Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations
Solutions of the qKZ equation associated with the quantum affine algebra Uq(^sl2) and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of q-Wakimoto modules are shown to coincide with those of Tarasov a...
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irk-123456789-1490322019-02-21T01:24:02Z Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations Kuroki, K. Nakayashiki, A. Solutions of the qKZ equation associated with the quantum affine algebra Uq(^sl2) and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of q-Wakimoto modules are shown to coincide with those of Tarasov and Varchenko. 2008 Article Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations/ K. Kuroki, A. Nakayashiki // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 23 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81R50; 20G42; 17B69 http://dspace.nbuv.gov.ua/handle/123456789/149032 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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Solutions of the qKZ equation associated with the quantum affine algebra Uq(^sl2) and its two dimensional evaluation representation are studied. The integral formulae derived from the free field realization of intertwining operators of q-Wakimoto modules are shown to coincide with those of Tarasov and Varchenko. |
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Kuroki, K. Nakayashiki, A. |
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Kuroki, K. Nakayashiki, A. Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations Symmetry, Integrability and Geometry: Methods and Applications |
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Kuroki, K. Nakayashiki, A. |
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Kuroki, K. |
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Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
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Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
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Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
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Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
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Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
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free field approach to solutions of the quantum knizhnik-zamolodchikov equations |
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Інститут математики НАН України |
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2008 |
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Free Field Approach to Solutions of the Quantum Knizhnik-Zamolodchikov Equations/ K. Kuroki, A. Nakayashiki // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 23 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT kurokik freefieldapproachtosolutionsofthequantumknizhnikzamolodchikovequations AT nakayashikia freefieldapproachtosolutionsofthequantumknizhnikzamolodchikovequations |
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1837476095253807104 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 049, 13 pages
Free Field Approach to Solutions
of the Quantum Knizhnik–Zamolodchikov Equations
Kazunori KUROKI † and Atsushi NAKAYASHIKI ‡
† Department of Mathematics, Kyushu University, Hakozaki 6-10-1, Fukuoka 812-8581, Japan
E-mail: ma306012@math.kyushu-u.ac.jp
‡ Department of Mathematics, Kyushu University, Ropponmatsu 4-2-1, Fukuoka 810-8560, Japan
E-mail: 6vertex@math.kyushu-u.ac.jp
Received February 18, 2008, in final form May 27, 2008; Published online June 03, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/049/
Abstract. Solutions of the qKZ equation associated with the quantum affine algebra
Uq(ŝl2) and its two dimensional evaluation representation are studied. The integral formulae
derived from the free field realization of intertwining operators of q-Wakimoto modules are
shown to coincide with those of Tarasov and Varchenko.
Key words: free field; vertex operator; qKZ equation; q-Wakimoto module
2000 Mathematics Subject Classification: 81R50; 20G42; 17B69
1 Introduction
In 1992 I. Frenkel and N. Reshetikhin [7] had developed the theory of intertwining operators
for quantum affine algebras and had shown that the matrix elements of intertwiners satisfy the
quantized Knizhnik–Zamolodchikov (qKZ) equations.
The theory of intertwiners and qKZ equations was successfully applied to the study of solvable
lattice models [9] (and references therein). As to the study of solutions of the qKZ equations,
bases are constructed by Tarasov and Varchenko [22] in the form of multi-dimensional hyper-
geometric integrals in the case of Uq(ŝl2). However solutions of the qKZ equations for other
quantum affine algebras are not well studied [21].
The method of free fields is effective to compute correlation functions in conformal field theo-
ry (CFT) [3], in particular, solutions to the Knizhnik–Zamolodchikov (KZ) equations [18, 19].
A similar role is expected for those of quantum affine algebras. Unfortunately it is difficult to
say that this expectation is well realized, as we shall explain below.
Free field realizations of quantum affine algebras are constructed by Frenkel and Jing [6]
for level one integrable representations of ADE type algebras and by Matsuo [15], Shiraishi [16]
and Abada et al. [1] for representations with arbitrary level of Uq(ŝl2). The latter results are
extended to Uq(ŝlN ) in [2]. Free field realizations of intertwiners are constructed based on these
representations in the case of Uq(ŝl2) [9, 10, 12, 15, 4].
The simplicity of the Frenkel–Jing realizations makes it possible not only to compute matrix
elements but also traces of intertwining operators [9], which are special solutions to the qKZ
equations. The case of q-Wakimoto modules with an arbitrary level becomes more complex and
the detailed study of the solutions of the qKZ equations making use of it is not well developed.
In [15] Matsuo derived his integral formulae [14] from the formulae obtained by the free field
calculation in the simplest case of one integration variable. However it is not known in general
whether the integral formulae derived from the free field realizations recover those of [14, 23, 22]1.
1See Note 1 in the end of the paper.
mailto:ma306012@math.kyushu-u.ac.jp
mailto:6vertex@math.kyushu-u.ac.jp
http://www.emis.de/journals/SIGMA/2008/049/
2 K. Kuroki and A. Nakayashiki
The aim of this paper is to study this problem in the case of the qKZ equation with the value
in the tensor product of two dimensional irreducible representations of Uq(ŝl2). More general
cases will be studied in a subsequent paper.
There are mainly two reasons why the comparison of two formulae is difficult. One is that
the formulae derived from the free field calculations contain more integration variables than
in Tarasov–Varchenko’s (TV) formulae. This means that one has to carry out some integrals
explicitly to compare two formulae. The second reason is that the formulae from free fields
contains a certain sum. This stems from the fact that the current and screening operators are
written as a sum which is absent in the non-quantum case. Since TV formulae have a similar
structure to those for the solutions of the KZ equation [18, 19], one needs to sum up certain
terms explicitly for the comparison of two formulae. We carry out such calculations in the case
we mentioned.
The plan of this paper is as follows. In Section 2 the construction of the hypergeometric solu-
tions of the qKZ equation due to Tarasov and Varchenko is reviewed. The free field construction
of intertwining operators is reviewed in Section 3. In Section 4 the formulae for the highest to
highest matrix elements of some operators are calculated. The main theorem is also stated in
this section. The transformation of the formulae from free fields to Tarasov–Varchenko’s formu-
lae is described in Section 5. In Section 6 the proof of the main theorem is given. Remaining
problems are discussed in Section 7. The appendix contains the list of the operator product
expansions which is necessary to derive the integral formula.
2 Tarasov–Varchenko’s formula
Let V (1) = Cv0 ⊕ Cv1 be a two-dimensional irreducible representation of the algebra Uq(sl2),
and R(z) ∈ End(V (1)⊗2) be a trigonometric quantum R-matrix given by
R(z) (vε ⊗ vε) = vε ⊗ vε,
R(z) (v0 ⊗ v1) =
1− z
1− q2z
qv0 ⊗ v1 +
1− q2
1− q2z
v1 ⊗ v0,
R(z) (v1 ⊗ v0) =
1− q2
1− q2z
zv0 ⊗ v1 +
1− z
1− q2z
qv1 ⊗ v0,
Let p be a complex number such that |p| < 1 and Tj denote the multiplicative p-shift operator
of zj ,
Tjf(z1, . . . , zn) = f(z1, . . . , pzj , . . . , zn).
The qKZ equation for the V (1)⊗n-valued function Ψ(z1, . . . , zn) is
TjΨ = Rj,j−1(pzj/zj−1) · · ·Rj,1(pzj/z1)κ
1−hj
2 Rj,n(zj/zn) · · ·Rj,j+1(zj/zj+1)Ψ, (1)
where Rij(z) signifies that R(z) acts on the i-th and j-th components, κ is a complex parameter,
κ
1−hj
2 acts on the j-th component as
κ
1−hj
2 vε = κεvε.
Let us briefly recall the construction of the hypergeometric solutions [22, 20] of the equation (1).
In the remaining part of the paper we assume |q| < 1. We set
(z)∞ = (z; p)∞, (z; p)∞ =
∞∏
j=0
(1− pjz), θ(z) = (z)∞(pz−1)∞(p)∞.
Free Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 3
Let n and l be non-negative integers satisfying l ≤ n. For a sequence (ε) = (ε1, . . . , εn) ∈
{0, 1}n satisfying ]{i|εi = 1} = l let
w(ε)(t, z) =
∏
a<b
ta − tb
q−2ta − tb
∑
1≤a1,...,al≤l
ai 6=aj(i6=j)
l∏
i=1
tai
tai − q−1zki
∏
j<ki
q−1tai − zj
tai − q−1zj
∏
i<j
q−2tai − taj
tai − taj
,
where {i|εi = 1} = {k1 < · · · < kl}.
The elliptic hypergeometric space Fell is the space of functions W (t, z) = W (t1, . . . , tl,
z1, . . . , zn) of the form
W = Y (z)Θ(t, z)
1∏n
j=1
∏l
a=1 θ(qta/zj)
∏
1≤a<b≤l
θ(ta/tb)
θ(q−2ta/tb)
satisfying the following conditions
(i) Y (z) is meromorphic on (C∗)n in z1,. . . ,zn, where C∗ = C\{0};
(ii) Θ(t, z) is holomorphic on (C∗)n+l in t1,. . . ,zn and symmetric in t1,. . . ,tl;
(iii) T t
aW/W = κqn−2l+4a−2, T z
j W/W = q−l, where T t
aW = W (t1, . . . , pta, . . . tl, z) and T z
j W =
W (t, z1, . . . , pzj , . . . zn).
Define the phase function Φ(t, z) by
Φ(t, z) =
n∏
i=1
l∏
a=1
(qta/zi)∞
(q−1ta/zi)∞
∏
a<b
(q−2ta/tb)∞
(q2ta/tb)∞
.
For W ∈ Fell let
I(w(ε),W ) =
∫
T̃l
l∏
a=1
dta
ta
Φ(t, z)w(ε)(t, z)W (t, z),
where T̃l is a suitable deformation of the torus
Tl = {(t1, . . . , tl)| |ti| = 1, 1 ≤ i ≤ l},
specified as follows [22].
Notice that the integrand has simple poles at
ta/zj = (psq−1)±1, s ≥ 0, 1 ≤ a ≤ l, 1 ≤ j ≤ n,
ta/tb = (psq2)±1, s ≥ 0, 1 ≤ a < b ≤ l.
The contour for the integration variable ta is a simple closed curve which rounds the origin in
the counterclockwise direction and separates the following two sets,
{psq−1zj , p
sq2tb | s ≥ 0, 1 ≤ j ≤ n, a < b },
{p−sqzj , p
−sq−2tb | s ≥ 0, 1 ≤ j ≤ n, a < b }.
Then
ΨW =
∑
(ε)
I(w(ε),W )vε1 ⊗ · · · ⊗ vεn ,
is a solution of the qKZ equation (1) for any W ∈ Fell.
4 K. Kuroki and A. Nakayashiki
3 Free field realizations
In this section we review the free field construction of the representation of the quantum affine
algebra Uq(ŝl2) of level k and intertwining operators. We mainly follow the notation in [10]. We
set
[x] =
qx − q−x
q − q−1
.
Let k be a complex number and {an, bn, cn, ã0, b̃0, c̃0, Qa, Qb, Qc|n ∈ Z\{0}} satisfy
[an, am] = δm+n,0
[2n][(k + 2)n]
n
, [ã0, Qa] = 2(k + 2),
[bn, bm] = −δm+n,0
[2n]2
n
, [b̃0, Qb] = −4,
[cn, cm] = δm+n,0
[2n]2
n
, [c̃0, Qc] = 4,
Other combinations of elements are supposed to commute. Set
N± = C[an, bn, cn| ± n > 0].
Then the Fock module Fr,s is defined to be the free N−-module of rank one generated by the
vector which satisfies
N+|r, s〉 = 0, ã0|r, s〉 = r|r, s〉, b̃0|r, s〉 = −2s|r, s〉, c̃0|r, s〉 = 2s|r, s〉.
We set
Fr = ⊕s∈ZFr,s.
A representation of the quantum affine algebra Uq(ŝl2) is constructed on Fr for any r ∈ C
in [16].
The right Fock module F †
r,s and F †
r are similarly defined using the vector 〈r, s| satisfying the
conditions
〈r, s|N− = 0, 〈r, s|ã0 = r〈r, s|, 〈r, s|b̃0 = −2s〈r, s|, 〈r, s|c̃0 = 2s〈r, s|.
Remark 1. We change the definition of |r, s〉 in [10]. Namely we use
|r, s〉 = exp
(
r
2(k + 2)
Qa + s
Qb + Qc
2
)
|0, 0〉.
Let us introduce field operators which are relevant to our purpose. For x = a, b, c let
x(L;M,N |z : α) = −
∑
n6=0
[Ln]xn
[Mn][Nn]
z−nq|n|α +
Lx̃0
MN
log z +
L
MN
Qx,
x(N |z : α) = x(L;L,N |z : α) = −
∑
n6=0
xn
[Nn]
z−nq|n|α +
x̃0
N
log z +
1
N
Qx.
The normal ordering is defined by specifying N+, ã0, b̃0, c̃0 as annihilation operators and N−,
Qa, Qb, Qc as creation operators. With this notation let us define the operators
J−(z) : Fr,s −→ Fr,s+1, φ(l)
m (z) : Fr,s −→ Fr+l,s+l−m, S(z) : Fr,s −→ Fr−2,s−1,
Free Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 5
by
J−(z) =
1
(q − q−1)z
(
J−+ (z)− J−− (z)
)
,
J−µ (z) =: exp
(
a(µ)
(
q−2z;−k + 2
2
)
+ b
(
2|q(µ−1)(k+2)z;−1
)
+ c
(
2|q(µ−1)(k+1)−1z; 0
))
:,
a(µ)
(
q−2z;−k + 2
2
)
= µ
{
(q − q−1)
∞∑
n=1
aµnz−µnq(2µ− k+2
2
)n + ã0 log q
}
,
S(z) =
−1
(q − q−1)z
(S+(z)− S−(z)) ,
Sε(z) =: exp
(
−a
(
k + 2|q−2z;−k + 2
2
)
− b
(
2|q−k−2z;−1
)
− c
(
2|q−k−2+εz; 0
))
:,
φ
(l)
l (z) =: exp
(
a
(
l; 2, k + 2|qkz;
k + 2
2
))
:,
φ
(l)
l−r(z) =
1
[r]!
∮ r∏
j=1
duj
2πi
[
· · ·
[[
φ
(l)
l (z), J−(u1)
]
ql
, J−(u2)
]
ql−2
, . . . , J−(ur)
]
ql−2r+2
,
where
[r]! = [r][r − 1] · · · [1], [X, Y ]q = XY − qY X,
and the integral in φ
(l)
l−r(z) signifies to take the coefficient of (u1 · · ·ur)−1.
The operator J−(z) is a generating function of a part of generators of the Drinfeld realization
Uq(ŝl2) at level k. While the operators φ
(l)
m (z) are conjectured to determine the intertwining
operator for Uq(ŝl2) modules [10, 15]
φ(l)(z) : Wr −→ Wr+l ⊗ V (l)
z , φ(l)(z) =
l∑
m=0
φ(l)
m (z)⊗ v(l)
m ,
where Wr is a certain submodule of Fr specified as a kernel of a certain operator, called q-
Wakimoto module [15, 12, 13, 11, 1], V (l) is the irreducible representation of Uq(sl2) with spin l/2
and V
(l)
z is the evaluation representation of Uq(ŝl2) on V (l).
In this paper we exclusively consider the case l = 1 and set
φ+(z) = φ
(1)
0 , φ−(z) = φ
(1)
1 , v0 = v
(1)
0 , v1 = v
(1)
1 .
The operator S(z) commutes with Uq(ŝl2) modulo total difference. Here modulo total
difference means modulo functions of the form
k+2∂zf(z) :=
f(qk+2z)− f(q−(k+2)z)
(q − q−1)z
.
Remark 2. The intertwining properties of φ(l)(z) for l ∈ Z are not proved in [10] as pointed out
in [15]. However the fact that the matrix elements of compositions of φ(l)(z)’s and S(t)’s satisfy
the qKZ equation modulo total difference can be proved in a similar way to Proposition 6.1
in [15] using the result of Konno [11] (see (4)).
Let
|m〉 = |m, 0〉 ∈ Fm,0, 〈m| = 〈m, 0| ∈ F †
m,0.
6 K. Kuroki and A. Nakayashiki
They become left and right highest weight vectors of Uq(ŝl2) with the weight mΛ1 + (k−m)Λ0
respectively, where Λ0, Λ1 are fundamental weights of ŝl2. Consider
F (t, z) = 〈m + n− 2l|φ(1)(z1) · · ·φ(1)(zn)S(t1) · · ·S(tl)|m〉 (2)
which is a function taking the value in V (1)⊗n. Let
∆j =
j(j + 2)
4(k + 2)
, s =
1
2(k + 2)
.
Set
F̂ =
(
n∏
i=1
z
∆m+n−2l+1−i−∆m+n−2l−i
i
)
F =
(
n∏
i=1
z
s(m+n−2l−i+ 3
2
)
i
)
F, (3)
Let the parameter p be defined from k by
p = q2(k+2).
We assume |p| < 1 as before. Then the function F̂ satisfies the following qKZ equation modulo
total difference of a function [15, 8, 7, 10, 11]
T z
j F̂ = R̂j,j−1(pzj/zj−1)· · ·R̂j,1(pzj/z1)q−(m+n/2−l+1)hj R̂j,n(zj/zn)· · ·R̂j,j+1(zj/zj+1)F̂ , (4)
where
R̂(z) = ρ(z)R̃(z), R̃(z) = C⊗2R(z)C⊗2,
ρ(z) = q1/2 (z−1; q4)∞(q4z−1; q4)∞
(q2z−1; q4)2∞
, (z;x)∞ =
∞∏
i=0
(1− xiz), Cvε = v1−ε.
4 Integral formulae
Define the components of F (t, z) by
F (t, z) =
∑
(ν)∈{0,1}n
F (ν)(t, z)v(ν), v(ν) = vν1 ⊗ · · · ⊗ vνn ,
where (ν) = (ν1, . . . , νn). By the weight condition F (ν)(t, z) = 0 unless the condition
]{i|νi = 0} = l
is satisfied. We assume this condition once for all. Notice that
φ+(z) =
1
(q − q−1)
∮
du
2πiu
[φ−(z), J−+ (u)− J−− (u)]q,
S(t) =
−1
(q − q−1)t
(S+(t)− S−(t)).
Let
{i|νi = 0} = {k1 < · · · < kl},
Free Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 7
and
F
(ν)
(ε)(µ)(t, z|u) = 〈m + n− 2l|φ−(z1) · · · [φ−(zk1), J
−
µ1
(u1)]q · · · [φ−(zkl
), J−µl
(ul)]q · · ·φ−(zn)
× Sε1(t1) · · ·Sεl
(tl)|m〉.
Then F (ν)(t, z) can be written as
F (ν)(t, z) = (−1)l(q − q−1)−2l
l∏
a=1
t−1
a
∑
εi,µj
l∏
i=1
(εiµi)
∫
Cl
l∏
j=1
duj
2πiuj
F
(ν)
(ε)(µ)(t, z|u),
where C l is a suitable deformation of the torus Tl specified as follows.
The contour for the integration variable ui is a simple closed curve rounding the origin in the
counterclockwise direction such that qk+3zj (1 ≤ j ≤ n), q−2uj (i < j), q−µi(k+2)ta (1 ≤ a ≤ l)
are inside and qk+1zj (1 ≤ j ≤ n), q2uj (j < i) are outside.
By the operator product expansions (OPE) of the products of φ−(z), J−µ (u), Sε(t) in the
appendix, one can compute the function F
(ν)
(ε)(µ)(t, z|u) explicitly. In order to write down the
formula we need some notation. Set
ξ(z) =
(pz−1; p, q4)∞(pq4z−1; p, q4)∞
(pq2z−1; p, q4)∞
, (z; p, q)∞ =
∞∏
i=0
∞∏
j=0
(1− piqjz).
Then
F
(ν)
(ε)(µ)(t, z|u) = f
(ν)
(µ)(t, z|u)Φ(t, z)G(ν)
(ε)(µ)(t, z|u),
where
f
(ν)
(µ)(t, z|u) = (1− q2)lq
∑l
i=1(n+m−2l−ki+i)µi
n∏
i=1
(
qkzi
)s(m+n−l−i)
∏
i<j
ξ(zi/zj)
×
l∏
a=1
(q−2ta)4s(a−1)−2ms,
G
(ν)
(ε)(µ)(t, z|u) = Ĝ
(ν)
(ε)(µ)(t, z|u)
∏
a<b
qεbtb − qεata
tb − q−2ta
,
Ĝ
(ν)
(ε)(µ)(t, z|u) =
l∏
i=1
ui(zki
− qµi−2−kui)
(zki
− q−1−kui)(ui − qk+3zki
)
l∏
j=1
∏
i<kj
zi − qµj−2−kuj
zi − q−1−kuj
×
l∏
j=1
∏
kj<i
uj − qk+2−µjzi
uj − qk+3zi
∏
i<j
ui − qµi−µjuj
ui − q−2uj
∏
i,a
ui − q−µi(k+1)−εata
ui − q−µi(k+2)ta
.
Let
G
(ν)
(µ)(t, z) =
∑
ε1,...,εn=±
n∏
j=1
εj
∫
Cl
l∏
j=1
duj
2πiuj
G
(ν)
(ε)(µ)(t, z).
The main theorem in this paper is
Theorem 1. If (µ) 6= (−l) = (−, . . . ,−), G
(ν)
(µ)(t, z) = 0. For (µ) = (−l) we have
G
(ν)
(−l)
(t, z) = q−2l+ 1
2
l(l−1)−
∑l
i=1 ki(q − q−1)lw(−ν)(t, z),
where (−ν) = (1− ν1, . . . , 1− νn).
8 K. Kuroki and A. Nakayashiki
It follows that F (ν)(t, z) is given by
F (ν)(t, z) = (−1)lq−(n+m+2−2l)l+ksn(m+n−l)− 1
2
ksn(n+1)+4sl(m−l+1)
×
n∏
i=1
z
s(m+n−l−i)
i
∏
i<j
ξ(zi/zj)
l∏
a=1
t2s(2a−2−m)−1
a Φ(t, z)w(−ν)(t, z).
5 Transformation to Tarasov–Varchenko’s formulae
We describe a transformation from F , which satisfies (4), to Ψ, which satisfies (1). The parame-
ter κ is also determined as a function of l, m, n.
For a solution Ĝ of (4) let
G̃ =
n∏
i=1
z
−s(m+n
2
−l+1)
i
(
n∏
i=1
zi)s/2
∏
i<j
ξ(zi/zj)
−1
C⊗nĜ. (5)
One can easily verify that G̃ satisfies (1) with κ = q2l−2−n−2m using
ξ(pz)
ξ(z)
=
(z−1; q4)∞(q4z−1; q4)∞
(q2z−1; q4)2∞
.
Let F̂ be defined by (3) and F̃ by (5). Then
F̃ = (−1)lq−(n+m+2−2l)l+ksn(m+n−l)− 1
2
ksn(n+1)+4sl(m−l+1)
×
n∏
i=1
z
s(m+n−2l−i)
i
l∏
a=1
t2s(2a−2−m)−1
a Φ(t, z)
∑
w(ν)(t, z)⊗ v(ν).
For W ∈ Fell let
W̃ = W
(
n∏
i=1
z
s(m+n−2l−i)
i
l∏
a=1
t2s(2a−2−m)
a
)−1
. (6)
Then the condition (iii) for W is equivalent to the following conditions,
T t
aW̃/W̃ = 1, T z
j W̃/W̃ = ql−m−n+j .
To sum up we have
Proposition 1. For any W ∈ Fell
Ψ̃W =
∫
T̃l
l∏
a=1
dta
2πi
F̃ (t, z)W̃ (t, z),
is a solution of the qKZ equation (1), where F̃ is defined by (5) with F̂ and F being given in (3)
and (2) and W̃ is defined by (6).
Free Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 9
6 Proof of Theorem
Let A± = {j|µj = ±}. Suppose that the number of elements in A± is r± and write A± = {l±1 <
· · · < l±
r±}. Set A = A−, r = r− and li = l−i for simplicity. Let
I
(ν)
(ε)(µ)(t, z) =
∫
Cl
l∏
j=1
duj
2πiuj
Ĝ
(ν)
(ε)(µ)(t, z|u).
Lemma 1. We have
I
(ν)
(ε)(µ)(t, z) = (−q−2)r(q − q−1)r
l∑
a1,...,ar=1
ai 6=aj(i6=j)
r∏
i=1
δεai ,+
r∏
i=1
tai
zkli
− qtai
r∏
j=1
∏
i<klj
zi − q−1taj
zi − qtaj
×
r∏
i=1
∏
a 6=ai,...,ar
tai − q−1−εata
tai − ta
.
Proof. We first integrate in the variables uj , j ∈ A+ in the order ul+1
, . . . , ul+
r+
. Let us consider
the integration in ul+1
. We denote the integration contour in ui by Ci. The only singularity of
the integrand outside Cl+1
is ∞. Thus the integral is calculated by taking residue at ∞. Since
the integrand is of the form
dul+1
ul+1
H(ul+1
),
where H(u) is holomorphic at ∞. Then∫
C
l+1
du
2πiu
H(u) = − Res
u=∞
du
u
H(u) = lim
u−→∞
H(u).
In this way the integral in ul+1
is calculated. After this integration the integrand as a function
of ul+2
has a similar structure. Therefore the integration with respect to ul+2
is carried out in
a similar way and so on. Finally we get
I
(ν)
(ε)(µ)(t, z) = (−1)r+
Res
u
l+
r+
=∞
· · · Res
u
l+1
=∞
Ĝ
(ν)
(ε)(µ)(t, z|u)
=
∫
Cn−r+
∏
j∈A
duj
2πiuj
∏
i∈A
−q−3−kui
zki
− q−1−kui
∏
i<kj
j∈A
zi − q−3−kuj
zi − q−1−kuj
∏
i<j
i,j∈A
ui − uj
ui − q−2uj
×
∏
i∈A
a
ui − qk+1−εata
ui − qk+2ta
.
Here Cn−r+
is specified by similar conditions to C l, where ul+i
1 ≤ i ≤ r+ are omitted. We
denote the right hand side of this equation other than
∫
Cn−r+
∏
j
duj
2πiuj
by I
(ν)+
(ε)(µ)(t, z).
Next we integrate with respect to the remaining variables uj , j ∈ A in the order ulr ,. . . ,ul1 .
Let us consider the integration with respect to ulr . The poles of the integrand inside Clr is
10 K. Kuroki and A. Nakayashiki
qk+2ta, a = 1, . . . , l. Thus we have∫
Clr
dulr
2πiulr
I
(ν)+
(ε)(µ)(t, z) =
l∑
ar=1
Res
ulr=qk+2tar
I
(ν)+
(ε)(µ)(t, z)
=
l∑
ar=1
∏
i∈A
i6=lr
−q−3−kui
zki
− q−1−kui
∏
i<kj
j∈A\{lr}
zi − q−3−kuj
zi − q−1−kuj
∏
i<j<lr
i,j∈A
ui − uj
ui − q−2uj
∏
i∈A\{lr}
a
ui − qk+1−εata
ui − qk+2ta
× −q−1tar
zklr
− qtar
∏
i<klr
zi − q−1tar
zi − qtar
∏
i<klr
i∈A
ui − qk+2tar
ui − qktar
∏
a 6=ar
tar − q−1−εata
tar − ta
(1− q−1−εa)
= (1− q−2)
l∑
ar=1
δεar ,+
∏
i∈A
i6=lr
−q−3−kui
zki
− q−1−kui
∏
i<kj
j∈A\{lr}
zi − q−3−kuj
zi − q−1−kuj
∏
i<j<lr
i,j∈A
ui − uj
ui − q−2uj
×
∏
i∈A\{lr}
a6=ar
ui − qk+1−εata
ui − qk+2ta
−q−1tar
zklr
− qtar
∏
i<klr
zi − q−1tar
zi − qtar
∏
a 6=ar
tar − q−1−εata
tar − ta
.
The last expression as a function of ulr−1 has a similar form to I
(ν)+
(ε)(µ)(t, z) as a function of ulr .
Namely the poles inside Clr−1 are qk+2ta, a 6= ar. Thus the integral in ulr−1 is the sum of residues
at qk+2ta, a 6= ar and so on. Finally we get
I
(ν)
(ε)(µ)(t, z) =
l∑
ar=1
l∑
ar−1=1
ar−1 6=ar
· · ·
l∑
a1
a1 6=a2,...,ar
Res
ul1
=qk+2ta1
· · · Res
ulr=qk+2tar
I
(ν)+
(ε)(µ)(t, z)
= (−q−2)r(q − q−1)r
l∑
a1,...,ar=1
ai 6=aj(i6=j)
r∏
i=1
δεai ,+
r∏
i=1
tai
zkli
− qtai
r∏
j=1
∏
i<klj
zi − q−1taj
zi − qtaj
×
r∏
i=1
∏
a 6=ai,...,ar
tai − q−1−εata
tai − ta
.
Thus the lemma is proved. �
Recall that
G
(ν)
(ε)(µ)(t, z) = I
(ν)
(ε)(µ)(t, z)
∏
a<b
qεbtb − qεata
tb − q−2ta
.
For a set {a1, . . . , ar} let {b1, . . . , br+} be defined by
{1, . . . , l} = {ai} ∪ {bi}.
Then
(−q2)r(q − q−1)−rG
(ν)
(µ)(t, z)
=
l∑
a1,...,ar=1
ai 6=aj(i6=j)
r∏
i=1
tai
zkli
− qtai
r∏
i=1
∏
j<kli
zj − q−1tai
zj − qtai
∏
i>j
tai − q−2taj
tai − taj
r∏
i=1
r+∏
j=1
1
tai − tbj
∏
a<b
1
tb − q−2ta
×
r∏
i=1
δεai ,+
∑
εb1
,...,εb
r+
=±
r+∏
j=1
εbj
r∏
i=1
r+∏
j=1
(tai − q
−1−εbj tbj
)
∏
a<b
(qεbtb − qεata).
Free Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 11
Let us calculate the sum in {εbi
} assuming εai = +. Using
(ta − q−1−εbtb)(ta − q−1+εbtb) = (ta − tb)(ta − q−2tb),
and ∏
a<b
(qεbtb − qεata) =
∏
i,j
ai>aj
q(tai − taj )
∏
i,j
bi>bj
(qεbi tbi
− q
εbj tbj
)
×
∏
i,j
ai>bj
(qtai − q
εbj tbj
)
∏
i,j
bj>ai
(qεbj tbj
− qtai),
we have
∑
εb1
,...,εb
r+
=±
r+∏
j=1
εbj
r∏
i=1
r+∏
j=1
(tai − q
−1−εbj tbj
)
∏
a<b
(qεbtb − qεata)
= (−1)lr− 1
2
r(r−1)−
∑r
i=1 aiqrr++ 1
2
r(r−1)
r∏
i=1
r+∏
j=1
(tai − tbj
)(tai − q−2tbj
)
∏
i,j
ai>aj
(tai − taj )
×
∑
εb1
,...,εb
r+
=±
r+∏
j=1
εbj
∏
i,j
bi>bj
(qεbi tbi
− q
εbj tbj
). (7)
Lemma 2. For N ≥ 1 we have
∑
ε1,...,εN=±
N∏
j=1
εj
∏
i>j
(qεiti − qεj tj) = 0. (8)
Proof. Let
ai(ε) = t(1, qεti, (qεti)2, . . . , (qεti)N−1).
Then the left hand side of (8) is equal to
∑
ε1,...,εN=±
N∏
j=1
εj det (a1(ε1), . . . ,aN (εN )) = det
(∑
ε1
ε1a1(ε1), . . . ,
∑
εN
εNaN (εN )
)
. (9)
Since∑
ε
εai(ε) = t
(
0, (q − q−1)ti, . . . , (qN−1 − q−(N−1))tN−1
i
)
,
the right hand side of (9) is zero. �
By this lemma the right hand side of (7) becomes zero if r+ > 0. Consequently G
(ν)
(µ) = 0 for
r+ > 0. Suppose that r+ = 0. In this case r = l, li = i (1 ≤ i ≤ l) and
(−q2)l(q − q−1)−lG
(ν)
(−l)
(t, z)
=
∏
a<b
q(tb − ta)
tb − q−2ta
l∑
a1,...,al=1
ai 6=aj(i6=j)
l∏
i=1
tai
zki
− qtai
∏
j<ki
zj − q−1tai
zj − qtai
∏
i>j
tai − q−2taj
tai − taj
.
The theorem easily follows from this.
12 K. Kuroki and A. Nakayashiki
7 Concluding remarks
In this paper we study the solutions of the qKZ equation taking the value in the tensor product
of the two dimensional evaluation representation of Uq(ŝl2). The integral formulae are derived
for the highest to highest matrix elements for certain intertwining operators by using free field
realizations. The integrals with respect to u variables corresponding to the operator J−(u) are
calculated and the sum arising from the expression of J−(u) and the screening operator S(t)
is calculated. The formulae thus obtained coincide with those of Tarasov and Varchenko. The
calculations in this paper can be extended to the case where the vector space V (1)⊗n is replaced
by a tensor product of more general representations. It is an interesting problem to perform
similar calculations for other quantum affine algebras [2] and the elliptic algebras [13].
In Tarasov–Varchenko’s theory solutions of a qKZ equation are parametrized by elements of
the elliptic hypergeometric space Fell while the matrix elements are specified by intertwiners.
It is an interesting problem to establish a correspondence between intertwining operators and
elements of Fell. With the results of the present paper one can begin to study this problem.
Study in this direction will provide a new insight on the space of local fields and correlation
functions of integrable field theories and solvable lattice models. The corresponding problem in
CFT is studied in [5].
Appendix. List of OPE’s
Here we list OPE’s which are necessary in this paper. Almost all of them are taken from the
paper [10]. Let
C(z) =
(q−2z; p)∞
(q2z; p)∞
, (z)∞ = (z; p)∞.
Sε1(t1)Sε2(t2) = (q−2t1)4sqε1 t1 − qε2−ε1t2
t1 − q−2t2
C(t2/t1) : Sε1(t1)Sε2(t2) :, |q−2t2| < |t1|,
φ−(z)Sε(t) = (qkz)−s (qt/z)∞
(q−1t/z)∞
: φ−(z)Sε(t) :, |q−1t| < |z|,
J−µ (u)Sε(t) = q−µ u− q−µ(k+1)−εt
u− q−µ(k+2)t
: J−µ (u)Sε(t) :, |q−(k+2)t| < |u|,
φ−(z)J−µ (u) =
z − qµ−2−ku
z − q−1−ku
: φ−(z)J−µ (u) :, |u| < |qk+3z| for µ = −,
J−µ (u)φ−(z) = qµ u− qk+2−µz
u− qk+3z
: φ−(z)J−µ (u) :, |qk+1z| < |u| for µ = +,
[φ−(z), J−µ (u)]q =
(1− q2)u(z − qµ−2−ku)
(z − q−1−ku)(u− qk+3z)
: φ−(z)J−µ (u) :,
J−µ1
(u1)J−µ2
(u2) = q−µ1
u1 − qµ1−µ2u2
u1 − q−2u2
: J−µ1
(u1)J−µ2
(u2) :, |q−2u2| < |u1|,
φ−(z1)φ−(z2) = (qkz1)sξ(z1/z2) : φ−(z1)φ−(z2) :, |pz2| < |z1|.
Note 1. After completing the paper we were informed that H. Awata, S. Odake and J. Shiraishi
obtained a similar result to this paper. The result is reviewed in Shiraishi’s PhD thesis [17] in
which one statement is a conjecture. However the complete version containing all proofs for all
statements had not been published after all. We would like to thank J. Shiraishi for the kind
correspondence.
Free Field Approach to Solutions of the Quantum Knizhnik–Zamolodchikov Equations 13
Acknowledgements
We would like to thank Hitoshi Konno and Yasuhiko Yamada for valuable discussions and
comments. We are also grateful to Atsushi Matsuo for useful comments on the manuscript.
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http://arxiv.org/abs/hep-th/9209009
http://arxiv.org/abs/hep-th/9305146
http://arxiv.org/abs/hep-th/9305127
http://arxiv.org/abs/hep-th/9208066
http://arxiv.org/abs/hep-th/9209015
http://arxiv.org/abs/hep-th/9310108
http://arxiv.org/abs/hep-th/9407122
http://arxiv.org/abs/q-alg/9709013
http://arxiv.org/abs/hep-th/9212040
http://arxiv.org/abs/hep-th/9311040
1 Introduction
2 Tarasov-Varchenko's formula
3 Free field realizations
4 Integral formulae
5 Transformation to Tarasov-Varchenko's formulae
6 Proof of Theorem
7 Concluding remarks
Appendix. List of OPE's
References
|