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FREE FIELD REALIZATION OF VERTEX OPERATORS FOR LEVEL TWO MODULES OF Uq sl(2)

arXiv:math/9809078v1 [math.QA] 16 Sep 1998

YUJI HARA

Abstract. Free ?eld realization of vertex operators for level two modules of Uq sl(2) are shown through the free ?eld realization of the modules given by Idzumi in Ref.[4, 5]. We constructed types I and II vertex operators when the spin of the associated evaluation module is 1/2 and type II’s for the spin 1.

1. Introduction Vertex operators for the quantum a?ne algebra Uq sl(2) have played essential roles in the algebraic analysis of solvable lattice models since the pioneering works of [1, 2, 3]. In these works which analyze the XXZ model, type I vertex operators are identi?ed with half in?nite transfer matrices as their representation-theoretical counter part and type II vertex operators are interpreted as particle creation operators. To perform concrete computation such as a trace of composition of vertex operators, we need free ?eld realization of modules and operators. In the said example of the XXZ model, the integral expressions of n-point correlation functions which are special cases of the traces are obtained through bosonization of level one module of Uq sl(2) . Motivated by these results, Idzumi [4, 5] constructed level two modules and type I vertex operators accompanied by spin 1 evaluation modules for Uq sl(2) in terms of bosons and fermions and then calculated correlation functions of a spin 1 analogue of the XXZ model. The purpose of this paper is to extend Idzumi’s free ?eld realization to other kinds of vertex operators i.e. type I and II vertex operators for the level two modules associated with the evalution module of spin 1/2 and the type II’s for the spin 1. The results are given in Section 3 and their derivation is discussed in the ?rst case in Section 4. The results together with Ref.[4, 5] give the complete set of vertex operators for level two module of Uq sl(2) and enable one to calculate form factors of the spin 1 analogue of the XXZ model. Recently Jimbo and Shiraishi [7] showed a coset-type construction for the deformed Virasoro algebra with the vertex operators for Uq sl(2) . They constructed a primary operator for the deformed Virasoro algebra as coset type composition of vertex operators which may be denoted as Uq sl(2)

k

⊕ Uq sl(2)

1

/Uq sl(2)

k+1

. We hope that our results will be helpful

1

for extending this work to the deformed supersymmetric Virasoro algebra through Uq sl(2) k ⊕ Uq sl(2)

2

/Uq sl(2)

k+2

.

2. Free field realization of level two module 2.1. Convention. In the following we will use U to denote the quantum a?ne algebra Uq sl(2) . Unless mentioned, we follow the notations of Ref.[4, 5]. As for the free ?eld representation, we slightly modify the convention. The quantum a?ne algebra U is an associative algebra with unit 1 generated by ei , fi (i = 0, 1), q h (h ∈ P ? ) with relations q 0 = 1, q h q h = q h+h , q h ei q ?h = q

? h,αi

′ ′

ei , q h fi q ?h = q ? h,αi fi ,

ti ? t?1 i [ei , fi ] = δij , (ti = q hi ) ?1 q?q e3 ej ? [3]e2 ej ei + [3]ei ej e2 ? ej e3 = 0, i i i i fi3 fj ? [3]fi2 fj fi + [3]fi fj fi2 ? fj fi3 = 0, where P = ZΛ0 + ZΛ1 + Zδ is the weight lattice of the a?ne Lie algebra sl(2) and P ? is the dual lattice to P with the dual basis {h0 , h1 , d} to {Λ0 , Λ1 , δ} with respect to the natural pairing , : P × P ? → Z. We also use current type generators introduced by Drinfeld [11] [ak .al ] = δk+l,0 [2k] γ k ? γ ?k , k q ? q ?1

Kak K ?1 = ak , Kx± K ?1 = q ±2 x± , k k [ak , x± ] = ± l [2k] ?|k|/2 ± γ xk+l , k

x± x± ? q ±2 x± x± = q ±2 x± x± ? x± x± , k+l l l k+l k l+1 l+1 k [x+ , x? ] k l where ψk , and ?k are de?ned as ψk z ?k = K exp (q ? q ?1 )

k 0

?

=

γ

k?l 2

ψk+l ? γ 2 φk+l , q ? q ?1 ak z ?k ,

k 1

l?k

[A,B]=AB-BA

2

φk z k = K ?1 exp ?(q ? q ?1 )

k 0 k 1

a?k z k .

The relation between two types of generators are t1 = K, t0 = γK ?1 , e1 = x+ , e0 t1 = x? , f1 = x? , t?1 f1 = x?1 . 0 1 0 1 0 The higest weight module and the evaluation module are described compactly in Ref.[4]. Commutation and anti commutation relations of bosons and fermions are given by [am , an ] = δm+n,0 [2m]2 , m {φm , φn }? = δm+n,0 ηm , ηm = q 2m + q ?2m . where m, n ∈ Z + 1/2 or ∈ Z for Neveu-Schwarz-sector or Ramond-sector respectively. Fock spaces and vacuum vectors are denoted as F a , F φ φN S (z) =

1 n∈Z+ 2 NS

, F φ and |vac ,|NS , |R for the boson φR z ?n . n

n∈Z

R

and NS and R fermion respectively. Fermion currents are de?ned as φN S z ?n , φR (z) = n

Q = Zα is the root lattice of sl2 and F [Q] be the group algebra. We use ? as [?, α] = 2. 2.2. V (2Λ0 ), V (2Λ1 ). The highest weight module V (2Λ0) is identi?ed with the Fock space (1)

φ φ F+ = F a ? (Feven ? F [2Q]) ⊕ (Fodd ? eα F [2Q]) , (0)

NS NS

subscripts even and odd represent the number of fermions. The highest weight vector is |vac ? |NS ? 1. V (2Λ1 ) is (2)

φ φ F? = F a ? (Feven ? eα F [2Q]) ⊕ (Fodd ? F [2Q]) (0)

NS NS

with the highest weight vector being |vac ? |NS ? eα . Note that F (0) = F? ⊕ F+ . F (0) = F a ? F φ

NS

(0)

(0)

? F [Q].

The operators are realized in the following manner. γ = q2, x± (z) =

m∈Z

?

K = q? ,

1 1

± ± x± z ?m = E< (z)E> (z)φN S (z)e±α z 2 ± 2 ? , m

{A,B}=AB-BA

3

± E< (z) = exp (±

am ?m ?m a?m ?m m ± q z ), E> (z) = exp (? q z ), [2m] [2m] m>0 m>0

and (3) (4) ? 2 (λ, λ) φNS a d=? + ? mNm ? kNk , 8 4 m=1 k>0

a Nm = ∞

m a?m am , [2m]2

φ Nk

NS

=

1 NS NS φ φ ηm ?m m

(m > 0),

where the higest weight vector of the module should be substituted for λ of (3). 2.3. V (Λ0 + Λ1 ). The module V (Λ0 + Λ1 ) is identi?ed with (5) where φR |R = |R . 0 The highest weight vector is identi?ed with |vac ? |R ? e 2 . Operators are constructed in the same way as before except that subscripts for fermion sector are R instead of NS. 3. Free field realizations of vertex operators Let V, V ′ be level two modules and Vz

V ΦV

′ ,k α

F (1) = F a ? F φ ? e 2 F [Q],

R

α

(k)

be a spin k/2 evaluation module of U. Vertex

operators we will consider are U-linear maps of the following kinds [8, 9] (6) (7) (z) : V ?→ V ′ ? Vz(k) ,

Ψk,V (z) : V ?→ Vz(k) ? V ′ . V

′

Vertex operators of the form (6,7) are called type I and II respectively. Components of vertex operators are de?ned as

k

′ Φ(z)V ,k V

k

=

n=0

Φn (z) ? un ,

′ Ψ(z)k,V V

=

n=0

un ? Ψn (z).

3.1. type I Vertex Operators for level 2 and spin 1/2. We show free ?eld realization of type I vertex operators of the following kinds (8) (9) where i = 0 or 1. ΦΛ0 +Λ1 ,1 (z) : V (2Λi ) ?→ V (Λ0 + Λ1 ) ? Vz(1) , 2Λi Φ2Λi ,1 1 (z) : V (Λ0 + Λ1 ) ?→ V (2Λi) ? Vz(1) Λ0 +Λ

4

Under the free ?eld realization of level 2 modules reviewed in Secton 2, the explicit forms of the components of the vertex operators in (8) are (10) (11) Φ1 (z) = BI,< (z)BI,> (z)?R S (z)eα/2 (?q 4 z)?/4 , N Φ0 (z) = dw ? ? BI,< (z)E< (w)BI,> (z)E> (w)?R S (z)φN S (w) N 2πi w 4 ;q ∞ 3 1 w q5z ?α/2 4 ?/4 ??/2 4 3 ?2 q z ×e (?q z) w (?q zw ) , + w 4 ?3 1 ? q 5 z/w ; q ∞ 1 ? q w/z qz

∞

(12) (13)

BI,< (z) = exp

n=1 ∞

[n]a?n 5 n (q z) , [2n]2 [n]an 3 ?n (q z) . [2n]2

BI,> (z) = exp ?

n=1

The integrand of Φ0 (z) has poles only at w = q 5 z, q 3 z except for w = 0, ∞ and the contour of integration encloses w = 0, q 5 z, details are discussed in Sec.4. For those of (9) we just replace ?R S (z) with ?N S (z) in (10,11). N R The fermionic part ?(z)’s are maps between di?erent fermion sectors and satisfy φN S (w)?(z)N S = R ?q 4 z w

1/2

(14)

w 4 ;q ∞ q3z w 4 ;q ∞ qz

q7z 4 ;q ∞ w ?(z)N S φR (w) R q5z 4 ;q ∞ w

and exactly the same equation except subscripts for fermion sectors are exchanged. This kind of mapping for fermions ?rst appeared in high-energy phisics theory as “fermion emission vertex operator”[6, 10]. Their free ?eld realizations are (15) Y =?

m>n≥0

?R S (z) = NS| eY |R , N Xm,n ?R ?R z m+n ? ?m ?n

k>l≥0

Xk+1/2,l+1/2 ?N S ?N S z ?k?l?1 k+1/2 l+1/2

(16)

+

m≥0 k≥0

Xm,?k?1/2 ?R ?N S z m?k?1/2 ?m k+1/2

5

(17) Y′ =

k>l≥0

?N S (z) = R| eY |NS , R

NS k+l+1 Xk+1/2,l+1/2 ?N S + ?k?1/2 ??l?1/2 z m>n≥0 R k?m+1/2 X?k?1/2,m ?N S ?k?1/2 ?m z k≥0 m≥0 5m R γm q , φ?m ηm ?3m R γm q φm ηm

′

Xm,n ?R ?R z ?m?n m n

(18)

?

(19) (20)

?R 0

=

φR , 0

?R ?m

=

?R m

=

(m > 0), (k > 0),

?N S = φ N S k+1/2 k+1/2

γk q ?3k?2 (?(?1)1/2 ), ηk+1/2 Xk,l =

NS ?N S ?k?1/2 = φ?k?1/2

γk q 5k+2 (?1)1/2 ηk+1/2

q 4k ? q 4l , 1 ? q 4(k+l) (q 2 z; q 4 )∞ = (z; q 4 )∞

∞

(21)

(q 2 ; q 4 )n γn = 4 4 , (q ; q )n

γn z n .

n=0

(15,17) are to mean that a matrix element is given by

R

out|?R S (z)|in N

NS R

= R out| ? NS| eY |R ? |in ∈ F φ , |in

R

NS,

for |out

NS

∈ Fφ

NS

.

? We de?ne the normalized vertex operators Φ(z)’s as follows ? Λ0 + Λ1 |Φ1 (z)|2Λ0 = 1, ? Λ0 + Λ1 |Φ0 (z)|2Λ1 = 1, and these are given by (22) (23) (24) (25) ? ΦΛ0 +Λ1 ,1 (z) = Φ(z), 2Λ0 ? 1 ,1 Φ2Λ+Λ1 (z) = (?q 4 z)?1/4 Φ(z), Λ0 ? Φ2Λ0 ,1 1 (z) = (?q 4 z)1/4 Φ(z), Λ0 +Λ ? ΦΛ0 +Λ1 ,1 (z) = (?q 6 z)?1/2 Φ(z). 2Λ1 ? 2Λ1|Φ1 (z)|Λ0 + Λ1 = 1, ? 2Λ0|Φ0 (z)|Λ0 + Λ1 = 1,

3.2. type II Vertex Operators for level 2 and spin 1/2. We consider type II vertex operators of the following kind (26) (27)

1,Λ Ψ2Λi0 +Λ1 (z) : V (2Λi) ?→ Vz(1) ? V (Λ0 + Λ1 ),

Ψ1,2Λi 1 (z) : V (Λ0 + Λ1 ) ?→ Vz(1) ? V (2Λi ). Λ0 +Λ

6

Explicit forms of the components are as follows. (28) (29) Ψ0 (z) = BII,< (z)BII,> (z)?(q ?2 z)e?α/2 (?q 2 z)??/4 , Ψ1 (z) = dw + + BII,< (z)E< (w)BII,> (z)E> (w)?(q ?2z)φ(w) 2πi w 4 ;q ∞ 1 w q3z qz , + × eα/2 (?q 2 z)??/4 w ?/2 (?q 2 zw 3 )? 2 qw ?3 ; q 4 ∞ 1 ? q w/z 1 ? qz/w z ∞ [n]a?n (qz)n , BII,< (z) = exp ? 2 [2n] n=1

∞

(30) (31)

BII,> (z) = exp

n=1

[n]an 3 ?n (q z) . [2n]2

The integrand of Ψ1 (z) has poles only at w = q 3 z, qz except for w = 0, ∞ and the contour of integration encloses w = 0, qz. Subscripts for fermion sectors are abbreviated. Normalized vertex operators are de?ned by the conditions ? Λ0 + Λ1 |Ψ1 (z)|2Λ0 = 1, ? Λ0 + Λ1 |Ψ0 (z)|2Λ1 = 1, and these are given by (32) (33) (34) (35) ? Ψ1,Λ0 +Λ1 (z) = (?q)?1 Ψ(z), 2Λ0 ? Ψ1,2Λ1 1 (z) = ?(?q 6 z)?1/4 Ψ(z), Λ0 +Λ ? Ψ1,2Λ0 1 (z) = (?q 2 z)1/4 Ψ(z), Λ0 +Λ ? Ψ1,Λ0 +Λ1 (z) = (?q 2 z)1/2 Ψ(z). 2Λ1 ? 2Λ1|Ψ1 (z)|Λ0 + Λ1 = 1, ? 2Λ0|Ψ0 (z)|Λ0 + Λ1 = 1,

3.3. type II Vertex Operators for level 2 and spin 1. When the spin of the evaluation module is 1, the type II vertex operators do not contain any fermion emission vertex operators. (36) (37) Ψ2,2Λi (z) : V (2Λi ) ?→ Vz(2) ? V (2Λi), 2Λi Ψ2,Λ0 +Λ1 (z) : V (Λ0 + Λ1 ) ?→ Vz(2) ? V (Λ0 + Λ1 ). Λ0 +Λ1

Explicit form of the components are as follows. (38) (39) Ψ0 (z) = FII,< (z)FII,> (z)e?α (?q 2 z)??/2+1 , Ψ1 (z) = w dw + + FII,< (z)E< (w)FII,> (z)E> (w)φ(w) 2πi ?q 2 z q4z 1 + × w ?1/2 w z , 1? 4 w 1? q z w

7

?/2

The integration contour encircles poles w = 0, z but the pole w = q 4 z lies outside of it. dw2 dw1 + + + + Ψ2 (z) = (40) FII,< (z)E< (w1 )E< (w2 )FII,> (z)E> (w1 )E> (w2 ) 2πi 2πi w1 w2 ?/2 q4z 1 × eα + (w1 w2 )?1/2 w1 z ?q 2 z 1? 4 w1 1 ? q z w1 w1 1? 2 ?2 w1 ? q w2 q w2 × [2]?1 : φ(w1 )φ(w2) : w2 + z ?q 2 z 1 ? 4 1? q w1 w2 w2 w1 w1 1/2 (w1 w2 )1/2 1 ? 1? w1 w2 w2 + , ? q 2 w1 q 2 w2 w2 z 2z 1 ? 1? ?q 1? 4 ) 1? w1 q z w2 w2 We have to prepare two contours because of the fermionic part and one is for the term including

w2 : φ(w1 )φ(w2 ) : and the other is for the rest. The former satis?es | q4 w1 | < 1, |w2| > |z| and the

same condition satis?ed by the contour for P si1 with substitution w = w1 . The latter satis?es |q 2 w2 | < |w1 | < |q ?2w2 | and the same conditions as P si1 with w = w1 , w2 . a?m (41) (qz)m , FII,< (z) = exp ? [2m] m>0 (42) Under the normailzation ? 2Λ0|Ψ0 (z)|2Λ1 = 1, ? 2Λ1|Ψ2 (z)|2Λ0 = 1, FII,> (z) = exp am 3 ?m . (q z) [2m] m>0

? Λ0 + Λ1 |Ψ1 (z)|Λ0 + Λ1 = 1, ? Ψ(z)’ are given by (43) (44) (45) ? Ψ2,2Λ0 (z) = Ψ(z), 2Λ1 ? 2,Λ0 +Λ ΨΛ0 +Λ1 1 (z) = ?(?q 2 z)?1/2 Ψ(z), ? Ψ2,2Λ1 (z) = (?q 4 z)?1 Ψ(z). 2Λ0 4. Derivation

Λ0 +Λ Taking Φ2Λi 1 ,1 (z) as an example, we discuss the derivation of the results in the previous

section. Other cases can be treated in almost the same way. 4.1. General structure of Φ0 (z) and Φ1 (z). Calculating ?(x)Φ(z) = Φ(z)x

8

for x = Chevalley generators of U and an , we get 0 = [Φ1 (z), x+ ], 0 KΦ1 (z) = [Φ0 (z), x+ ], 0 0 = x? Φ0 (z) ? qΦ0 (z)x? , 0 0 (46) Φ0 (z) = Φ1 (z)x? ? qx? Φ1 (z), 0 0 0 = Φ0 (z)x? ? qx? Φ0 (z), 1 1 (47) q 3 zΦ0 (z) = Φ1 (z)x? ? q ?1 x? Φ1 (z), 1 1 (qzK)?1 Φ1 (z) = [Φ0 (z), x+ ], ?1 0 = [Φ1 (z), x+ ], ?1 (48) KΦ1 (z)K ?1 = qΦ1 (z), KΦ0 (z)K ?1 = q ?1 Φ0 (z), [m] Φ1 (z), m [m] [a?m , Φ1 (z)] = (q 3 z)?m (50) Φ1 (z). m From (48,49,50), we can speculate the form of Φ1 (z) as (49) [am , Φ1 (z)] = (q 5 z)m Φ1 (z) = BI,< (z)BI,> (z)?R S (z)eα/2 y ? . N To determine y and the fermionic part ?R S (z), we impose the following conditions on Φ1 (z) N Φ1 (z)x? ? qx? Φ1 (z) = (q 3 z)?1 (Φ1 (z)x? ? q ?1 x? Φ1 (z)), 0 0 1 1 0 = [Φ1 (z), x+ (w)], which can be easily seen from (46,47) and the proposition of Section 4.4 of Ref.[12]. Then we have (10,14) Φ1 (z) = BI,< (z)BI,> (z)?R S (z)eα/2 (?q 4 z)?/4 , N φR (w)?R S (z) = N ?q 4 z w

1/2

w 4 ;q ∞ q3z w 4 ;q ∞ qz

q7z 4 ;q ∞ w ?R S (z)φN S (w). N q5z 4 ;q ∞ w

9

Φ1 (z) can be calculated through (46) Φ0 (z) = dw 1 {Φ1 (z)x? (w) ? qx? (w)Φ1 (z)} 2πi w dw ? ? = BI,< (z)E< (w)BI,>(z)E> (w)?R S (z)φN S (w) N 2πi w 4 ;q ∞ 3 w q5z ?α/2 4 ?/4 ??/2 4 3 ?1 q z , + ×e (?q z) w (?q zw ) 2 w ?3 1 ? q 5 z/w ; q 4 ∞ 1 ? q w/z qz

To determine the contour of integration we have to ?nd the poles of ?R S (z)φN S (w) and this N can be seen from w 4 w 4 ;q ∞ ;q ∞ w 1/2 qz qz , NS|?N S (z)φR (w)|R = R|?R S (z)φN S (w)|NS = w R N w 4 . ?q 4 z ; q4 ∞ ;q ∞ q3z q3z w 4 ;q ∞ q3z R NS in the integrand has no poles and the contour Hence as a composite ?N S (z)φ (w) w ; q4 ∞ qz 5 is the one encloses w = 0, q z. 4.2. Fermion emission vertex operator. In Ref.[6], Eqn.(15) appears in the study of the Ising model and its free ?eld realization is given without any details. Thus we give the exposition of its derivation? . The main point of derivating free ?eld realization of the fermion emission vertex operator ?R S (z) (15,16) is to expand ?R S (z) as N N ?R S (z) = N

K,L

aK,L φR1 φR2 · · · |R NS|φN S φN S · ··, k k l1 l2 K = {ki }, L = {li },

and to calculate the coe?cients aK,L . After normalizing φn suitably to ?n (19,20), we see “aK,L /(normalization factor)” are identi?ed with Pfa?ans of Xk,l . With the aid of a relation satis?ed by Pfa?an ω ∧n = n!Pf(bij )x1 ∧ x2 · · · ∧x2n , where xk (1 ≤ k ≤ 2n) is a Grassmann variable and ω=

1≤i<j≤2n

bij xi ∧ xj ,

we get (15,16).

?

We are indebted to M.Jimbo for explaining the details of Ref.[6].

10

Wick’s theorem can be generalized to the present situation and we only need to calculate oneand two-point correlation functions for aK,L. To calculate these, we rewrite (14) and introduce auxiliary operators (51) (52) (53) ? ? φN S (w)?N S (q ?4 ) = ?N S (q ?4 )φR (w), R R (qw ?1 ; q 4 )∞ ? φN S (w) = (?1)?1/2 w 1/2 3 ?1 4 φN S (w), (q w ; q )∞ (qw; q 4 )∞ R ? φR (w) = 3 φ (w) = f+ (w)φR (w), (q w; q 4)∞ ? NS|φN S = 0 (n < 0), n and this enables us to see that ? ? ? ? NS|?N S (q ?4 )φR (z)φR (w)|NS = NS|φN S (z)?N S (q ?4 )φR (w)|NS R R contains only negative (positive) powers of z (w). On the other hand the expectation value of q2w w ? ? {φR (z), φR (w)} = f+ (z)f+ (w) δ +δ 2 z q z δ(z) =

n∈Z

we set ?(z = q 4 ) for simplicity. They are de?ned to satisfy ? φR |R = 0 (n > 0), n ? φR |R = |R , 0

,

zn,

with respect to NS|?N S (q ?4 ) and |R is R ? ? ? ? NS|?N S (q ?4 )φR (z)φR (w)|NS + NS|?N S (q ?4 )φR (w)φR (z)|NS R R = f+ (z)f+ (w) δ w q2w +δ 2 z q z 1 ? qw 1 ? q ?1 w + ?1 1 ? q 2 w/z 1 ? q ?2 w/z NS|?N S (q ?4 )φR φR |R z ?n w ?m R n m

n,m∈Z

where we normalize NS|?N S (q ?4 )|R = 1. And we get R ? ? NS|?N S (q ?4 )φR (z)φR (w)|R = R

Expanding the last line of the following equation as in C NS|?N S (q ?4 )φR (z)φR (w)|R = R = we have (54) NS|?N S (q ?4 )φR φR |R = Xm,n γn γm q n+m (n, m ≥ 0). R ?n ?m

1 ? qw 1 ? q ?1 w 1 + ?1 , f+ (z)f+ (w) 1 ? q 2 w/z 1 ? q ?2 w/z

11

Similar calculation yields (55) (56) NS|φN S ?N S (q ?4 )φR |R = ?(?1)1/2 X?k?1/2,n γn γk q n+k (n, k ≥ 0), k+1/2 R ?n NS|φN S φN S ?N S (q ?4 )|R = ?Xl+1/2,k+1/2 γl γk q l+k (k, l ≥ 0). k+1/2 l+1/2 R

z-dependence of ?R S (z) is recovered with the equation N (57) ζ d ?R S (z)ζ ?d N

i R NS

= ?R S (ζ ?1z), N

ζ ?d φi (z)ζ d = φi (ζz), i|di = di|i = 0, where di ’s are the fermionic part of d of (3) di = ?

k>0 φ kNk , (i = NS or R)

i

i

and satisfy [di , φi ] = nφn . n To derive (57), we multiply (14) by ζ d , ζ ?d

R NS

from left and right respectively.

Acknowledgement

The author thanks M.Jimbo, H.Konno, S.Odake and J.Shiraishi for helpful discussions. He also thanks A.Kuniba for warm encourragement. Appendix A. boson Followings are useful formulae for normal ordering bosons. We set (z)∞ = (z; q 4 )∞ for brevity.

? BI,> (z)E< (w) = ? E> (w)BI,< (z) = + BI,> (z)E< (w)

(qw/z)∞ ? E (w)BI,> (z), (q ?1 w/z)∞ < (q 9 z/w)∞ ? BI,< (z)E> (w), (q 7 z/w)∞

(q ?3 w/z)∞ + = ?1 E (w)BI,> (z), (q w/z)∞ < (q 5 z/w)∞ + + E> (w)BI,< (z) = 7 BI,< (z)E> (w), (q z/w)∞ (q ?1 w/z)∞ + E (w)BII,> (z), (q ?3 w/z)∞ < (q 3 z/w)∞ + BII,< (z)E> (w), (qz/w)∞

+ BII,> (z)E< (w) = + E> (w)BII,< (z) =

12

? BII,> (z)E< (w) = ? E> (w)BII,< (z) =

(q ?1 w/z)∞ ? E (w)BII,> (z), (qw/z)∞ <

(q 3 z/w)∞ ? BII,< (z)E> (w), 5 z/w) (q ∞ w ? ? FII,> (z)E< (w) = (1 ? 2 )E< (w)FII,>(z), q z q2z ? ? E> (w)FII,<(z) = (1 ? )FII,< (z)E> (w), w 1 + FII,> (z)E< (w) = E + (w)FII,> (z), 1 ? q ?4 w/z < 1 + ? E> (w)FII,< (z) = FII,< (z)E> (w), 1 ? z/w 1 ? ? + E + (w2 )E> (w1 ), E> (w1 )E< (w2 ) = 1 ? w2 /w1 < 1 + ? + E> (w2 )E< (w1 ) = E ? (w1 )E> (w2 ), 1 ? w1 /w2 < Appendix B. fermion For ?N S (z), we show the equations corresponding to the ones from (51) to (56) R (58) (59) (60)

′

? ′ ? ′ φR (w)?R S (q ?4 ) = ?R S (q ?4 )φN S (w), N N (q/w; q 4)∞ R ? ′ φR (w) = 3 φ (w), (q /w; q 4)∞ (qw; q 4)∞ N S ? ′ φN S (w) = (?1)1/2 w ?1/2 3 φ (w), (q w; q 4)∞ ? R|φR = 0 (n < 0), n ? R|φR = R|, 0

′

?N ′ φn S |NS = 0 (n > 0),

?1 ?R′ (z)φR′ (w)?R (q ?4 )|NS = 1 ? q/z + 1 ? q /z ? 1 ? R|φ NS 1 ? q 2 w/z 1 ? q ?2 w/z

R|φRφR ?R S (q ?4 )|NS = Xn,m γn γm q n+m (n, m ≥ 0), n m N

1/2 R|φR ?R S (q ?4 )φN S X?k?1/2,n γn γk q n+k (n, k ≥ 0), n N ?k?1/2 |NS = (?1) NS l+k R|?R S (q ?4 )φN S (k, l ≥ 0) N ?k?1/2 φ?l?1/2 |NS = Xl+1/2,k+1/2 γl γk q

13

Appendix C. Calculation of Eqn.(54) We show details of calculation of (54). From (21) NS|?N S (q ?4 )φR (z)φR (w)|R R = =

k 0,l 0

1 1 ? qw 1 ? q ?1 w + ?1 f+ (z)f+ (w) 1 ? q 2 w/z 1 ? q ?2 w/z γk (qz)k γl (qw)l

a 0

(1 ? qw)

q2w z

a

+ (1 ? w/q)

w q2z

a

?1

=

0 a m

γn+a γm?a ηa q n+m z n w m ?

0 a m?1

γn+a γm?a?1 (q 2a + q ?2(a+1) )q n+m z n w m ? γn γm z n w m

Hence the equation to be proved is Xn,m γn γm =

0 a m

γn γm ηa ?

0 a m?1

γn+a γm?a?1 (q 2a + q ?2(a+1) ) ? γn γm z n w m ,

which is equivalent to (61) Xn,m = 1 + (1 ? t?1 )(1 + t2n )

1 a 2

(t1+2n ; t2 )a?1 (t2m?2a+2 ; t2 )a ta (t2+2n ; t2 )a?1 (t2m?2a+1 ; t2 )a 1 ? t2(n+a) m

where we set t = q . It can be proved by induction with respect to k that the summation over a = m, m ? 1, · · ·, m ? k yields

1+2n 2 ; t )m?k?1 m?k (t t 2+2n ; t2 ) (t m?k?1 k 2j (t2k+2 ; t2 )m?k j=0 t . (t2k+1 ; t2 )m?k 1 ? t2(n+k) t2m ?t2n . 1?t2(n+m)

Setting k = m ? 1 we can see that the right hand side of (61) is equal to References

[1] B.Davies, O.Foda, M.Jimbo, T.Miwa and A.Nakayashiki, Diagonalization of the XXZ Hamiltonian by vertex operators, Commun. Math. Phys. 151, 89 (1993). [2] M.Jimbo, K.Miki, T.Miwa and A.Nakayashiki, Correlation functions of the XXZ model for ? < ?1, Phys.Lett.A, 168, 256 (1992). [3] M.Jimbo, T.Miwa, Algebraic Analysis of Solvable Lattice Models, American Mathematical Society, 1993. [4] M.Idzumi, Level two irreducible representations of Uq sl (2) , vertex operators and their correlations, Int.J.Mod.Phys. A9, 4449 (1994). [5] M.Idzumi, Correlation functions of the spin-1 analog of the XXZ model, hep-th/9307129. [6] O.Foda, M.Jimbo, T.Miwa, K.Miki and A.Nakayashiki, Vertex operators in solvable lattice models, J.Math.Phys. 35, 13 (1994). [7] M.Jimbo and J.Shiraishi, A Coset-type construction for the deformed Virasoro algebra, Lett. Math. Phys. 43, 173 (1998). [8] I.B.Frenkel, N.Yu.Reshetikhin, Quantum a?ne algebras and holonomic di?erence equations, Commun.Math.Phys. 146, 1 (1992).

14

[9] E.Date, M.Jimbo, M.Okado, Crystal base and q-vertex operators, Commun.Math.Phys. 155, 47 (1993). [10] D.Friedan, Z.Qiu and S.Shenker, Superconformal invariance in two dimensions and the tricritical Ising model, Phys.Lett.B 151, 37 (1985); E.F.Corrigan and D.I.Olive, Fermion-meson vertices in dual theories, Nuovo Cim. 11A, 749 (1972); E.F.Corrigan and P.Goddard, Gauge conditions in the dual fermion model, Nuovo Cim. 18A, 339 (1973); M.Kato and S.Matsuda, Null Field Construction in Conformal and Superconformal Algebras, Adv.Std.Pure Math. 16, 205 (1988). [11] V.G.Drinfeld, A new realization of Yangians and quantized a?ne algebras, Soviet Math. Doklady 36, 212 (1988). [12] V.Chari and A.Pressley, Quantum A?ne Algebra, Commun. Math. Phys. 142, 261 (1991). Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Tokyo 153, Japan E-mail address: ss77070@komaba.ecc.u-tokyo.ac.jp

15

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2

The PWM

Affine Algebra

P2 (4) and then by expressing P in terms

BASES-

In exact solvable lattice models 1]- 6], the

that has a

On

1 SUPER

We consider the

use directly the QCD

m (7) Equivalently, given oscillator

Twisted

A