a4}. A symmetric bilinear form ( ai, aj ) = a"iym is defined by
a sym =
0 -1 0
-1 0 1
0 1 0
1 0 -- 1
1 -1
0 1
--1 0
0 1
-1 ) 1 0 .
( 1)
1 0
In terms of the basis { r l , r z , r 3 , r 4 } with the bilinear form (ri,'rj) = --(--1)i•i,j the classical roots &i = - ( - 1 ) i ( r i + ri+l),i = 1,2,3, &4 = rl - r4 and Ai = Y'~-5=lrj 12~j41 "/'J"Introducing an affine root a and an affine weight Ao with (Ao, Ao) = (& 6) = (ri, Ao) = (6, ri) = 0 and (Ao,8) = 1 the set of simple roots is given by a0 = - &l - &2 - &3 and ai = &i for i = 1,2, 3,4. The free abelian group P = @4=oZAi + Z 6
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w i t h Ai = zii q- Ao for i = l, 2, 3 and A4 = A4 is called the weight lattice. Via ( , ) its dual lattice P* = 04__oZhi 4- Zd can be identified with P by setting ai = hi and d = A0. The quantum deformation Uq (si(212)) of the universal enveloping algebra U(sl(212 ) ) is introduced as the unital Z2-graded associative algebra generated by the Chevalley generators el, fi, qh,, i = 0 . . . . . 3 through the defining relations
qh,qhj = qhjqh,, qh'ejq-h' = q( .... J)ej, qhi -- q-h, [ei, fi] =•id q -_ ~
qh,fjq-h, = q-(~,,aj) fi '
(3)
,
[el,e3] = [e0, e2} = [ f l , f3] = [f0, f2] = 0,
[[eo, e,]q,[eo,e3]q] =0,
[[fo,f,]q ,,[fo, f3]q-,] [[fl,f2]q-i, [f,,fo]q ,] [[f2,fl]q-l,[f2, f3]q-~] [[f3,f2]q-l,[f3,fo]q-,]
[[e,,e2]q, [e,,eo]q]
=0, [[e2, ej]q,[e2,e3]q] =0, [[e3,e2]q,[e3,eo]q] =0,
(2)
(4) =0,
=0, =0, =0.
(5)
In (3) and (4) [, ] denotes the usual Lie superbracket [x,y] = x y - (-1)14l:'lyx. Due to the special choice of simple roots the Z2 grading ]. I : Uq(M(212)) ~ Z2 assigns the value 1 to each generator ei, fi and the value zero to @. The q-deformed supercommutators in the Serre relations (5) are defined by
[el, ej ]q = eie.i -+-q(a"aJ) e.iei, [fi, fi]u-~ = f i f i + q-(""aJ) fJfi.
(6)
In the homogeneous picture a grading operator d is defined by
[",ei] :(~i,oei,
[d, Yi] = --~i,oYi,
[d,q h'] : 0 .
(7)
The algebra Uq(J)(212)) admits a comultiplication A and an antipode S:
A(ei)=qhi®ei+ei®l,
A(fi)=fi®q-hi+l®fi,
A(q:~h,) = q~h, ® q±h,
(8)
and the antipode
S(ei) = --q-h'ei,
S(fi) = - f i @ ,
S(q ~:h~) = q:Fhq
(9)
The antipode is a Zz-graded algebra antiautomorpbism, which states the property
S(xy) = (-l)[xl'lYIS(y)S(x)
(10)
for any two homogeneous elements x, y of a quantum affine superalgebra. A further peculiarity of a graded Hopf algebra is the supercommutativity of the tensor product:
(v ® w) (x ® y) = (-1)lwl'lxlvx ® wy.
(11)
A realization of Uq(~(212)) particularly convenient for the purpose of bosonization is provided by Drinfeld's generators [ 18]. The bosonization of vertex operators related
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to the evaluation modules introduced in the next section will require two sets D(a) of Drinfeld's generators { E ~ , H~,a, q4-hj, y± l } with a = + 1, m C Z, n C Z - {0}, i = 1,2, 3 and j = l, 2, 3,4. The generators of each set satisfy the defining relations qn(C%a/ ) _ q - n ( a j , % , ) y n _ y - n
[ H~,a, H~',,a ] = ~.+n'.o
n(q_q-l) qh~ ,~i,=t=q - h~ _ ,~+ ( ,~. aj ) l~i,-4~--'~nl,¢l
- - "1
j
i,q-
[Hn,a, Em,a] = ±
q_q-I
'
~ l l l ~ a "j
q n(ai'a)) -- q - n ( a i ' a j )
-~
~q ,
"'
a
e~fm,aYT~n'
a t i i,+ '.t '- . 6.,,q )et,+ [Em.a,Em'.a] . . . aq_ 1 [.,~(m-m \J" - - m + m ' . a - t,~,--~(m--m')!lti,--,n+m',a)"~
(12)
with J'+ Z-" = qahj exp ( a ( q _ q - l ) Z H J , a z - n ) , qtn,a
Z n)0
n>0
Z ~ l ~ _ n j,,a z n =q-ahj
exp(-a(q _
(13)
nJn,aZn)
q-l)Z
n)O
n>O
and i,--Ii',--b [Em,~,E'm,,~] =0
for (ai, ai,) = 0 ,
(14)
p~,4I~l ,~ .a- ,-,4-a(al,ai') lg ,4-El,q_ i~,~ ,-I- igt,q~ m + l , a ~ m J , a ~ '4 ~m',a m+l,a ~m'+l,a~m,a •
4
i r
•
4
.
_
,.4-a(eti,ai,)17~,&17 i t ,-I- = 0, tt ~m,a~mt+l •
(15) [Em,a,Em,.a]q±,
, [Ek,a,Ek,,a]q±,
[Ek, a, Em,,a]q± , [E~,'a, E k ' , a ] q ±
h-
= O. (16)
The Z2 grading is given by IE&,~I = 1 and IH~,~I = Iq+hj[ = ITI = 0. Analogously to (6) the deformed supercommutators in the Serre-relation (16) are defined by •
r ~" + E' t ~m,a,
.i
"+ ]J q,
mt,a
.
.¢
!
.t
,
= lgt' 4-1g' ,-4- ..1- n ( ai,ai' ) Igt '4-Ig','4~m,a~mt,a | '4 ~mt,a~m,a ,
q,
~
q+ l •
(17)
Drinfeld's realization of Uq ( ~ ( m In) ) has been obtained in [ 20]. The Chevalley generators are related to the sets D(a) by the formulae
ei = Eo,i,+a,
i,fi = Eo.~
for i = 1,2, 3,
qho = yq--hl--h2--h3
[E~),a ,El, ~ ]q. fo
= qa(hl+h2+h~)
--
q-~[E~,-~,~.~,-~ w3,-~.-~(h,+h:+h3) , , tal,a Jq"L'0, a ]~/
1,+ 2,+ [[E_l,a, EO,a ]q-.,
~_ qa(hl+h2+h3) fl- gTl,+
k t~-l,a'
K,2,+ l
Eo,a3,+] q - " p3,+
~O,a Jq-"~O,a
_a r~3,+ r ~ 1 , +
~:,2,+~
"~
-- q l:0,a t t Z - l , a ' C~O,a l q - ° ) .
(18)
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R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
By means of the identification (18) the grading, coproduct and antipode of Drinfeld's of Uq(sl(2]2)) can be obtained from (7), (8) and (9) whose definitions for qhi apply to qh,, too. The following partial description of A(H/~) proves to be sufficient to derive expressions for the vertex operators: A(H~,I)_~Htn,, ®,y2 + " ~ ® H',1 modN_ ® N+,
A(Hin,1)=H i_~,] ® y - 73,, + y_~ ® Hi-n,J modU_ ® U+, A(H~ ,- ,)_- Hni _ I ® y-~ + y~" ® H.._~ modN+ ® N _ , A(Hin,-l)--Hi-n,-l®Y-~"+Y~®Hi-n,-lm°dN+®N-
n>0, i=1,2,3. (19)
Here N+ are left Q ( q ) [ y + , ~'~.a, i,+ ~0-.,a, i,- . n E Z/>0] modules generated by {E~ +, m E Z}. The definition of A(H~:.,~) is provided by the construction of the universal R-matrix [ 19] for Uq(~(212)). Following [21] an automorphism Dz of Uq(gl(Zl2))®C[z, z -'] and maps A z , A z, : Uq(~(212) ) ®C[z, z -1] ~ Uq(gl(212) ) ®Uq(gl(212) ) ®C[z, z -i] are introduced as O~(En,a) i,:L = Z nr,i,± 15n,a ,
oz(/L,.)
d:(x) = (D z ® id)d(x),
= Z . iHn, a,
Dz(a)=a,
A'z(X) = (Dz ® id)A'(x)
(20) (21)
with d~(x) = o-o d(x), tr(x ® y) = (-1)Ixl.I;'ly ® x. The spectral-dependent universal R-matrix is the unique element R ( z ) E Uq(gl(212))+Uq(~(212)) ® C[ [z ] ], where is the completion of ® over C[ [q - 1 ] ] such that
~ ( z ) A z (x) = A~z(x)7~(z),
(22)
(A z ® id) (7~(w)) = n13(zw)~23(w), (id ® Zlz) (~(zw)) = ~13(z)TClz(zw)
(23)
with 7~12(z) = ~ ( z ) @ 1,7~23 = 1 ® ~ ( z ) , ~ 3 ( z ) = (o-®id)Te23(z). In order to satisfy F-Xl. (22) for any x E Uq(s)(2[2)), Te(z) has to contain H4~,,+ (or H4n,_). Then Eqs. (23) imply the definition of A(~'4,'~1 ) by means of the following expression. Writing 4
4
(L+ ( zq_ly_~ ) ) = ]-IaP'I "±( z ) = I-[ ( ~ i=1
i=1
~±n,,Z
(24)
n/>0
the coproduct of L + (z) is given by
A(L±( z ) ) = (I ® L+ ( zq~('+I)c') ) 4
× ( l ® l + L a i l ( z q :1:-
½( izF1)cZ)®B~ (zq½(ld:l)cl))
i=2
x(L±(zqJ (1~:|)c2) ® 1),
(25)
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where c1 = c ® 1, c2 = 1 ® c with qC = y and
A~(z)=(q-q-1)
Z
E~-~(Zy½(I~q))Tn'
n=½(l+l)
A ~ ( z ) = - ( q - q -l)
~
[E2;,E2'-]q-,(zY½('~:')) ~:",
n=½(l+l)
a ~ ( z ) = ( q - q -1) ~
[[EI"2,E2"-]q-,,E3o'-]q_I(Z~/½(I~I)) :qzn,
,,=½0+l)
B~(z)=-(q-q-l)
Z
EL+(ZY½(I+I))
n=½0:Fl)
B~(z)=(q-q-l)
Z
[E02'+'I~'I'+] g"l(14"l)/::Fn ~,4-nJq~,~f ! ,
n=½(l=gl)
B~(z)=(q-q-')
Z
1,+ [E°3'+'[ E;2,+ 'E-bn]q]q (z~1½(l'4-1,)~n"
(26)
n=½(l=F1)
Making use of (13), (18) and (8) the expressions for A(H4~n,+) can be obtained. The antipode relates the two sets of Drinfeld's generators according to i,+
S(Em,1) = --
T--m -hi Ei,+
q
m,-l'
S ( E mi,+ ,_I) =
--Y-m q-hi--i,+ P-'m,l'
i,-s(<,,,) -m q ,z,,,,_l, =-y
s ( < , i,-_,)
s ( < , , ) = --y - " JHn,_ 1,
S ( h ' j° _ , ) = - ~ '
=
..,,
-n < ,j, ,
Vm, i = 1,2,3 Vn 4= 0, j = 1,2,3,4 (27)
Applied to ( 2 4 ) - ( 2 6 ) Eqs. (27) yield A(H4~n,_I). Due to ( 2 4 ) - ( 2 6 ) , (27) and (10), Eq. (19) holds true also for i = 4. Further details as well as an explicit expression of 7~(z ) in terms of Drinfeld's generators will be given in [ 13 ]. Finally, the commutators of the homogeneous grading operator are given by
[d, Em,a i'+]J --- mEre,a, i,± [d, <,o] = nW;,o.
(28
3. An evaluation m o d u l e from an infinite-dimensional Uq(gl(2]2)) module
Generally, one has to take into account the direct sum of several modules of a nonaffine quantum algebra Uq(g) [22] to obtain an irreducible level-zero representation of the affine quantum algebra Uq(~). In the case of Uq(~l(N)) however an evaluation homomorphism permits to lift any Uq(sl(N))-module to a Uq(s'l(N)) module [23]. For the quantum superalgebra Uq(9(N]M)) there exists an evaluation homomorphism
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p : Uq(f'I(NIM)) , U q ( s l ( N I M ) ) ® C[z, z -I ] as well [24]. For Uq(gl(NIN)) the action of elements not contained in Uq(sl(N]N)) needs to be introduced separately. Specializing to N = M = 2 p is given by A
p(ei)=ei,
P ( f i ) = fi,
f o r / = 1,2,3,
p(q+hj) =q+hj, f o r j = 1,2,3,4, p( q±ho ) = qT( hl+h2+h3), p(eo) = q - h 4 ( f 3 f 2 f l + q-l f 3 f l f 2 -- q f 2 f l f 3 -- f l f 2 f 3 ) , P(fO) = (ele2e3 + qe2e~e3 -- q-le3ele2 -- e3e2el)q h4.
(29)
In the following the evaluation homomorphism will be applied to obtain affinizations of particular infinite-dimensional representations of Uq(gl(2]2)) which in the limit q ~ 1 coincide with the modules discussed in [25]. To elucidate the structure of the gl(2NI2N) modules called V and V* in [25], let us focus on the subalgebra gl(N[N) ® gl(N]N). If N = 1, the latter is generated by { e l , f ~ , e 3 , f 3 , h j } at q = 1. V and V* are one-dimensional irreducible representations of the first factor gl(NIN) containing the elements v0,0 and v~,0, respectively. The action of the second factor on v0,0 and v(~,0 vanishes completely. The Uq(gl(212)) module Vq is conveniently parametrized by a pair of labels ( i , p ) , i = 0, 1,2, 3 and p = 0, 1,2 . . . . In this basis the Z2 grading yields 1 for vl,p, v2,p and 0 for Vo,p, V3,p. The weights are given by
hlvo,p hlVl,p hlU2,p hlv3,/,
= = = =
-pro,p, - ( p -}- l)Ul,p, - ( p -[- ])U2,p, - ( p + 1)V3,p,
h2vo,p = - ( p + 1)VO,p, h4vo,p = - ( p + l)V0,p,
h2Ul,p = - ( p + l)Ol,p, h2o2,p = - ( p + l)u2,p, h2g3,p = -pv3,p,
h4Vl,p = - ( p + 3)Ol,p, h4o2,p = - ( p + l)o2,p, h4v3,p = - ( p + 2)v3,p.
(30)
For any element vi,p of the module ( h i -+- h3)vi,p = 0 holds. The elements v0,o and v~,o satisfy
eivo,o = fiv~,o = 0
for i = 1,2, 3.
(31 )
Vq is spanned by the elements fi~ . . . fiNvo,o where (il . . . . . iN = l, 2, 3) as visualized by the diagrams below. Alternating applications of f l , f2 yield the elements VO,p and V2,p: f2 U2,0 ~ UO,O f2 U2,1
U2,2 (
f2
U0,3
(
f2
II,
U0,2
~ fl UO,1
(32)
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The remainder of the module forms a similar sequence: U3,0 /33,1 f2 U3,2 (
f2
f2 <
I fl V1,0
I fl U1,1
(
(33)
V1,2
Both sequences are related by the action of f3 as f3VO,p = - - [ p ] V l , p - 1 ,
f3VZ,p = --V3,t,,
(34)
where the standard notation [k] = is used. Replacing fi by ei and vi4, by vi4, yields the corresponding diagrams of the module Vq*. The detailed description of the basis of Vq and Vq can be found in the appendix. With a suitable normalization Vq* can be viewed as the dual module Vq s of Vq by (xv*,v) = - ( - l ) l x l ' M ( v * , S ( x ) v } with the dual basis (v[,t,, Ui,.p, } = (~i.i,(~p,p,. Due to the infiniteness of Vq and Vq there exists no isomorphism between the two modules. In the homogeneous picture the evaluation modules Vz = Vq ® C(z, z -1 ) and V.* = Vq ® C(z, z - l ) are equipped with a Uq(~(212) )-module structure via
ei(vi,p ® z") = eivi,p ® z "+''°,
fi(Ui,p @ z n) = fiui, p @ Z n-6i'° ,
ei(vi*q, ® z') =eivi*,p @ z "+''°,
fi(vi*,p ® z') = fivi*,p ® z n-~''° ,
Wt(Ui, p ® Z n) = rl(~ -- ( p dr- 1) (z~1 + ,~ -- z~3 + a4)
-~-t~i,0 (z~ 1 -- zt3) -- 2t~i, lZ~4 + ~/,3 (z~ -- z~4) ,
wt(v;,p ®z o)
(p + 1)(a,
- a3 +a4)
-6i,0 ( / t l - / i 3 ) + 28i,~zi4- 8i,3 (zi2- A4)
(35)
and n 4m , - I (vo,o @ z n) -- -
--
q2m -[m] m
VO,O @
[m] . zn+m Hn.,,l(v~,o®Z")=--m-VO,o®
zn+m '
Vm ~ O,
(36)
where in (36) Drinfeld's generators are chosen as required by the construction of vertex operators in Section 5.
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4. B o s o n i z a t i o n
Constructing the quantum affine algebra Uq(~(2)) at level one and its vertex operators through bosonization allows to derive an integral formula for n-point correlation functions of local fields [26]. Several realizations of Uq(sl(2)) at general level k and its vertex operators in terms of three sets of bosonic modes have been proposed in the literature [27-30]. The relations among them are clarified in [ 15]. A generalization A of the bosonization scheme to Uq(sl(n)) at level one has been used to calculate the one-point-function of vertex models associated to the vector representation in [31]. For A Uu ( s l ( n ) ) at general level k a suitable deformation of the Wakimoto free field realization of sl(n) has been constructed in [32]. A bosonization scheme for U q ( ' S ' I ( N + I i M + I ) ) with the standard choice of simple roots can be found in [ 11,12]. Guided by the procedure explained in detail for Uq(~(2)) in [15] a bosonization A of level-one Uq(gl(212)) with the system of simple roots chosen above is obtained in two steps. The construction leads to a deformation of the bosonized realization of ~(212) by bilinears in commuting and anticommuting ghost fields bi(z),ci(z ) and fli(z ), yi(z ), i = 1,2. Their non-vanishing operator product expansions are given by ci( z ) bi( w ) = - b i ( z ) c i ( w ) -
yi( z ) fli( w) = fli( z )yi( w) =
I Z -w'
1
(37)
Z--W
In terms of the ghost fields, the Chevalley generators ~i, J~ of ~(212) are written
~l(z) =- : cl(z)fll(z) e2(z)
-- " ~ / I ( Z ) C 2 ( Z )
:, :,
g3(z) = : b 2 ( z ) y 2 ( z ) :,
fl(z) = : yl(z)bl(z) /2(Z)
= :b2(z)t~l(Z)
:, :,
(38)
f 3 ( z ) = : f l 2 ( z ) c 2 ( z ) :.
A bosonization of b i(z ), ci(z ) and fli(z ), Yi ( Z ) into commuting free fields following [33,34] yields the limit q --+ 1 of the bosonization scheme outlined below. The affine quantum superalgebra is conveniently reexpressed in terms of the generating functions (or currents) = V
TM
l p j , + 7 --n
n~>0
n>~O
E~;+(z) = ~-~' E~,a -m-l mEZ
Then the defining relations (12) are rewritten as
i,+
j,-
"tFa (Z)qta
(z - wyq(m'~J))(z - w ~y - i q -("''~j)) J'-• i~~--W '+ wyq-(CmaJ))(Z-wy-lq('~"aJ)) ~a ( )~a (Z),
( w ) = (Z
(39)
R. Gade/NuclearPhysicsB 500 [PM] (1997) 547-564
557
Fi'+( "~E j'+ (w~ - q±a(al,aj) Z - WTT~q q:a(ai'aj) E~,+(w)q:i.+(z) Z
-
wym ~q+(a~,aj) a
~i.-(z)E~,+(w ) q+a(.... s) zz -- wy+~q wy±~q +a(~'~j) 4-a(~''~j)Ea., (w)~a (z),
(40)
6,,; (~i,+(w3/~) E'i;+(z)E~-(w) "" w ( q : q - ' ) \ z ----w--yy
(41)
~ ' - (w7-½))
z--~y_~
,
EIi+(z)E~'+(w) = -E{'+(w)EI;+(z) for (ai,c~/) = 0, (wq+a(a"~J) - z)E~+(z)EJa'+(w ) = (zqma(~,~J) - w)EJd+(w)E~+(Z),
[IEX,+(
I
+[[E2'+(Z2),Ela'+(Wl)]qa: , , [E]'+(z|),E3'+(w2)]q±d =0,
(42)
where use of the symbol ~ indicates the omission of terms regular in the limit z -~ wy+. Realizing the quantum current superalgebra as a deformation of (38) implies 3 / ~ q-1 at level one. First, bosonization of the currents ~ ' + (z) is achieved by means of the four deformed free fields
~;+ ( z ) = ~ - i q ~ 0 1 n z + i n÷o--(-~q~nZ Z q+a~ i -n ,
j = 1,2,3,4,
(43)
where the boson oscillators {q~J,q~Jn}fulfill the deformed Heisenberg algebra
n
[,p/,
=
(44)
By means of {~,q~i} the currents ~ ' + ( z ) are realized as
q:J'+"t z ) = q -aij(~-i~°+') exp ( - a ( q - q - '
) E i J ( ~ -iq~+')z -n) , n>0
~J'-(Z )=q ai~(~°-i~+') exp ( a ( q - q - ' ) E iJ(q~j-n -i~°J-+~n)Z") ' for j = 1,2,3 n>0
(45) and ~,,'4 + ( z ) = q-a(i~o~+~og) exp (_a(
q _ q-~ ) ~-~(iq~'. +~pa)z - . ) , n>0
~4"--(Z ) =qa(i~°~+~°~)exp(a(q-q-1) ~-~(i~pln+~O4n)zn ) . n>0
The currents
E~'+(z) depend on the fields q~+(z) in the following way:
(46)
R. Gade/Nuclear Physics B 500 [PM] (1997)547-564
558
E:J'+(Z) =exp(-t-iJ+l(q~+ ( z ) - i~a+l'+(z)))exp(Sj,,Trq~2)xJ':k(z),
j = 1,2,3. (47)
It is easily verified that the expressions (45), (46) and (47) fulfill the relations (40) with y = q-J. The cocycles exp(Sj,37"r~o02) are needed in order to obtain the right signs in the remaining relations. Second, X j+ ( z ) have to be constructed such that their operator product expansions insure (41) with Y = q-1 and (42). As can be inferred from the bosonization of gl(212), two more fields are necessary to complete the expressions for E~'± ( z ) . A suitable choice of the fields is m
flJ(z)=flJ
-
i f l ~ l n z + i ~ - ~ -z-n
J
j = 1,2
(48)
n~O
with [/T,/3ok] = -i~j,k,
[/3J,/3~,] = -n~.+m,OSj.k.
(49)
Then the X)'+(z) are realized as XI,+(z) _
z(q
1
q-l)
(exp (/3' ( q - ' z ) )
- exp(/3' ( q z ) ) ) ,
XJ'-(Z) = exp(-/3 j (z)), X2'+(Z) = X 3 ' - ( z ) - z(q -1q - l ) ( e x p ( / 3 2 ( q _ , z ) ) - exp(/32(qz))) ' X 2'- ( z ) = -X3'+ ( z ) = exp(-,82 ( z ) ) .
(50)
Part of the level-one weight modules V(A) of Uq(fl(212)) with the decomposition V(A) = ®.ee V ( A ) . , V(A)~ = {v ~ V(A) [ qhv = q(h'~)v Vh E P*} include submodules reducing to ~ ( 1 , l) modules of the lower discrete series [14,35] in the classical limit q --+ 1. These modules contain a vector va E V(A) such that eigA = O, i = 0 , 1 , 2 , 3 and qhJgA = q(hj'A)vA f o r j = 0 , 1 , 2 , 3 , 4 , H4,_IvA = 0 for n > 0 and V(A) = Urn. An example is furnished by V(A2,4) with (h2, A2,4) = ( h 4 , A2,4) = - 1 and (hi,A2.4) = ( h 3 , A 2 , 4 ) = 0. The vacuum representation V(Ao) is spanned H = H n~,-l''" 4 H nM,_lVAo, 4 by f i , . . . f i N V~,, il . . . . iN = 0, 1,2,3 where VAo nk < 0 and qhJvAo = UAo for j = 1,2,3,4. Analogously, a module V(A*) containing an ~ ( 1 , 1 ) representation of the upper discrete series for q --~ 1 has an element distinguished by fiVA* = 0, i = 1,2,3 and qhJvA* = q(hj'A*)uA* for j = 0 , 1 , 2 , 3 , 4 , H~,jVA* = 0 l b r n > 0 and V(A*) = UVA.. The corresponding example is given by V(A~, 4) with (h2, A~,4) = ( h a , A~,4) = 1, ( h l , A ~ , 4 ) = ( h 3 , h ~ , 4 ) = 0. The reason for the choice of Drinfeld's generators H4,+ made here will become apparent in the following section. In terms of the oscillator algebras the vectors va2.~ and VA;.4 are realized by /)A2.4 exp(~o' - i~o2)[0), ~-~
VA;,4 = exp(--(q~' -- i~,2))[0),
(51)
where 10) denotes the vacuum state defined by ~oil0} =/3.k10) = 0 Vn > 0. Bosonizafion associates the vacuum state 10) to va0.
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
559
5. The free hoson realization of a type I vertex operator Vertex operators of type I are defined as intertwiners of U-modules in the following way [36]:
4~AV(z) : v ( a )
~ voz) ® vz.
(52)
Here Vz and V(,~), V(/x) denote an evaluation representation and highest weight modules of U, respectively. The vertex operators obey the intertwining condition
• ~V(z)ox=zl(x)oqg~V(z)
VxcU.
(53)
The vertex operator construction, first introduced and studied in [21] provides a method to construct eigenvectors of the corner transfer matrix [36] and thereby allow, at least in principle, for an evaluation of correlation functions. For quantum affine algebras a general criterion about the existence of vertex operators with finite-dimensional modules V has been proven in [37]. Not disposing of such result in the case at hand, one may resort to construct bosonized expressions for suitable operators satisfying the intertwining condition in a first step. Then, studying the action of the bosonized currents and vertex operators on the weight spaces of V(A) succeeding UA or UA. will yield a hint on the existence of vertex operators. To this aim, vertex operators associated to the evaluation modules Vz and Vz* described in Section 3 are introduced as formal series ~,u. V ~a (Z) = ~
~,u V t~A ,i ,p ( Z )
@Ui,p,
~ *p. V* *;~ (Z) = ~
i=0,l,2,3 p=0,1,2.... ~p. V
. g~V*(Z)® ~A,i,p Ui,p '
(54)
i=0,1,2,3 p--O,I.2....
~
V
qb2t,i,p(Z)=~rlzAA,i.p,nZ--n' ncg
a~*/~ V*
.7.*/z V*
~h,i,p (Z) = ~ tPh,i,p,nZ nEZ
--n
.
(55)
The coefficients in the last line are linear maps
,~"A,i,p,n •" v ( a ) -.11. t~,i,p.n : V ( , ~ )
,
v(iz)
)
V(ix),
(56)
subject to the intertwining conditions
(-1)l(i'l"Hxl~ ~)gAA,iV, p,nXU® (Ui,p ® Z -n) i,p ,n
® (n,,, ® z - ° ) =
i,p,n
t~)lzA,Vp,nU® (Ui,p ~ z - n ) ) ,
=A(X)(~-'~
i,p,n k2_. i,p,n
®
® z -n (57)
for all x E Uq(~(212)) and v E V(A), where I(0,p)l = 1(3,p)l = 0 and I ( l , p ) l = 1(2,p)l = 1. Operators ~b(z) = ~-~ipt~ip(Z) ~ Ui,p -- ~i,p,nf~i,p,n ~ (Ui,p ~ Z -n) and ~b*(z) = ~i.,, &i*,,,( z ) ® vi*,p = ~-~i,p,nqbi*,p,n® (U~,p ® z-n) satisfying the intertwining condition
560
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
can be determined to a l~ge extend by the method used in [ 10] to construct the type-one vertex operators of Uq(sl(2)) at level one. In order to obtain bosonized expressions for the vertex operators, the restrictions due to the intertwining condition with x = qd, qhi, ei ' fi are reformulated in terms of the Drinfeld's generators of Uq ( s'l(2t2)). Eq. (19) indicates the choice of generators: the set Dr is appropriate to derive equations for the bottom component ~b~.0 (z), while analogous equations for the top component ~0,0(z) are obtained using D_j. The intertwining conditions for qhj and qd read q-hJqbi,p ( Z )qhj = qX(j;i,P) ~i.p ( Z ), q -h~cb*Ti,p
(z)qhj=q-~r(J;i'P)(b[,p(Z),
j = 1,2,3,4,
q-d q~i,p( Z ) qd = fbi,p ( qz ), q - d q~* d- * i,p (z)q - ~i,l , (qz),
(58)
where hjvi.p = K(j;i,p)vi,p with K(j;i,p) listed in (30). Making use of (18), (12), (13) with a = -1 and (19), (36) one finds for the top component ~b0,0: [En~:_i,, ~bo.o(z)] = [E3:_i,, q~o,o(z)] =0,
E~;-_jfbo,o(z) - q-~qbo,o(z)E~;-_ I =0 Vn
(59)
and 1
[Hn _l , q~O,O( Z ) ] =[n._,.4,o.o(z)] 3
=o.
v,,,
2 )] H4 [n] [Hn, - l ' ~ b 0 ' 0 ( z = [ n,-I '~b0,0(z)] = q-~(q2z)nfbo,o(Z), n [ H 2- n - 1 , ~ 0 , 0 ( Z ) ]
n>0,
[n] - ~ 4 =[H-n,-I,OO,O(Z)]= q (q2z) - n ~b0,0(z), n
n > 0. (60)
Choosing a = 1 one derives for the bottom component ~ . 0 ( z ) fut-~n,l = " + , "V-J0,0 * (z)]
3,± ,, = [H°,, = [H°., ,6oo(z)] 2,+
* [H1.,,,¢bo.o(Z)]=[H3,,dP~.o(Z)] [H2,,, ~b0,0(z)] •
o,
Vn.
(61)
[n]q-~nznqb~o(Z),
H4 =[
=
=-
[H 2-n,1 , q~;,0(Z)] = [ H4_ . , t , ~ ; , o ( Z ) ] =
n
n>0,
.
[n]q~z-"q~;,o(Z), n
n>0.
(62)
J fix the dependence of ~bo,o(z) and q~,o(Z) on q~J and The commutators with H.,+ for n # 0. The classical limit q --+ 1 indicates the choice of the solutions to (59) -(62): ~bo,o(z) = exp ( - (~o~+ (q2z) ~b;,o(Z) =exp(~p{'-(q-'z)
- iq~2-+(q2z)) ) e x p ( - l n q ( ¢ ~ - i~p2) + 7-rcp3), -i~@-(q-'z))exp(crq~3).
(63)
All other components of ~b(z) and if* (z) are determined from remaining restrictions imposed by the intertwining condition:
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
561
4'2,p ( Z ) =q-(p+l)[4'O,p(Z ),e2]q-h2 ,
4'O,p+l(z) - 1--------~q [p + 1 -~p+l) [4'2,p(Z) el]q -h', 4'l,p =,/~p+l r-" -h3 , [q)0,p+l(Z), e 3]q
4'3,,,- p +1 1-----5[4'2,p (Z), e3] q-h3
(64)
and
4'L,(z) -- [4';,.(z), f2]q_,.,,, . 4'0,p+l(Z) =
1 , ] [ p + 1] [4'2,p(z),fl. q-,,,~l,'
* * Z ),f3]qp+,, 4'1,1,(z) = [4'0,p+l(
4',~,p(z) =
1 [p + 1] [4'~,p(Z),f3]qe~.
(65)
In (65) the commutators are defined as [x,y]q~:~p+, = x y - (-1)lxllylq+(p+l)yx with I4,*0,pl = 4 I' * 3,pl = 0 and I4'*t,pl = 4 I' * 2,pl = 1. Eqs. (63) and (65) can be used to examine the action of 4'*(z) on VAo as well as on its first descendants. This yields a structure consistent with the existence of a homomorphism V(A0) ~ V(A2,4) (~ Vz. Thus one is led to conjecture the existence of the following vertex operator: t~v(a2,4) V* ( V(Ao) Z) : V(Ao) ~
,.
V(A2,4) ® V£
(66)
Identifying =V(ao) ,~v(~2.4)v" (z) with 4'. (z) the bosonic realization of the vertex operator is given by (63) and (65).
6. S u m m a r y The quantum affine superalgebra Uq(~(212)) has been considered with emphasis on a pair of evaluation modules related to infinite-dimensional representations of gl(212). A free field realization of the current superalgebra and the coefficients in the formal expansion of vertex operators involving these evaluation has been obtained. Certainly, a number of issues need to be resolved to gain sufficient insight into the representation theory of the algebra including non-integrable modules. In particular, vertex operators intertwining modules of the continuous series have to be taken into account in order to provide the framework for an analysis of vertex models. Work along these lines is in progress.
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
562
Acknowledgements The author wishes to thank M.R. Zirnbauer for several illuminating discussions regarding his ideas developed in [7]. She is grateful to T. Miwa for encouragement and helpful comments during her participation at the semester "Integrable Systems" in Autumn '96 in Paris.
Appendix A The following relations fix the basis of the U~q(~(212)) module Vq: hlvo,p = -pvo,p, hevo,p = - ( p +
hlVl,p = - ( p
1)vo,p,
h3vo,p = pro,p,
1)nO,p, - ( p + 1)V2,p, - ( p + 1)v2,p, (p + 1)v2,p, - ( p + 1)V2,p,
+
l)Vl,p,
h2Vl,p = - ( p + l )vl,t~, h3Vl,p = (p +
1)Vl,p,
h4vo,p = - ( p +
h4Vl,p = - ( p + 3)Vl,p,
hlV2,p =
hlV3,p = - ( p +
h2v2,p = h3v2,p = hav2,p =
l)v3,p,
h2v3,p = -pv3,p, h3v3,p = (p +
1)V3,p, 2)V3,p,
h4v3,p = - ( p +
eoVo,p = q~- [p + 1 ] Vl,p,
eovl,p = O,
elUo,p = - - [ p ] v 2 , p - l ,
elvl,p = U3,p,
e2uo, p = O,
e2Ul,p = O,
e3uo, p = O,
e3Vl,p = --VO,p+l,
foVo,i, = O,
fovl,p = q-2VO,p,
flVO,p = O,
f l v l , p = O,
f2vo,p = - [ p
+ 1]V2,p,
(A.1)
f2vl,p = V3,p+l,
f3vo,p = - - [ p ] V l , p - l ,
f3Vl,p = O,
eov2, p = q2V3,p+l,
eOv3,p = O,
elU2,p ---- O,
elV3,p = O,
e2V2,p = Vo,p,
e2V3,p = - - [ p ] C l , p - l ,
e3V2,p = O,
e3V3,p = -- [p + 1 ] V2,p,
foV2,p = 0,
fov3,p = q - 2 [ p ] v 2 , p - 1 ,
flV2,p = vo,p+l,
flV3,p = - - [ P + 1 ] v l , p ,
f2v2,p = O,
f2v3,p = O,
f 3v2,1, = --V3,p,
f 3V3,p = O.
(A.2)
Here p (eo) and P ( f o ) are briefly written as eo and fo. For Vq* the basis is given by
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564 * hlVO,p = P V *O,p,
* hlVl,p
h2v~,p = ( p + 1)v~,p,
h2Vl, p = ( p "k- 1)vl, p,
h3v~,p
h3v~,t, = - ( p
-b 1)Vl,* p,
h4v~,p = (p +
3)v~,p, 1)V~,p,
=
--pro,* p,
h4v~,p = (p -~- 1)v~),p,
1)V~,p,
hlV~,p = ( p +
~
* (P + 1)Vl,p,
hlV~,p = ( p +
hiP2,* p = (p + 1 ) V2,p, *
* = PV3,p, * h2v3,p
hgv~,p = - ( p
h3V~,p = - ( p
+ 1)V~,p,
h4v~,p = ( p + 2 ) V ~ , p ,
eovo, p = O,
* eOvl,p = --Vo,p,
elV~,p = O,
eivT, p = O,
•
g¢
(A.3)
q- 1)v~,p,
h4v~,p = (p-Jr l)v~,p,
e2Vo, p = --[p + 1 ]V2, p,
563
e2Vl,p = --V3,p+l, .~
e3vo,p = [ P ] V l , p _ 1,
e3vl, p = O,
foV~,p = [p + 1]Vl,p,
foV~,p = O,
flV;,p = [p]O~,p_ 1,
flV~,p = v~,v,
f 2 *V2,p = O,
f2Vl,* p = O,
• = O, f3Uo,p
f3Vl,p = --V0,p+ 1,
eoV2,p = O,
eoIo3,p ~. [P]V2,p_l,
elU2,p = VO,p+1,
elY3, p = [p + 1 ] V 1 , p ,
e2v2, p = O,
e2v3,p* ~ O,
e3V2,p -= -v3, p,
e3V3,p ~ O,
foV~,p -- -V3,p+ , * 1
e3v3,p* ~ O,
fi V~,p = O,
fiV~,p = O,
f 2V~,p = -v~,p ,
f 2v~,p = - [ p ] V~,p_ , ,
f3v~,l, = 0,
f30~,p = [p q- llV~,p.
(A.4)
References [11 M.D. Gould, J.R. Links and Y.-Z. Zhang, Twisted quantum affine super algebras Uq[sl(212)(2)], Uq[osp(2[2)] invariant R-matrices and a new integrable electronic model, cond-mat/9611014. 12] A. Ludwig, M. Fisher, R. Shankar and G. Grinstein, Phys. Rev. B 50 (1994) 7526. [3] R. Gade, Anderson localization for sublattice models, Nucl. Phys. B 398 (1993) 499. [4] A.A. Nersesyan, A.M. Tsvelik and F. Wenger, Phys. Rev. Lett. 72 (1994) 2628. [ 51 C. Murdy, C. Chamon and X.G. Wen, Two-dimensional conformal field theory for disordered systems at criticality, Nucl. Phys. B 466 (1996) 383. 161 L. Rozansky and H. Saleur, Quantum field theory for the multi- variable Alexander-Conway polynomial, Nucl. Phys. B 376 (1992) 461, [ 7 ] M.R. Zirnbauer, Towards a theory of the integer quantum Hall transition: continuum limit of the ChalkerCoddington model, cond-mat/9701024.
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
564
18] D. Bernard, (Perturbed) conformal field theory applied to 2d disordered systems: An introduction, hep-th/9509137. [9] R.B. Zhang, Symmetrizable quantum affine superalgebras and their representations, q-alg/9612020. [ 101 M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, vol. 85, AMS, 1994. I 11 I K. Kimura, J. Shiraishi and J. Uchiyama, A level-one representation of the quantum affine superalgebra Uq( sl( M + 11N + 1 ) ), q-alg/9605047. [ 121 H. Awata, S. Odake and J. Shiraishi, q-difference realization of Uq( sl(M[ N) ) and its application to free boson realization of Uq(sl(211 ) ), q-alg/9701032. [ 131 R. Gade, in preparation. [14] L.J. Dixon, J. Lykken and M.E. Peskin, N = 2 superconformal symmetry and SO(2,1 ) current algebra, Nucl. Phys. B 325 (1989) 329. [151 A.H. Bougourzi, Uniqueness of the bosonization of the Uq(,~(2)k) quantum current algebra, Nucl. Phys. B 404 (1993) 457+ [ 16[ V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977) 8. 1171 S.M. Khoroshkin and V.N. Tolstoy, Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan-Weyl realizations for quantum affine algebras, hep-th/9404036. [ 181 V.G. Drinfeld, A new realization of Yangians and quantized affine algebras, Sov. Math. Dokl. 36 (1988) 212. [ 191 V.G. Drinfeld, Quantum groups, Proc. ICM-86 (Berkeley), Vol. 1,798, AMS (1987) 1201 H. Yamane, On defining relations of the Lie superalgebras and their quantized universal enveloping superalgebms, q-alg/9603015. 1211 I.B. Frenkel and N.Yu Reshetikhin, Quantum affine algebras and holonomic q-difference equations, Comm. Math. Phys. 146 (1992) 1. [221 G.W. Delius and Y.-Z. Zhang, Finite-dimensional representations of quantum affine algebras, J. Phys. A 28 (1995) 1915. 1231 M. Jimbo, A q-Analogue of U(gl( N + 1) ), Hecke Algebra, and the Yang-Baxter Equation, Lett. Math. Phys. 10 (1985) 63. 1241 R.B. Zhang, Universal L-operator and invariants of the quantum supergroup Uq(gl(mln)), J. Math. Phys. 33 (1992) 1970. 125] M.R. Zirnbauer, Towards a theory of the integer quantum Hall transition: from the non-linear sigma model to superspin chains, Ann. der Phys. 3 (1994) 513. 1261 M. Jimbo, K. Miki, I". Miwa and A. Nakayashiki, Correlation functions of the XXZ-model for zl < - 1, Phys. Lett. A 168 (1991) 256. 127] A. Abada, A.H. Bougourzi and M.A. El Gradechi, Deformation of the Wakimoto construction, Mod. Phys. Lett. A 8 (1993) 715. [281 A. Matsuo, Free Field Representation of quantum affine algebra Uq(~ll2), Phys. Lett. B 308 (1993) 260. 129] J. Shiraishi, Free boson representation of Uq(sl(2)), Phys. Lett. A 171 (1992) 243. [301 A. Kato, Y.-H. Quano and J. Shiraishi, Free boson representation of q-vertex operators and their correlation functions, Comm. Math. Phys. 157 (1993) 119. [31 I Y. Koyama, Staggered polarization of vertex models with Uq(~?l(n))-symmetry, Comm. Math. Phys. 164 (1994) 277. 132 ] H. Awata, S. Odake and J. Shiraishi, Free boson realization of Uq(,~IN), Comm. Math. Phys. 162 (1994) 61. [33 ] D. Friedan, E. Martinec and S. Shenker, Conformal invariance, supersymmetry and string theory, Nucl. Phys. B 271 (1986) 93. [ 341 E Bouwknegt, A. Ceresole, J.G. McCarthy and P. van Nieuwenhnizen, Extended Sugawara construction for the superalgebras SU ( M+ 1IN+ 1). I. Free-field representation and bosonization of super Kac-Moody currents, Phys. Rev. D 39 (1989) 2971. 1351 P.A. Griffin and O.F. Hernandez, Feigin-Fuchs derivation of SU( 1,1 ) parafermion characters, Nucl. Phys. 13356 (1991) 287. 136 ] 13. Davies, O. Foda, M. Jimbo, T. Miwa and A. Nakayashiki, Diagonalization of the XXZ Hamiltonian by vertex operators, Comm. Math. Phys. 151 (1993) 89. 1371 E. Date, M. Jimbo and M. Okado, Crystal base and q vertex operators, Comm. Math. Phys. 155 (1993) 47. A
Nuclear Physics B 500 [PM] (1997) 547-564
On the quantum affine superalgebra Uq( at level one
(2 2) )
R. G a d e Max Planck Institute for Physics of Complex Systems, 01187 Dresden, Germany Received 21 March 1997; accepted 17 June 1997
Abstract
Motivated by applications to two-dimensional disordered s~stems this study focuses on evaluation modules ~ , V~* of the quantum current superalgebra Uq(gl(212 ) ) obtained from deformations of infinite-dimensional gl(212) representations. A free field realization of Uq(~(212) ) associated to the appropriate non-standard root system is given. The intertwining conditions for vertex operators related to the evaluation modules Vz and V~* are explored to provide bosonized expressions for the coefficients in their formal series expansions. Finally, the existence of one vertex operator is conjectured. (~) t997 Elsevier Science B.V. Keywords: Quantum affine superalgebras; Two-dimensional disordered systems
1. Introduction
Recently, integrable systems with an underlying quantum affine superalgebra have been studied in context with supersymmetric models of strongly correlated electrons. Ground state properties and excitations of various models obtained from R-matrices associated to finite-dimensional representations of quantum superalgebras are analyzed by means of the Bethe ansatz method (for references, see Ref. [ 1 ] ). Further applications of (quantum) affine superalgebras to condensed matter physics include models for the description of two-dimensional disordered systems. Fixed point theories based on 2 ( N I N ) - K a c - M o o d y superalgebras govern the critical behaviour of two-dimensional I Present address: Institut fur Physik, UniversiRit Augsburg, Memminger Strasse 6, D-86135 Augsburg, Germany. E-mail: [email protected] 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0550-321 3 (97)00407-0
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electronic systems in the presence of random abelian [3,2] or non-abelian [4,5] vector gauge fields. Systems of this type have been proposed to model a particular universality class of integer quantum Hall transitions [2] or to describe dirty d-wave superconductors [4]. A detailed investigation of the g ) ( l l l ) current algebra can be found in [6]. In a more recent attempt to provide an analytically tractable model for the generic integer quantum Hall transition, a supersymmetric vertex model constructed from two g1(212) modules V and V* has been suggested [7]. Containing su(l, 1) representations from the lower and upper discrete series as submodules, V and V* are infinite-dimensional. Another affine superalgebra relevant to the description of disordered systems such as the random Dirac theory is o-~fi(2Nl2N) (see Ref. [8] and references therein). In view of these applications an investigation of models related to the corresponding quantum affine superalgebras seems promising. In general, knowledge about the representation theory of quantum affine superalgebras is still incomplete. However, the irreducible integrable highest weight representations of quantum atfine superalgebras have been classified and shown to be deformations of their classical counterparts [9]. Based on the representation theory of quantum affine algebras and the related vertex operators a method to analyse various classes of solvable lattice models in the thermodynamic limit has been developed [ 10]. The approach permits to derive integral formulas for correlation functions making use of free field representations of the vertex operators and highest weight modules of the algebra. A step towards extending the method to quantum affine superalgebras has been made in [11[. Starting from a free boson realization of Uq(sl(MIN)), M ~ N, at level one operators obeying the same commutation relations as vertex operators associated to N + M-dimensional Uq(sI(MIN)) modules are constructed. For Uq(sl(211 )), based on an analysis of the Fock spaces the existence of the corresponding vertex operators is conjectured. Subsequently a free field realization of Uq(sl(2[ 1)) for arbitrary levels has been presented [ 12]. In this paper an investigation of Uq(fl(2]2)) is started. To be able to construct the universal R-matrix Uq(gl(212)) has to be considered rather than Uq(sl(2]2)) [13]. As stressed in [ 1], a Lie superalgebra has many inequivalent systems of simple roots which yield different Hopf algebras upon deformation. Motivated by the physical applications intended in [7], all simple roots are chosen odd in contrast to the standard simple root system used by the authors of [ 11 ]. In the classical limit q --~ 1 the currents corresponding to the even roots generate a ( i , 1) ® t~(2). So far the current algebra k~(1, 1) has rarely been studied. Representations of ~ ( 1 , 1) can be build up on the base of the unitary representations of su(1, 1) [14]. Thus part of the Uq(gl(212)) modules contain deformations of the ~k(1,1) representations related to the discrete series 79~+~ or D ~-~ or to the continuous series C as submodules. If vertex operators associated to the modules V and V* exist, the homomorphisms are expected to involve affine representations of this type. A free field realization of Uq(~(212)) with the particular choice of simple roots is provided making use of the procedure presented in [ 15] for Uo(s)(2) ) at arbitrary level. Examples for weight modules are given, a more complete account of level one representations is relegated to a separate publication. The lack of A
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a general criterion on the existence of vertex operators for quantum affine superalgebrascomplicates further analysis considerably. A classification of vertex operators for Uq(gl(212) ) is far beyond the scope of this paper. As exemplified in a number of cases (see references in Section 4), bosonized expressions for vertex operators are obtained from their commutators with Drinfeld's generators of the quantum affine algebra as dictated by the intertwining condition. This procedure is carried over to Uq(gl(2f2)) to construct operators subject to the same commutators as vertex operators associated to infinite-dimensional Uq(gl(212)) modules if the latter contain an dement annihilated by the Chevalley generators fi, i = 1,2, 3. If instead the infinite-dimensional module is generated from an element v such that eiu = 0 for i = 1,2, 3, another set of Drinfeld generators is required. It is related to the standard set by the antipode. As bosonization follows from the Drinfeld realization, this implies the introduction of two free field realizations of the quantum current superalgebra. Finally, the existence of a vertex operator related to the module V* is conjectured. The paper is organized as follows. In Section 2, the definitions of Uq(})(212)) and Uq(~(212)) are given in terms of Chevalley's and Drinfeld's generators together with the relation between the latter. Evaluation modules obtained from deformations of the gl(212) modules of Ref. [7] are explained in Section 3. The free field realization is considered in Section 4. Section 5 addresses the issue of vertex operators. Section 6 contains a summary. The basis of the modules Vq and Vq obtained from V and V* by deformation is described in Appendix A.
2. The quantum affine superalgebra Uq(/~(212)) For a given (affine) Lie superalgebra several inequivalent systems of simple roots exist [ 16]. The quantum superalgebras obtained for different systems of simple roots are related by twistings [ 17]. Motivated by the physical applications mentioned in the introduction, the choice adopted in this paper differs from the standard set considered in most of the literature on the subject. Denote the set of simple roots by //{a0, a l , a2, as, a4}. A symmetric bilinear form ( ai, aj ) = a"iym is defined by
a sym =
0 -1 0
-1 0 1
0 1 0
1 0 -- 1
1 -1
0 1
--1 0
0 1
-1 ) 1 0 .
( 1)
1 0
In terms of the basis { r l , r z , r 3 , r 4 } with the bilinear form (ri,'rj) = --(--1)i•i,j the classical roots &i = - ( - 1 ) i ( r i + ri+l),i = 1,2,3, &4 = rl - r4 and Ai = Y'~-5=lrj 12~j41 "/'J"Introducing an affine root a and an affine weight Ao with (Ao, Ao) = (& 6) = (ri, Ao) = (6, ri) = 0 and (Ao,8) = 1 the set of simple roots is given by a0 = - &l - &2 - &3 and ai = &i for i = 1,2, 3,4. The free abelian group P = @4=oZAi + Z 6
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w i t h Ai = zii q- Ao for i = l, 2, 3 and A4 = A4 is called the weight lattice. Via ( , ) its dual lattice P* = 04__oZhi 4- Zd can be identified with P by setting ai = hi and d = A0. The quantum deformation Uq (si(212)) of the universal enveloping algebra U(sl(212 ) ) is introduced as the unital Z2-graded associative algebra generated by the Chevalley generators el, fi, qh,, i = 0 . . . . . 3 through the defining relations
qh,qhj = qhjqh,, qh'ejq-h' = q( .... J)ej, qhi -- q-h, [ei, fi] =•id q -_ ~
qh,fjq-h, = q-(~,,aj) fi '
(3)
,
[el,e3] = [e0, e2} = [ f l , f3] = [f0, f2] = 0,
[[eo, e,]q,[eo,e3]q] =0,
[[fo,f,]q ,,[fo, f3]q-,] [[fl,f2]q-i, [f,,fo]q ,] [[f2,fl]q-l,[f2, f3]q-~] [[f3,f2]q-l,[f3,fo]q-,]
[[e,,e2]q, [e,,eo]q]
=0, [[e2, ej]q,[e2,e3]q] =0, [[e3,e2]q,[e3,eo]q] =0,
(2)
(4) =0,
=0, =0, =0.
(5)
In (3) and (4) [, ] denotes the usual Lie superbracket [x,y] = x y - (-1)14l:'lyx. Due to the special choice of simple roots the Z2 grading ]. I : Uq(M(212)) ~ Z2 assigns the value 1 to each generator ei, fi and the value zero to @. The q-deformed supercommutators in the Serre relations (5) are defined by
[el, ej ]q = eie.i -+-q(a"aJ) e.iei, [fi, fi]u-~ = f i f i + q-(""aJ) fJfi.
(6)
In the homogeneous picture a grading operator d is defined by
[",ei] :(~i,oei,
[d, Yi] = --~i,oYi,
[d,q h'] : 0 .
(7)
The algebra Uq(J)(212)) admits a comultiplication A and an antipode S:
A(ei)=qhi®ei+ei®l,
A(fi)=fi®q-hi+l®fi,
A(q:~h,) = q~h, ® q±h,
(8)
and the antipode
S(ei) = --q-h'ei,
S(fi) = - f i @ ,
S(q ~:h~) = q:Fhq
(9)
The antipode is a Zz-graded algebra antiautomorpbism, which states the property
S(xy) = (-l)[xl'lYIS(y)S(x)
(10)
for any two homogeneous elements x, y of a quantum affine superalgebra. A further peculiarity of a graded Hopf algebra is the supercommutativity of the tensor product:
(v ® w) (x ® y) = (-1)lwl'lxlvx ® wy.
(11)
A realization of Uq(~(212)) particularly convenient for the purpose of bosonization is provided by Drinfeld's generators [ 18]. The bosonization of vertex operators related
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to the evaluation modules introduced in the next section will require two sets D(a) of Drinfeld's generators { E ~ , H~,a, q4-hj, y± l } with a = + 1, m C Z, n C Z - {0}, i = 1,2, 3 and j = l, 2, 3,4. The generators of each set satisfy the defining relations qn(C%a/ ) _ q - n ( a j , % , ) y n _ y - n
[ H~,a, H~',,a ] = ~.+n'.o
n(q_q-l) qh~ ,~i,=t=q - h~ _ ,~+ ( ,~. aj ) l~i,-4~--'~nl,¢l
- - "1
j
i,q-
[Hn,a, Em,a] = ±
q_q-I
'
~ l l l ~ a "j
q n(ai'a)) -- q - n ( a i ' a j )
-~
~q ,
"'
a
e~fm,aYT~n'
a t i i,+ '.t '- . 6.,,q )et,+ [Em.a,Em'.a] . . . aq_ 1 [.,~(m-m \J" - - m + m ' . a - t,~,--~(m--m')!lti,--,n+m',a)"~
(12)
with J'+ Z-" = qahj exp ( a ( q _ q - l ) Z H J , a z - n ) , qtn,a
Z n)0
n>0
Z ~ l ~ _ n j,,a z n =q-ahj
exp(-a(q _
(13)
nJn,aZn)
q-l)Z
n)O
n>O
and i,--Ii',--b [Em,~,E'm,,~] =0
for (ai, ai,) = 0 ,
(14)
p~,4I~l ,~ .a- ,-,4-a(al,ai') lg ,4-El,q_ i~,~ ,-I- igt,q~ m + l , a ~ m J , a ~ '4 ~m',a m+l,a ~m'+l,a~m,a •
4
i r
•
4
.
_
,.4-a(eti,ai,)17~,&17 i t ,-I- = 0, tt ~m,a~mt+l •
(15) [Em,a,Em,.a]q±,
, [Ek,a,Ek,,a]q±,
[Ek, a, Em,,a]q± , [E~,'a, E k ' , a ] q ±
h-
= O. (16)
The Z2 grading is given by IE&,~I = 1 and IH~,~I = Iq+hj[ = ITI = 0. Analogously to (6) the deformed supercommutators in the Serre-relation (16) are defined by •
r ~" + E' t ~m,a,
.i
"+ ]J q,
mt,a
.
.¢
!
.t
,
= lgt' 4-1g' ,-4- ..1- n ( ai,ai' ) Igt '4-Ig','4~m,a~mt,a | '4 ~mt,a~m,a ,
q,
~
q+ l •
(17)
Drinfeld's realization of Uq ( ~ ( m In) ) has been obtained in [ 20]. The Chevalley generators are related to the sets D(a) by the formulae
ei = Eo,i,+a,
i,fi = Eo.~
for i = 1,2, 3,
qho = yq--hl--h2--h3
[E~),a ,El, ~ ]q. fo
= qa(hl+h2+h~)
--
q-~[E~,-~,~.~,-~ w3,-~.-~(h,+h:+h3) , , tal,a Jq"L'0, a ]~/
1,+ 2,+ [[E_l,a, EO,a ]q-.,
~_ qa(hl+h2+h3) fl- gTl,+
k t~-l,a'
K,2,+ l
Eo,a3,+] q - " p3,+
~O,a Jq-"~O,a
_a r~3,+ r ~ 1 , +
~:,2,+~
"~
-- q l:0,a t t Z - l , a ' C~O,a l q - ° ) .
(18)
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By means of the identification (18) the grading, coproduct and antipode of Drinfeld's of Uq(sl(2]2)) can be obtained from (7), (8) and (9) whose definitions for qhi apply to qh,, too. The following partial description of A(H/~) proves to be sufficient to derive expressions for the vertex operators: A(H~,I)_~Htn,, ®,y2 + " ~ ® H',1 modN_ ® N+,
A(Hin,1)=H i_~,] ® y - 73,, + y_~ ® Hi-n,J modU_ ® U+, A(H~ ,- ,)_- Hni _ I ® y-~ + y~" ® H.._~ modN+ ® N _ , A(Hin,-l)--Hi-n,-l®Y-~"+Y~®Hi-n,-lm°dN+®N-
n>0, i=1,2,3. (19)
Here N+ are left Q ( q ) [ y + , ~'~.a, i,+ ~0-.,a, i,- . n E Z/>0] modules generated by {E~ +, m E Z}. The definition of A(H~:.,~) is provided by the construction of the universal R-matrix [ 19] for Uq(~(212)). Following [21] an automorphism Dz of Uq(gl(Zl2))®C[z, z -'] and maps A z , A z, : Uq(~(212) ) ®C[z, z -1] ~ Uq(gl(212) ) ®Uq(gl(212) ) ®C[z, z -i] are introduced as O~(En,a) i,:L = Z nr,i,± 15n,a ,
oz(/L,.)
d:(x) = (D z ® id)d(x),
= Z . iHn, a,
Dz(a)=a,
A'z(X) = (Dz ® id)A'(x)
(20) (21)
with d~(x) = o-o d(x), tr(x ® y) = (-1)Ixl.I;'ly ® x. The spectral-dependent universal R-matrix is the unique element R ( z ) E Uq(gl(212))+Uq(~(212)) ® C[ [z ] ], where is the completion of ® over C[ [q - 1 ] ] such that
~ ( z ) A z (x) = A~z(x)7~(z),
(22)
(A z ® id) (7~(w)) = n13(zw)~23(w), (id ® Zlz) (~(zw)) = ~13(z)TClz(zw)
(23)
with 7~12(z) = ~ ( z ) @ 1,7~23 = 1 ® ~ ( z ) , ~ 3 ( z ) = (o-®id)Te23(z). In order to satisfy F-Xl. (22) for any x E Uq(s)(2[2)), Te(z) has to contain H4~,,+ (or H4n,_). Then Eqs. (23) imply the definition of A(~'4,'~1 ) by means of the following expression. Writing 4
4
(L+ ( zq_ly_~ ) ) = ]-IaP'I "±( z ) = I-[ ( ~ i=1
i=1
~±n,,Z
(24)
n/>0
the coproduct of L + (z) is given by
A(L±( z ) ) = (I ® L+ ( zq~('+I)c') ) 4
× ( l ® l + L a i l ( z q :1:-
½( izF1)cZ)®B~ (zq½(ld:l)cl))
i=2
x(L±(zqJ (1~:|)c2) ® 1),
(25)
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where c1 = c ® 1, c2 = 1 ® c with qC = y and
A~(z)=(q-q-1)
Z
E~-~(Zy½(I~q))Tn'
n=½(l+l)
A ~ ( z ) = - ( q - q -l)
~
[E2;,E2'-]q-,(zY½('~:')) ~:",
n=½(l+l)
a ~ ( z ) = ( q - q -1) ~
[[EI"2,E2"-]q-,,E3o'-]q_I(Z~/½(I~I)) :qzn,
,,=½0+l)
B~(z)=-(q-q-l)
Z
EL+(ZY½(I+I))
n=½0:Fl)
B~(z)=(q-q-l)
Z
[E02'+'I~'I'+] g"l(14"l)/::Fn ~,4-nJq~,~f ! ,
n=½(l=gl)
B~(z)=(q-q-')
Z
1,+ [E°3'+'[ E;2,+ 'E-bn]q]q (z~1½(l'4-1,)~n"
(26)
n=½(l=F1)
Making use of (13), (18) and (8) the expressions for A(H4~n,+) can be obtained. The antipode relates the two sets of Drinfeld's generators according to i,+
S(Em,1) = --
T--m -hi Ei,+
q
m,-l'
S ( E mi,+ ,_I) =
--Y-m q-hi--i,+ P-'m,l'
i,-s(<,,,) -m q ,z,,,,_l, =-y
s ( < , i,-_,)
s ( < , , ) = --y - " JHn,_ 1,
S ( h ' j° _ , ) = - ~ '
=
..,,
-n < ,j, ,
Vm, i = 1,2,3 Vn 4= 0, j = 1,2,3,4 (27)
Applied to ( 2 4 ) - ( 2 6 ) Eqs. (27) yield A(H4~n,_I). Due to ( 2 4 ) - ( 2 6 ) , (27) and (10), Eq. (19) holds true also for i = 4. Further details as well as an explicit expression of 7~(z ) in terms of Drinfeld's generators will be given in [ 13 ]. Finally, the commutators of the homogeneous grading operator are given by
[d, Em,a i'+]J --- mEre,a, i,± [d, <,o] = nW;,o.
(28
3. An evaluation m o d u l e from an infinite-dimensional Uq(gl(2]2)) module
Generally, one has to take into account the direct sum of several modules of a nonaffine quantum algebra Uq(g) [22] to obtain an irreducible level-zero representation of the affine quantum algebra Uq(~). In the case of Uq(~l(N)) however an evaluation homomorphism permits to lift any Uq(sl(N))-module to a Uq(s'l(N)) module [23]. For the quantum superalgebra Uq(9(N]M)) there exists an evaluation homomorphism
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p : Uq(f'I(NIM)) , U q ( s l ( N I M ) ) ® C[z, z -I ] as well [24]. For Uq(gl(NIN)) the action of elements not contained in Uq(sl(N]N)) needs to be introduced separately. Specializing to N = M = 2 p is given by A
p(ei)=ei,
P ( f i ) = fi,
f o r / = 1,2,3,
p(q+hj) =q+hj, f o r j = 1,2,3,4, p( q±ho ) = qT( hl+h2+h3), p(eo) = q - h 4 ( f 3 f 2 f l + q-l f 3 f l f 2 -- q f 2 f l f 3 -- f l f 2 f 3 ) , P(fO) = (ele2e3 + qe2e~e3 -- q-le3ele2 -- e3e2el)q h4.
(29)
In the following the evaluation homomorphism will be applied to obtain affinizations of particular infinite-dimensional representations of Uq(gl(2]2)) which in the limit q ~ 1 coincide with the modules discussed in [25]. To elucidate the structure of the gl(2NI2N) modules called V and V* in [25], let us focus on the subalgebra gl(N[N) ® gl(N]N). If N = 1, the latter is generated by { e l , f ~ , e 3 , f 3 , h j } at q = 1. V and V* are one-dimensional irreducible representations of the first factor gl(NIN) containing the elements v0,0 and v~,0, respectively. The action of the second factor on v0,0 and v(~,0 vanishes completely. The Uq(gl(212)) module Vq is conveniently parametrized by a pair of labels ( i , p ) , i = 0, 1,2, 3 and p = 0, 1,2 . . . . In this basis the Z2 grading yields 1 for vl,p, v2,p and 0 for Vo,p, V3,p. The weights are given by
hlvo,p hlVl,p hlU2,p hlv3,/,
= = = =
-pro,p, - ( p -}- l)Ul,p, - ( p -[- ])U2,p, - ( p + 1)V3,p,
h2vo,p = - ( p + 1)VO,p, h4vo,p = - ( p + l)V0,p,
h2Ul,p = - ( p + l)Ol,p, h2o2,p = - ( p + l)u2,p, h2g3,p = -pv3,p,
h4Vl,p = - ( p + 3)Ol,p, h4o2,p = - ( p + l)o2,p, h4v3,p = - ( p + 2)v3,p.
(30)
For any element vi,p of the module ( h i -+- h3)vi,p = 0 holds. The elements v0,o and v~,o satisfy
eivo,o = fiv~,o = 0
for i = 1,2, 3.
(31 )
Vq is spanned by the elements fi~ . . . fiNvo,o where (il . . . . . iN = l, 2, 3) as visualized by the diagrams below. Alternating applications of f l , f2 yield the elements VO,p and V2,p: f2 U2,0 ~ UO,O f2 U2,1
U2,2 (
f2
U0,3
(
f2
II,
U0,2
~ fl UO,1
(32)
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The remainder of the module forms a similar sequence: U3,0 /33,1 f2 U3,2 (
f2
f2 <
I fl V1,0
I fl U1,1
(
(33)
V1,2
Both sequences are related by the action of f3 as f3VO,p = - - [ p ] V l , p - 1 ,
f3VZ,p = --V3,t,,
(34)
where the standard notation [k] = is used. Replacing fi by ei and vi4, by vi4, yields the corresponding diagrams of the module Vq*. The detailed description of the basis of Vq and Vq can be found in the appendix. With a suitable normalization Vq* can be viewed as the dual module Vq s of Vq by (xv*,v) = - ( - l ) l x l ' M ( v * , S ( x ) v } with the dual basis (v[,t,, Ui,.p, } = (~i.i,(~p,p,. Due to the infiniteness of Vq and Vq there exists no isomorphism between the two modules. In the homogeneous picture the evaluation modules Vz = Vq ® C(z, z -1 ) and V.* = Vq ® C(z, z - l ) are equipped with a Uq(~(212) )-module structure via
ei(vi,p ® z") = eivi,p ® z "+''°,
fi(Ui,p @ z n) = fiui, p @ Z n-6i'° ,
ei(vi*q, ® z') =eivi*,p @ z "+''°,
fi(vi*,p ® z') = fivi*,p ® z n-~''° ,
Wt(Ui, p ® Z n) = rl(~ -- ( p dr- 1) (z~1 + ,~ -- z~3 + a4)
-~-t~i,0 (z~ 1 -- zt3) -- 2t~i, lZ~4 + ~/,3 (z~ -- z~4) ,
wt(v;,p ®z o)
(p + 1)(a,
- a3 +a4)
-6i,0 ( / t l - / i 3 ) + 28i,~zi4- 8i,3 (zi2- A4)
(35)
and n 4m , - I (vo,o @ z n) -- -
--
q2m -[m] m
VO,O @
[m] . zn+m Hn.,,l(v~,o®Z")=--m-VO,o®
zn+m '
Vm ~ O,
(36)
where in (36) Drinfeld's generators are chosen as required by the construction of vertex operators in Section 5.
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4. B o s o n i z a t i o n
Constructing the quantum affine algebra Uq(~(2)) at level one and its vertex operators through bosonization allows to derive an integral formula for n-point correlation functions of local fields [26]. Several realizations of Uq(sl(2)) at general level k and its vertex operators in terms of three sets of bosonic modes have been proposed in the literature [27-30]. The relations among them are clarified in [ 15]. A generalization A of the bosonization scheme to Uq(sl(n)) at level one has been used to calculate the one-point-function of vertex models associated to the vector representation in [31]. For A Uu ( s l ( n ) ) at general level k a suitable deformation of the Wakimoto free field realization of sl(n) has been constructed in [32]. A bosonization scheme for U q ( ' S ' I ( N + I i M + I ) ) with the standard choice of simple roots can be found in [ 11,12]. Guided by the procedure explained in detail for Uq(~(2)) in [15] a bosonization A of level-one Uq(gl(212)) with the system of simple roots chosen above is obtained in two steps. The construction leads to a deformation of the bosonized realization of ~(212) by bilinears in commuting and anticommuting ghost fields bi(z),ci(z ) and fli(z ), yi(z ), i = 1,2. Their non-vanishing operator product expansions are given by ci( z ) bi( w ) = - b i ( z ) c i ( w ) -
yi( z ) fli( w) = fli( z )yi( w) =
I Z -w'
1
(37)
Z--W
In terms of the ghost fields, the Chevalley generators ~i, J~ of ~(212) are written
~l(z) =- : cl(z)fll(z) e2(z)
-- " ~ / I ( Z ) C 2 ( Z )
:, :,
g3(z) = : b 2 ( z ) y 2 ( z ) :,
fl(z) = : yl(z)bl(z) /2(Z)
= :b2(z)t~l(Z)
:, :,
(38)
f 3 ( z ) = : f l 2 ( z ) c 2 ( z ) :.
A bosonization of b i(z ), ci(z ) and fli(z ), Yi ( Z ) into commuting free fields following [33,34] yields the limit q --+ 1 of the bosonization scheme outlined below. The affine quantum superalgebra is conveniently reexpressed in terms of the generating functions (or currents) = V
TM
l p j , + 7 --n
n~>0
n>~O
E~;+(z) = ~-~' E~,a -m-l mEZ
Then the defining relations (12) are rewritten as
i,+
j,-
"tFa (Z)qta
(z - wyq(m'~J))(z - w ~y - i q -("''~j)) J'-• i~~--W '+ wyq-(CmaJ))(Z-wy-lq('~"aJ)) ~a ( )~a (Z),
( w ) = (Z
(39)
R. Gade/NuclearPhysicsB 500 [PM] (1997) 547-564
557
Fi'+( "~E j'+ (w~ - q±a(al,aj) Z - WTT~q q:a(ai'aj) E~,+(w)q:i.+(z) Z
-
wym ~q+(a~,aj) a
~i.-(z)E~,+(w ) q+a(.... s) zz -- wy+~q wy±~q +a(~'~j) 4-a(~''~j)Ea., (w)~a (z),
(40)
6,,; (~i,+(w3/~) E'i;+(z)E~-(w) "" w ( q : q - ' ) \ z ----w--yy
(41)
~ ' - (w7-½))
z--~y_~
,
EIi+(z)E~'+(w) = -E{'+(w)EI;+(z) for (ai,c~/) = 0, (wq+a(a"~J) - z)E~+(z)EJa'+(w ) = (zqma(~,~J) - w)EJd+(w)E~+(Z),
[IEX,+(
I
+[[E2'+(Z2),Ela'+(Wl)]qa: , , [E]'+(z|),E3'+(w2)]q±d =0,
(42)
where use of the symbol ~ indicates the omission of terms regular in the limit z -~ wy+. Realizing the quantum current superalgebra as a deformation of (38) implies 3 / ~ q-1 at level one. First, bosonization of the currents ~ ' + (z) is achieved by means of the four deformed free fields
~;+ ( z ) = ~ - i q ~ 0 1 n z + i n÷o--(-~q~nZ Z q+a~ i -n ,
j = 1,2,3,4,
(43)
where the boson oscillators {q~J,q~Jn}fulfill the deformed Heisenberg algebra
n
[,p/,
=
(44)
By means of {~,q~i} the currents ~ ' + ( z ) are realized as
q:J'+"t z ) = q -aij(~-i~°+') exp ( - a ( q - q - '
) E i J ( ~ -iq~+')z -n) , n>0
~J'-(Z )=q ai~(~°-i~+') exp ( a ( q - q - ' ) E iJ(q~j-n -i~°J-+~n)Z") ' for j = 1,2,3 n>0
(45) and ~,,'4 + ( z ) = q-a(i~o~+~og) exp (_a(
q _ q-~ ) ~-~(iq~'. +~pa)z - . ) , n>0
~4"--(Z ) =qa(i~°~+~°~)exp(a(q-q-1) ~-~(i~pln+~O4n)zn ) . n>0
The currents
E~'+(z) depend on the fields q~+(z) in the following way:
(46)
R. Gade/Nuclear Physics B 500 [PM] (1997)547-564
558
E:J'+(Z) =exp(-t-iJ+l(q~+ ( z ) - i~a+l'+(z)))exp(Sj,,Trq~2)xJ':k(z),
j = 1,2,3. (47)
It is easily verified that the expressions (45), (46) and (47) fulfill the relations (40) with y = q-J. The cocycles exp(Sj,37"r~o02) are needed in order to obtain the right signs in the remaining relations. Second, X j+ ( z ) have to be constructed such that their operator product expansions insure (41) with Y = q-1 and (42). As can be inferred from the bosonization of gl(212), two more fields are necessary to complete the expressions for E~'± ( z ) . A suitable choice of the fields is m
flJ(z)=flJ
-
i f l ~ l n z + i ~ - ~ -z-n
J
j = 1,2
(48)
n~O
with [/T,/3ok] = -i~j,k,
[/3J,/3~,] = -n~.+m,OSj.k.
(49)
Then the X)'+(z) are realized as XI,+(z) _
z(q
1
q-l)
(exp (/3' ( q - ' z ) )
- exp(/3' ( q z ) ) ) ,
XJ'-(Z) = exp(-/3 j (z)), X2'+(Z) = X 3 ' - ( z ) - z(q -1q - l ) ( e x p ( / 3 2 ( q _ , z ) ) - exp(/32(qz))) ' X 2'- ( z ) = -X3'+ ( z ) = exp(-,82 ( z ) ) .
(50)
Part of the level-one weight modules V(A) of Uq(fl(212)) with the decomposition V(A) = ®.ee V ( A ) . , V(A)~ = {v ~ V(A) [ qhv = q(h'~)v Vh E P*} include submodules reducing to ~ ( 1 , l) modules of the lower discrete series [14,35] in the classical limit q --+ 1. These modules contain a vector va E V(A) such that eigA = O, i = 0 , 1 , 2 , 3 and qhJgA = q(hj'A)vA f o r j = 0 , 1 , 2 , 3 , 4 , H4,_IvA = 0 for n > 0 and V(A) = Urn. An example is furnished by V(A2,4) with (h2, A2,4) = ( h 4 , A2,4) = - 1 and (hi,A2.4) = ( h 3 , A 2 , 4 ) = 0. The vacuum representation V(Ao) is spanned H = H n~,-l''" 4 H nM,_lVAo, 4 by f i , . . . f i N V~,, il . . . . iN = 0, 1,2,3 where VAo nk < 0 and qhJvAo = UAo for j = 1,2,3,4. Analogously, a module V(A*) containing an ~ ( 1 , 1 ) representation of the upper discrete series for q --~ 1 has an element distinguished by fiVA* = 0, i = 1,2,3 and qhJvA* = q(hj'A*)uA* for j = 0 , 1 , 2 , 3 , 4 , H~,jVA* = 0 l b r n > 0 and V(A*) = UVA.. The corresponding example is given by V(A~, 4) with (h2, A~,4) = ( h a , A~,4) = 1, ( h l , A ~ , 4 ) = ( h 3 , h ~ , 4 ) = 0. The reason for the choice of Drinfeld's generators H4,+ made here will become apparent in the following section. In terms of the oscillator algebras the vectors va2.~ and VA;.4 are realized by /)A2.4 exp(~o' - i~o2)[0), ~-~
VA;,4 = exp(--(q~' -- i~,2))[0),
(51)
where 10) denotes the vacuum state defined by ~oil0} =/3.k10) = 0 Vn > 0. Bosonizafion associates the vacuum state 10) to va0.
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
559
5. The free hoson realization of a type I vertex operator Vertex operators of type I are defined as intertwiners of U-modules in the following way [36]:
4~AV(z) : v ( a )
~ voz) ® vz.
(52)
Here Vz and V(,~), V(/x) denote an evaluation representation and highest weight modules of U, respectively. The vertex operators obey the intertwining condition
• ~V(z)ox=zl(x)oqg~V(z)
VxcU.
(53)
The vertex operator construction, first introduced and studied in [21] provides a method to construct eigenvectors of the corner transfer matrix [36] and thereby allow, at least in principle, for an evaluation of correlation functions. For quantum affine algebras a general criterion about the existence of vertex operators with finite-dimensional modules V has been proven in [37]. Not disposing of such result in the case at hand, one may resort to construct bosonized expressions for suitable operators satisfying the intertwining condition in a first step. Then, studying the action of the bosonized currents and vertex operators on the weight spaces of V(A) succeeding UA or UA. will yield a hint on the existence of vertex operators. To this aim, vertex operators associated to the evaluation modules Vz and Vz* described in Section 3 are introduced as formal series ~,u. V ~a (Z) = ~
~,u V t~A ,i ,p ( Z )
@Ui,p,
~ *p. V* *;~ (Z) = ~
i=0,l,2,3 p=0,1,2.... ~p. V
. g~V*(Z)® ~A,i,p Ui,p '
(54)
i=0,1,2,3 p--O,I.2....
~
V
qb2t,i,p(Z)=~rlzAA,i.p,nZ--n' ncg
a~*/~ V*
.7.*/z V*
~h,i,p (Z) = ~ tPh,i,p,nZ nEZ
--n
.
(55)
The coefficients in the last line are linear maps
,~"A,i,p,n •" v ( a ) -.11. t~,i,p.n : V ( , ~ )
,
v(iz)
)
V(ix),
(56)
subject to the intertwining conditions
(-1)l(i'l"Hxl~ ~)gAA,iV, p,nXU® (Ui,p ® Z -n) i,p ,n
® (n,,, ® z - ° ) =
i,p,n
t~)lzA,Vp,nU® (Ui,p ~ z - n ) ) ,
=A(X)(~-'~
i,p,n k2_. i,p,n
®
® z -n (57)
for all x E Uq(~(212)) and v E V(A), where I(0,p)l = 1(3,p)l = 0 and I ( l , p ) l = 1(2,p)l = 1. Operators ~b(z) = ~-~ipt~ip(Z) ~ Ui,p -- ~i,p,nf~i,p,n ~ (Ui,p ~ Z -n) and ~b*(z) = ~i.,, &i*,,,( z ) ® vi*,p = ~-~i,p,nqbi*,p,n® (U~,p ® z-n) satisfying the intertwining condition
560
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
can be determined to a l~ge extend by the method used in [ 10] to construct the type-one vertex operators of Uq(sl(2)) at level one. In order to obtain bosonized expressions for the vertex operators, the restrictions due to the intertwining condition with x = qd, qhi, ei ' fi are reformulated in terms of the Drinfeld's generators of Uq ( s'l(2t2)). Eq. (19) indicates the choice of generators: the set Dr is appropriate to derive equations for the bottom component ~b~.0 (z), while analogous equations for the top component ~0,0(z) are obtained using D_j. The intertwining conditions for qhj and qd read q-hJqbi,p ( Z )qhj = qX(j;i,P) ~i.p ( Z ), q -h~cb*Ti,p
(z)qhj=q-~r(J;i'P)(b[,p(Z),
j = 1,2,3,4,
q-d q~i,p( Z ) qd = fbi,p ( qz ), q - d q~* d- * i,p (z)q - ~i,l , (qz),
(58)
where hjvi.p = K(j;i,p)vi,p with K(j;i,p) listed in (30). Making use of (18), (12), (13) with a = -1 and (19), (36) one finds for the top component ~b0,0: [En~:_i,, ~bo.o(z)] = [E3:_i,, q~o,o(z)] =0,
E~;-_jfbo,o(z) - q-~qbo,o(z)E~;-_ I =0 Vn
(59)
and 1
[Hn _l , q~O,O( Z ) ] =[n._,.4,o.o(z)] 3
=o.
v,,,
2 )] H4 [n] [Hn, - l ' ~ b 0 ' 0 ( z = [ n,-I '~b0,0(z)] = q-~(q2z)nfbo,o(Z), n [ H 2- n - 1 , ~ 0 , 0 ( Z ) ]
n>0,
[n] - ~ 4 =[H-n,-I,OO,O(Z)]= q (q2z) - n ~b0,0(z), n
n > 0. (60)
Choosing a = 1 one derives for the bottom component ~ . 0 ( z ) fut-~n,l = " + , "V-J0,0 * (z)]
3,± ,, = [H°,, = [H°., ,6oo(z)] 2,+
* [H1.,,,¢bo.o(Z)]=[H3,,dP~.o(Z)] [H2,,, ~b0,0(z)] •
o,
Vn.
(61)
[n]q-~nznqb~o(Z),
H4 =[
=
=-
[H 2-n,1 , q~;,0(Z)] = [ H4_ . , t , ~ ; , o ( Z ) ] =
n
n>0,
.
[n]q~z-"q~;,o(Z), n
n>0.
(62)
J fix the dependence of ~bo,o(z) and q~,o(Z) on q~J and The commutators with H.,+ for n # 0. The classical limit q --+ 1 indicates the choice of the solutions to (59) -(62): ~bo,o(z) = exp ( - (~o~+ (q2z) ~b;,o(Z) =exp(~p{'-(q-'z)
- iq~2-+(q2z)) ) e x p ( - l n q ( ¢ ~ - i~p2) + 7-rcp3), -i~@-(q-'z))exp(crq~3).
(63)
All other components of ~b(z) and if* (z) are determined from remaining restrictions imposed by the intertwining condition:
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
561
4'2,p ( Z ) =q-(p+l)[4'O,p(Z ),e2]q-h2 ,
4'O,p+l(z) - 1--------~q [p + 1 -~p+l) [4'2,p(Z) el]q -h', 4'l,p =,/~p+l r-" -h3 , [q)0,p+l(Z), e 3]q
4'3,,,- p +1 1-----5[4'2,p (Z), e3] q-h3
(64)
and
4'L,(z) -- [4';,.(z), f2]q_,.,,, . 4'0,p+l(Z) =
1 , ] [ p + 1] [4'2,p(z),fl. q-,,,~l,'
* * Z ),f3]qp+,, 4'1,1,(z) = [4'0,p+l(
4',~,p(z) =
1 [p + 1] [4'~,p(Z),f3]qe~.
(65)
In (65) the commutators are defined as [x,y]q~:~p+, = x y - (-1)lxllylq+(p+l)yx with I4,*0,pl = 4 I' * 3,pl = 0 and I4'*t,pl = 4 I' * 2,pl = 1. Eqs. (63) and (65) can be used to examine the action of 4'*(z) on VAo as well as on its first descendants. This yields a structure consistent with the existence of a homomorphism V(A0) ~ V(A2,4) (~ Vz. Thus one is led to conjecture the existence of the following vertex operator: t~v(a2,4) V* ( V(Ao) Z) : V(Ao) ~
,.
V(A2,4) ® V£
(66)
Identifying =V(ao) ,~v(~2.4)v" (z) with 4'. (z) the bosonic realization of the vertex operator is given by (63) and (65).
6. S u m m a r y The quantum affine superalgebra Uq(~(212)) has been considered with emphasis on a pair of evaluation modules related to infinite-dimensional representations of gl(212). A free field realization of the current superalgebra and the coefficients in the formal expansion of vertex operators involving these evaluation has been obtained. Certainly, a number of issues need to be resolved to gain sufficient insight into the representation theory of the algebra including non-integrable modules. In particular, vertex operators intertwining modules of the continuous series have to be taken into account in order to provide the framework for an analysis of vertex models. Work along these lines is in progress.
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
562
Acknowledgements The author wishes to thank M.R. Zirnbauer for several illuminating discussions regarding his ideas developed in [7]. She is grateful to T. Miwa for encouragement and helpful comments during her participation at the semester "Integrable Systems" in Autumn '96 in Paris.
Appendix A The following relations fix the basis of the U~q(~(212)) module Vq: hlvo,p = -pvo,p, hevo,p = - ( p +
hlVl,p = - ( p
1)vo,p,
h3vo,p = pro,p,
1)nO,p, - ( p + 1)V2,p, - ( p + 1)v2,p, (p + 1)v2,p, - ( p + 1)V2,p,
+
l)Vl,p,
h2Vl,p = - ( p + l )vl,t~, h3Vl,p = (p +
1)Vl,p,
h4vo,p = - ( p +
h4Vl,p = - ( p + 3)Vl,p,
hlV2,p =
hlV3,p = - ( p +
h2v2,p = h3v2,p = hav2,p =
l)v3,p,
h2v3,p = -pv3,p, h3v3,p = (p +
1)V3,p, 2)V3,p,
h4v3,p = - ( p +
eoVo,p = q~- [p + 1 ] Vl,p,
eovl,p = O,
elUo,p = - - [ p ] v 2 , p - l ,
elvl,p = U3,p,
e2uo, p = O,
e2Ul,p = O,
e3uo, p = O,
e3Vl,p = --VO,p+l,
foVo,i, = O,
fovl,p = q-2VO,p,
flVO,p = O,
f l v l , p = O,
f2vo,p = - [ p
+ 1]V2,p,
(A.1)
f2vl,p = V3,p+l,
f3vo,p = - - [ p ] V l , p - l ,
f3Vl,p = O,
eov2, p = q2V3,p+l,
eOv3,p = O,
elU2,p ---- O,
elV3,p = O,
e2V2,p = Vo,p,
e2V3,p = - - [ p ] C l , p - l ,
e3V2,p = O,
e3V3,p = -- [p + 1 ] V2,p,
foV2,p = 0,
fov3,p = q - 2 [ p ] v 2 , p - 1 ,
flV2,p = vo,p+l,
flV3,p = - - [ P + 1 ] v l , p ,
f2v2,p = O,
f2v3,p = O,
f 3v2,1, = --V3,p,
f 3V3,p = O.
(A.2)
Here p (eo) and P ( f o ) are briefly written as eo and fo. For Vq* the basis is given by
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564 * hlVO,p = P V *O,p,
* hlVl,p
h2v~,p = ( p + 1)v~,p,
h2Vl, p = ( p "k- 1)vl, p,
h3v~,p
h3v~,t, = - ( p
-b 1)Vl,* p,
h4v~,p = (p +
3)v~,p, 1)V~,p,
=
--pro,* p,
h4v~,p = (p -~- 1)v~),p,
1)V~,p,
hlV~,p = ( p +
~
* (P + 1)Vl,p,
hlV~,p = ( p +
hiP2,* p = (p + 1 ) V2,p, *
* = PV3,p, * h2v3,p
hgv~,p = - ( p
h3V~,p = - ( p
+ 1)V~,p,
h4v~,p = ( p + 2 ) V ~ , p ,
eovo, p = O,
* eOvl,p = --Vo,p,
elV~,p = O,
eivT, p = O,
•
g¢
(A.3)
q- 1)v~,p,
h4v~,p = (p-Jr l)v~,p,
e2Vo, p = --[p + 1 ]V2, p,
563
e2Vl,p = --V3,p+l, .~
e3vo,p = [ P ] V l , p _ 1,
e3vl, p = O,
foV~,p = [p + 1]Vl,p,
foV~,p = O,
flV;,p = [p]O~,p_ 1,
flV~,p = v~,v,
f 2 *V2,p = O,
f2Vl,* p = O,
• = O, f3Uo,p
f3Vl,p = --V0,p+ 1,
eoV2,p = O,
eoIo3,p ~. [P]V2,p_l,
elU2,p = VO,p+1,
elY3, p = [p + 1 ] V 1 , p ,
e2v2, p = O,
e2v3,p* ~ O,
e3V2,p -= -v3, p,
e3V3,p ~ O,
foV~,p -- -V3,p+ , * 1
e3v3,p* ~ O,
fi V~,p = O,
fiV~,p = O,
f 2V~,p = -v~,p ,
f 2v~,p = - [ p ] V~,p_ , ,
f3v~,l, = 0,
f30~,p = [p q- llV~,p.
(A.4)
References [11 M.D. Gould, J.R. Links and Y.-Z. Zhang, Twisted quantum affine super algebras Uq[sl(212)(2)], Uq[osp(2[2)] invariant R-matrices and a new integrable electronic model, cond-mat/9611014. 12] A. Ludwig, M. Fisher, R. Shankar and G. Grinstein, Phys. Rev. B 50 (1994) 7526. [3] R. Gade, Anderson localization for sublattice models, Nucl. Phys. B 398 (1993) 499. [4] A.A. Nersesyan, A.M. Tsvelik and F. Wenger, Phys. Rev. Lett. 72 (1994) 2628. [ 51 C. Murdy, C. Chamon and X.G. Wen, Two-dimensional conformal field theory for disordered systems at criticality, Nucl. Phys. B 466 (1996) 383. 161 L. Rozansky and H. Saleur, Quantum field theory for the multi- variable Alexander-Conway polynomial, Nucl. Phys. B 376 (1992) 461, [ 7 ] M.R. Zirnbauer, Towards a theory of the integer quantum Hall transition: continuum limit of the ChalkerCoddington model, cond-mat/9701024.
R. Gade/Nuclear Physics B 500 [PM] (1997) 547-564
564
18] D. Bernard, (Perturbed) conformal field theory applied to 2d disordered systems: An introduction, hep-th/9509137. [9] R.B. Zhang, Symmetrizable quantum affine superalgebras and their representations, q-alg/9612020. [ 101 M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics, vol. 85, AMS, 1994. I 11 I K. Kimura, J. Shiraishi and J. Uchiyama, A level-one representation of the quantum affine superalgebra Uq( sl( M + 11N + 1 ) ), q-alg/9605047. [ 121 H. Awata, S. Odake and J. Shiraishi, q-difference realization of Uq( sl(M[ N) ) and its application to free boson realization of Uq(sl(211 ) ), q-alg/9701032. [ 131 R. Gade, in preparation. [14] L.J. Dixon, J. Lykken and M.E. Peskin, N = 2 superconformal symmetry and SO(2,1 ) current algebra, Nucl. Phys. B 325 (1989) 329. [151 A.H. Bougourzi, Uniqueness of the bosonization of the Uq(,~(2)k) quantum current algebra, Nucl. Phys. B 404 (1993) 457+ [ 16[ V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977) 8. 1171 S.M. Khoroshkin and V.N. Tolstoy, Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan-Weyl realizations for quantum affine algebras, hep-th/9404036. [ 181 V.G. Drinfeld, A new realization of Yangians and quantized affine algebras, Sov. Math. Dokl. 36 (1988) 212. [ 191 V.G. Drinfeld, Quantum groups, Proc. ICM-86 (Berkeley), Vol. 1,798, AMS (1987) 1201 H. Yamane, On defining relations of the Lie superalgebras and their quantized universal enveloping superalgebms, q-alg/9603015. 1211 I.B. Frenkel and N.Yu Reshetikhin, Quantum affine algebras and holonomic q-difference equations, Comm. Math. Phys. 146 (1992) 1. [221 G.W. Delius and Y.-Z. Zhang, Finite-dimensional representations of quantum affine algebras, J. Phys. A 28 (1995) 1915. 1231 M. Jimbo, A q-Analogue of U(gl( N + 1) ), Hecke Algebra, and the Yang-Baxter Equation, Lett. Math. Phys. 10 (1985) 63. 1241 R.B. Zhang, Universal L-operator and invariants of the quantum supergroup Uq(gl(mln)), J. Math. Phys. 33 (1992) 1970. 125] M.R. Zirnbauer, Towards a theory of the integer quantum Hall transition: from the non-linear sigma model to superspin chains, Ann. der Phys. 3 (1994) 513. 1261 M. Jimbo, K. Miki, I". Miwa and A. Nakayashiki, Correlation functions of the XXZ-model for zl < - 1, Phys. Lett. A 168 (1991) 256. 127] A. Abada, A.H. Bougourzi and M.A. El Gradechi, Deformation of the Wakimoto construction, Mod. Phys. Lett. A 8 (1993) 715. [281 A. Matsuo, Free Field Representation of quantum affine algebra Uq(~ll2), Phys. Lett. B 308 (1993) 260. 129] J. Shiraishi, Free boson representation of Uq(sl(2)), Phys. Lett. A 171 (1992) 243. [301 A. Kato, Y.-H. Quano and J. Shiraishi, Free boson representation of q-vertex operators and their correlation functions, Comm. Math. Phys. 157 (1993) 119. [31 I Y. Koyama, Staggered polarization of vertex models with Uq(~?l(n))-symmetry, Comm. Math. Phys. 164 (1994) 277. 132 ] H. Awata, S. Odake and J. Shiraishi, Free boson realization of Uq(,~IN), Comm. Math. Phys. 162 (1994) 61. [33 ] D. Friedan, E. Martinec and S. Shenker, Conformal invariance, supersymmetry and string theory, Nucl. Phys. B 271 (1986) 93. [ 341 E Bouwknegt, A. Ceresole, J.G. McCarthy and P. van Nieuwenhnizen, Extended Sugawara construction for the superalgebras SU ( M+ 1IN+ 1). I. Free-field representation and bosonization of super Kac-Moody currents, Phys. Rev. D 39 (1989) 2971. 1351 P.A. Griffin and O.F. Hernandez, Feigin-Fuchs derivation of SU( 1,1 ) parafermion characters, Nucl. Phys. 13356 (1991) 287. 136 ] 13. Davies, O. Foda, M. Jimbo, T. Miwa and A. Nakayashiki, Diagonalization of the XXZ Hamiltonian by vertex operators, Comm. Math. Phys. 151 (1993) 89. 1371 E. Date, M. Jimbo and M. Okado, Crystal base and q vertex operators, Comm. Math. Phys. 155 (1993) 47. A