q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations
Matrix elements of intertwining operators between q-Wakimoto modules associated to the tensor product of representations of Uq(^sl2) with arbitrary spins are studied. It is shown that they coincide with the Tarasov-Varchenko's formulae of the solutions of the qKZ equations. The result generaliz...
Збережено в:
| Дата: | 2009 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2009
|
| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/149178 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations / K. Kuroki // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 13 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-149178 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1491782019-02-20T01:27:17Z q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations Kuroki, K. Matrix elements of intertwining operators between q-Wakimoto modules associated to the tensor product of representations of Uq(^sl2) with arbitrary spins are studied. It is shown that they coincide with the Tarasov-Varchenko's formulae of the solutions of the qKZ equations. The result generalizes that of the previous paper [Kuroki K., Nakayashiki A., SIGMA 4 (2008), 049, 13 pages]. 2009 Article q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations / K. Kuroki // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 13 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81R50; 20G42; 17B69 http://dspace.nbuv.gov.ua/handle/123456789/149178 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Matrix elements of intertwining operators between q-Wakimoto modules associated to the tensor product of representations of Uq(^sl2) with arbitrary spins are studied. It is shown that they coincide with the Tarasov-Varchenko's formulae of the solutions of the qKZ equations. The result generalizes that of the previous paper [Kuroki K., Nakayashiki A., SIGMA 4 (2008), 049, 13 pages]. |
| format |
Article |
| author |
Kuroki, K. |
| spellingShingle |
Kuroki, K. q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Kuroki, K. |
| author_sort |
Kuroki, K. |
| title |
q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
| title_short |
q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
| title_full |
q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
| title_fullStr |
q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
| title_full_unstemmed |
q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations |
| title_sort |
q-wakimoto modules and integral formulae of solutions of the quantum knizhnik-zamolodchikov equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/149178 |
| citation_txt |
q-Wakimoto Modules and Integral Formulae of Solutions of the Quantum Knizhnik-Zamolodchikov Equations / K. Kuroki // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 13 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT kurokik qwakimotomodulesandintegralformulaeofsolutionsofthequantumknizhnikzamolodchikovequations |
| first_indexed |
2025-07-12T21:35:16Z |
| last_indexed |
2025-07-12T21:35:16Z |
| _version_ |
1837478574485929984 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 027, 21 pages
q-Wakimoto Modules
and Integral Formulae of Solutions
of the Quantum Knizhnik–Zamolodchikov Equations
Kazunori KUROKI
Department of Mathematics, Kyushu University, Hakozaki 6-10-1, Fukuoka 812-8581, Japan
E-mail: ma306012@math.kyushu-u.ac.jp
Received October 31, 2008, in final form February 25, 2009; Published online March 07, 2009
doi:10.3842/SIGMA.2009.027
Abstract. Matrix elements of intertwining operators between q-Wakimoto modules associa-
ted to the tensor product of representations of Uq(ŝl2) with arbitrary spins are studied. It is
shown that they coincide with the Tarasov–Varchenko’s formulae of the solutions of the qKZ
equations. The result generalizes that of the previous paper [Kuroki K., Nakayashiki A.,
SIGMA 4 (2008), 049, 13 pages].
Key words: free field; vertex operator; qKZ equation; q-Wakimoto module
2000 Mathematics Subject Classification: 81R50; 20G42; 17B69
1 Introduction
In [8] the integral formulae of the quantum Knizhnik–Zamolodchikov (qKZ) equations [3] for
the tensor product of spin 1/2 representation of Uq(sl2) arising from q-Wakimoto modules have
been studied. The formulae are identified with those of Tarasov–Varchenko’s formulae. The aim
of this paper is to generalize the results to the case of tensor product of representations with
arbitrary spins.
It is known that certain matrix elements of intertwining operators between q-Wakimoto modu-
les satisfy the qKZ equation [3, 10]. Thus it is interesting to compute those matrix elements
explicitly. In [5] two kinds of intertwining operators were introduced, type I and type II. They
were defined according as the position of evaluation representations. In the application to the
study of solvable lattice models two types of operators have their own roles. Type I and type II
operators correspond to states and particles respectively. The properties of traces exhibit very
different structure. However as far as the matrix elements are concerned they are not expected
to be very different [5].
In [8] a computation of matrix elements has been carried out in the case of type I opera-
tor and the tensor product of 2-dimensional vector representation of Uq(sl2) generalizing the
result of [10] (see the previous paper [8]). In this paper we compute matrix elements for the
composition of the type I intertwining operators [5] associated to finite dimensional irreducible
representations of Uq(sl2). We perform certain multidimensional integrals and sums explicitly.
It is shown that the formulae thus obtained coincide with those of Matsuo [9], Tarasov and
Varchenko [13] without the term corresponding to the deformed cycles.
To obtain actual matrix elements of intertwining operators it is necessary to specify certain
contours of integration associated to screening operators. We do not consider this problem in
this paper. To find integration contours describing each composition of intertwining operators is
an important open problem. We also remark that the formulae for type II intertwining operators
are not obtained in this paper. The computation of them looks quite different from that for
mailto:ma306012@math.kyushu-u.ac.jp
http://dx.doi.org/10.3842/SIGMA.2009.027
http://dx.doi.org/10.3842/SIGMA.2008.049
2 K. Kuroki
type I case as opposed to the expectation. It is interesting to find the way to get a similar result
for matrix elements in the case of type II operators.
The paper is organized in the following manner. The construction of the solutions of the
qKZ equations due to Tarasov and Varchenko is reviewed in Section 2. In Section 3 a free field
construction of intertwining operator is reviewed. The formulae for the matrix elements of some
operators are calculated in Section 4. The main theorem of this paper is stated in this section.
In Section 5 the proof of the main theorem is given. The evaluation representation of Uq(ŝl2) is
explicitly described in Appendix A. Appendix B gives the explicit form of the R-matrix in special
cases. The explicit forms of the operators which appear in Section 3 are given in Appendix C.
Appendix D contains the list of OPE’s which is necessary to derive the integral formulae.
2 Tarasov–Varchenko’s formulae
We review Tarasov–Varchenko’s formula for solutions of the qKZ equations. In this paper
we assume that q is a complex number such that |q| < 1. We mainly follow the notation
of [13]. For a nonnegative integer l let V (l) =
⊕l
i=0 Cv
(l)
i be the l + 1 dimensional irreducible
Uq(sl2)-module and V
(l)
z = V (l)⊗C[z, z−1] the evaluation representation of Uq(ŝl2) on V (l). The
action of Uq(ŝl2) on V
(l)
z is given in Appendix A. Let l1 and l2 be nonnegative integers and
Rl1,l2(z) ∈ End(V (l1) ⊗ V (l2)) the trigonometric quantum R-matrix uniquely determined by the
following conditions:
(i) PRl1,l2(z) commutes with Uq(ŝl2),
(ii) PRl1,l2(z)
(
v
(l1)
0 ⊗ v
(l2)
0
)
= v
(l2)
0 ⊗ v
(l1)
0 ,
where P : V (l1) ⊗ V (l2) → V (l2) ⊗ V (l1) is a linear map given by
P (v ⊗ w) = w ⊗ v.
The explicit form of the R-matrix is given in Appendix B in case l1 = 1 or l2 = 1. We set
R̂li,lj (z) = ρli,lj (z)R̃li,lj (z), R̃li,lj (z) = (Cli ⊗ Clj )Rli,lj (z)(Cli ⊗ Clj ),
ρli,lj (z) = q
lilj
2
(qli+lj+2z−1; q4)∞(q−li−lj+2z−1; q4)∞
(q−li+lj+2z−1; q4)∞(qli−lj+2z−1; q4)∞
,
Clv
(l)
ε = v
(l)
l−ε
(
v(l)
ε ∈ V (l)
)
,
where for a complex number a with |a| < 1
(z; a)∞ =
∞∏
i=0
(
1− aiz
)
.
Let k be a complex number. We set
p = q2(k+2).
We assume that p satisfies |p| < 1. Let Tj denote the p-shift operator of zj ,
Tjf(z1, . . . , zn) = f(z1, . . . , pzj , . . . zn).
Let l1, . . . , ln and N be nonnegative integers. The qKZ equation for a Vl1 ⊗ · · · ⊗ Vln-valued
function Ψ(z1, . . . , zn) is
TjΨ = R̂j,j−1(pzj/zj−1) · · · R̂j,1(pzj/z1)κ
hj
2 R̂j,n(zj/zn) · · · R̂j,j+1(zj/zj+1)Ψ, (1)
q-Wakimoto Modules and Integral Formulae 3
where κ is a complex parameter, R̂i,j(z) signifies that R̂li,lj (z) acts on the i-th and j-th compo-
nents of the tensor product and κhj acts on j-th component as
κ
hj
2 v
(lj)
m = κ
lj−2m
2 v
(lj)
m .
We set
(z)∞ = (z; p), θ(z) = (z)∞
(
pz−1
)
∞(p)∞.
Consider a sequence (ν) = (ν1, . . . , νn) satisfying 0 ≤ νi ≤ li for all i and N =
n∑
i=1
νi. Let
r = ]{i | νi 6= 0}, {i | νi 6= 0} = {k(1) < · · · < k(r)} and ni = νk(i). We set
w(ν)(t, z) =
∏
a<b
ta − tb
q−2ta − tb
∑
Γ1t···tΓr={1,...,N}
|Γs|=ns(s=1,...,r)
∏
1≤i<j≤r
a∈Γi,b∈Γj
q−2ta − tb
ta − tb
×
∏
b∈Γs
(
tb
tb − q−lk(i)zk(i)
∏
j<k(i)
q−lj tb − zj
tb − q−ljzj
)
.
The elliptic hypergeometric space Fell is the space of functions W (t, z)=W (t1, . . . , tN , z1, . . . , zn)
of the form
W = Y (z)Θ(t, z)
1
n∏
j=1
N∏
a=1
θ(qlj ta/zj)
∏
1≤a<b≤N
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+N in t1, . . . , zn and symmetric in t1, . . . , tN ;
(iii) T t
aW/W = κq−2N+4a−2
n∏
i=1
qli , T z
j W/W = q−ljN , where T t
aW = W (t1, . . . , pta, . . . , tN , z)
and T z
j W = W (t, z1, . . . , pzj , . . . , zn).
Define the phase function Φ(t, z) by
Φ(t, z) =
(
N∏
a=1
n∏
i=1
(qlita/zi)∞
(q−lita/zi)∞
)(∏
a<b
(q−2ta/tb)∞
(q2ta/tb)∞
)
.
For W ∈ Fell let
I(w(ε),W ) =
∫
T̃N
N∏
a=1
dta
ta
Φ(t, z)w(ε)(t, z)W (t, z), (2)
where T̃N is a suitable deformation of the torus
TN = {(t1, . . . , tN ) | |ti| = 1, 1 ≤ i ≤ N},
specified as follows. The integrand has simple poles at
ta/zj =
(
psq−lj
)±1
, s ≥ 0, 1 ≤ a ≤ N, 1 ≤ j ≤ n,
4 K. Kuroki
ta/tb =
(
psq2
)±1
, s ≥ 0, 1 ≤ a < b ≤ N.
The contour of integration in ta is a simple closed curve which rounds the origin in the counter-
clockwise direction and separates the following two sets{
psq−ljzj , p
sq2tb|s ≥ 0, 1 ≤ j ≤ N, a < b
}
,{
p−sqljzj , p
−sq−2tb|s ≥ 0, 1 ≤ j ≤ N, a < b
}
.
Let L be a complex number and
κ = q
−2
(
L+
n∑
i=1
li
2
−N+1
)
.
Then
ΨW =
(
n∏
i=1
zai
i
)(∏
i<j
ξli,lj (zi/zj)
)∑
(ε)
I(w(−ε),W )v(l1)
ε1 ⊗ · · · ⊗ v(ln)
εn
(3)
is a solution of the qKZ equation (1) for any W ∈ Fell where (−ε) = (l1 − ε1, . . . , ln − εn) and
ai =
li
2(k + 2)
(
L +
n∑
j=1
lj −
li
2
−N + 1
)
,
ξli,lj (z) =
(
pqli+lj+2z−1; q4, p
)
∞
(
pq−li−lj+2z−1; q4, p
)
∞(
pqli−lj+2z−1; q4, p
)
∞
(
pq−li+lj+2z−1; q4, p
)
∞
,
(z; p, q) =
∞∏
i=0
∞∏
j=0
(1− piqjz).
3 Free field realizations
We briefly review the free field construction of the representation of the Uq(ŝl2) of level k
[1, 10, 11] and intertwining operators [2, 6, 7]. We mainly follow the notation of [6]. We set
[n] =
qn − q−n
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
[(k + 2)n][2n]
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].
Let r be a complex number and s an integer. The Fock module Fr,s is defined to be the free N−
module of rank one generated by the vector |r, s〉 satisfying
N+|r, s〉 = 0, ã0|r, s〉 = r|r, s〉, b̃0|r, s〉 = −2s|r, s〉, c̃0|r, s〉 = −2s|r, s〉.
q-Wakimoto Modules and Integral Formulae 5
We set
Fr = ⊕s∈ZFr,s.
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|.
Notice that Fr and F †
r have left and right Uq(ŝl2)-module structure respectively [10, 11].
Let
|L〉 = |L, 0〉 ∈ FL,0, 〈L| = 〈L, 0| ∈ F †
L,0.
They become left and right highest weight vectors of Uq(ŝl2) with the weight LΛ1 + (k − L)Λ0
respectively, where Λ0 and Λ1 are fundamental weights of ŝl2.
We consider operators
φ(l)
m (z) : Fr,s → Fr+l,s+l−m, J−(u) : Fr,s → Fr,s+1, S(t) : Fr,s → Fr−2,s−1,
the explicit forms of which are given in Appendix C. We set
φ
(l)
l (z) = φl(z)
for simplicity. The operator φ
(l)
m (z) is used to construct the vertex operator for Uq(ŝl2):
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 called q-Wakimoto module [10].
The operator J−(u) is a generating function of a part of generators of the Drinfeld realization
for Uq(ŝl2) at level k.
The operator S(t) commutes with Uq(ŝl2) modulo total differences. Here modulo total
differences means modulo functions of the form
k+2∂zf(z) =
f(qk+2z)− f(q−(k+2)z)
(q − q−1)z
.
Consider
F (t, z) = 〈L +
n∑
i=1
li − 2N |φ(l1)(z1) · · ·φ(ln)(zn)S(tN ) · · ·S(t1)|L〉
which is a function taking the value in V (l1) ⊗ · · · ⊗ V (ln). Let
4j =
j(j + 2)
4(k + 2)
.
Set
F̂ =
n∏
i=1
z
4
L+
n∑
j=i
lj−2N
−4
L+
n∑
j=i+1
lj−2N
i
F =
n∏
i=1
z
li
2(k+2)
(
L+
∑
i<j
lj−2N+
li+2
2
)
i
F.
Then the function F̂ (t, z) satisfies qKZ equation (1) with κ = q−2
(
L+
n∑
i=1
li
2
−N+1
)
modulo
total differences [10].
6 K. Kuroki
4 Integral formulae
Define the components of F (t, z) by
F (t, z) =
∑
νi∈{0,...,li}
1≤i≤n
F (ν)(t, z)v(l1)
ν1
⊗ · · · ⊗ v(ln)
νn
,
where (ν) = (ν1, . . . , νn). By the conditions on weights F (ν)(t, z) = 0 unless
n∑
i=1
(li − νi) = N
is satisfied. We assume this condition once for all. Let
]{i | νi 6= li} = r, {i | νi 6= li} = {k(1) < · · · < k(r)},
ni = lk(i) − νk(i) (1 ≤ i ≤ r).
The main result of this paper is
Theorem 1. We have
F (ν)(t, z) = A(ν)(t, z)
n∏
i=1
z
li
2(k+2)
(
L−2N+
∑
i<j
lj
)
i
(∏
i<j
ξli,lj (zi/zj)
)
Φ(t, z)w(−ν)(t, z),
where (−ν) = (l1 − ν1, . . . , ln − νn), ni = lk(i) − νk(i) and
A(ν)(t, z) = q−NLq
3N(N−1)
2
−
( n∑
i=1
li
)
N
q
1
2(k+2)
(
k
∑
i<j
lilj+k(L−2N)
n∑
i=1
li+4LN−4N(N−1)
)
×
(
1
q − q−1
)N ∑
(ν)
r∏
s=1
q
( r∑
t=s+1
nt
)
ns−lk(s)ns
{
r∏
s=1
ns−1∏
i=0
(
1− q2(lk(s)−i)
)}
×
(
N∏
a=1
t
2
k+2
(a−1)− 1
k+2
L−1
a
)
.
The formula for F (ν)(t, z) is of the form of (2), (3). More precisely in Tarasov–Varchenko’s
formula (2), (3), W can be written as
W =
n∏
i=1
z
li
2(k+2)
(
L−3N−
∑
j<i
lj+
∑
i<j lj
)
i
( N∏
a=1
ta
)
A(ν)(t, z)W ′
for suitable W ′. This W ′ specifies an intertwiner. In this paper we don’t consider the problem
on specifying W ′.
To prove Theorem 1 let us begin by writing down the formula obtained by the free field
description of operators φl(z), J−(u), S(t) given in Appendix C. Let (ε) = (ε1, . . . , εN ), (µ) =
(µ1,1, . . . , µ1,n1 , . . . , µr,nr) ∈ {0, 1}N . Then F (ν)(t, z) can be written as
F (ν)(t, z) = (−1)N
(
q − q−1
)−2N
r∏
i=1
1
[ni]!
N∏
a=1
t−1
a
q-Wakimoto Modules and Integral Formulae 7
×
∑
εi,µi1,i2
=±
N∏
i=1
εi
∮ ∏
1≤i1≤r
1≤i2≤ni1
µi1,i2
dui1,i2
2πiui1,i2
F
(ν)
(ε)(µ)(t, z|u),
where
F
(ν)
(ε)(µ)(t, z|u) =
〈
L +
n∑
i=1
li − 2N |φl1(z1) · · ·φlk(1)−1
(zk(1)−1)
× [. . . [φlk(1)
(zk(1)), J
−
µ1,1
(u1,1)]qlk(1) , J
−
µ1,2
(u1,2)]qlk(1)−2 . . . , J−µ1,n1
(u1,n1)]qlk(1)−2(n1−1) . . .
× [. . . [φlk(r)
(zk(r)), J
−
µr,1
(ur,1)]qlk(r) , J
−
µr,2
(ur,2)]qlk(r)−2 . . . , J−µr,nr
(ur,nr)]qlk(r)−2(nr−1)
× φlk(r)+1
(zk(r)+1) . . . φln(zn)SεN (tN ) . . . Sε1(t1)|L
〉
.
and the integrand in the right hand side signifies to take the coefficient of
( ∏
1≤i≤r
1≤j≤ni
ui,j
)−1
. For
the notation [x, y]q see Appendix C.
Let (m) = (m1, . . . ,mr), 0 ≤ mi ≤ ni. Then
∮ ∏
1≤i1≤r
1≤i2≤ni1
µi1,i2
dui1,i2
2πiui1,i2
F
(ν)
(ε)(µ)(t, z)
=
∑
0≤mi≤ni
1≤i≤r
(−1)
r∑
i=1
mi
(
r∏
i=1
qmilk(i)q−mi(ni−1)
[
ni
mi
])
×
∫
CN
∏
1≤i1≤r
1≤i2≤ni1
µi1,i2
dui1,i2
2πiui1,i2
F
(ν)
(ε)(µ)(m)(t, z|u),
where
F
(ν)
(ε)(µ)(m)(t, z|u) =
〈
L +
n∑
i=1
li − 2N |φl1(z1) · · ·φlk(1)−1
(zk(1)−1)
×
(
J−µ1,1
(u1,1) · · · J−µ1,m1
(u1,m1)φlk(1)
(zk(1))J
−
µ1,m1+1
(u1,m1+1) · · · J−µ1,n1
(u1,n1)
)
· · ·
×
(
J−µr,1
(ur,1) · · · J−µr,mr
(ur,mr)φlk(r)
(zk(r))J
−
µr,mr+1
(ur,mr+1) · · · J−µr,nr
(ur,nr)
)
× φlk(r)+1
(zk(r)+1) · · ·φln(zn)SεN (tN ) · · ·Sε1(t1)|L
〉
,
and CN is a suitable deformation of the torus TN specified as follows. We introduce the lexico-
graphical order
(i1, i2) < (j1, j2) ⇔ i1 < j1 or i1 = j1 and i2 < j2.
For a given (m) = (m1, . . . ,mr), 1 ≤ mi ≤ ni, we define
j < (i1, i2) ⇔ j < k(i1) or j = k(i1) and mi1 < i2,
j > (i1, i2) ⇔ j > k(i1) or j = k(i1) and mi1 ≥ i2.
8 K. Kuroki
The contour for the integration variable ui1,i2 is a simple closed curve rounding the origin in
the counterclockwise direction such that qlj+k+2zj ((i1, i2) < j), q−2uj1,j2 ((i1, i2) < (j1, j2)),
q−µi1,i2
(k+2)ta (1 ≤ a ≤ N) are inside, and q−lj+k+2zj ((i1, i2) > j), q2uj1,j2 ((j1, j2) < (i1, i2))
are outside. We denote it C(i1,i2).
Then
F
(ν)
(ε)(µ)(m)(t, z|u) = f (ν)(t, z)Φ(t, z)G(ν)
(ε)(µ)(m)(t, z|u),
where
f (ν)(t, z) =
∏
i<j
(qkzi)
lilj
2(k+2) ξli,lj (zi/zj)
{
n∏
i=1
(qkzi)
− Nli
k+2
}
×
{
n∏
i=1
(qkzi)
Lli
2(k+2)
}{
N∏
i=1
(q−2ti)
− L
k+2
}{∏
a<b
(q−2tb)
2
k+2
}
,
G
(ν)
(ε)(µ)(m)(t, z|u) = Ĝ
(ν)
(ε)(µ)(m)(t, z|u)
(∏
a<b
qεbtb − qεata
tb − q−2ta
)
,
Ĝ
(ν)
(ε)(µ)(m)(t, z|u) =
∏
(i1,i2)
qLµi1,i2
∏
(i1,i2)>j
zj − qµi1,i2
lj−k−2ui1,i2
zj − qlj−k−2ui1,i2
×
∏
(i1,i2)<j
qµi1,i2
lj
ui1,i2 − q−µi1,i2
lj+k+2zj
ui1,i2 − qlj+k+2zj
×
∏
(i1,i2)
1≤b≤N
q−µi1,i2
ui1,i2 − q−µi1,i2
(k+1)−εbtb
ui1,i2 − q−µi1,i2
(k+2)tb
×
∏
(i1,i2)<(j1,j2)
q−µi1,i2ui1,i2 − q−µj1,j2uj1,j2
ui1,i2 − q−2uj1,j2
.
For i, let A±
µ,i = {(i, j)|µi,j = ±}. The number of elements in A±
µ,i is a±i and A±
µ,i =
{`±i,1, . . . , `
±
i,a±i
} . We set a−i = ai, A−
µ,i = Aµ,i, Aµ = ∪r
i=1Aµ,i and
Ĵ
(ν)
(ε)(µ) =
∑
0≤mi≤ni
1≤i≤r
(−1)
r∑
i1
mi
{
r∏
i=1
qmilk(i)q−mi(ni−1)
[
ni
mi
]}
×
∫
CN
∏
(i1,i2)
µi1,i2
dui1,i2
2πiui1,i2
Ĝ
(ν)
(ε)(µ)(m).
See the beginning of the next section for the notation of the q-binomial coefficient
[
ni
mi
]
.
For a given (a) = (a1, . . . , ar), 1 ≤ ai ≤ ni, we define Ĵ
(ν)
(ε)(a) and J
(ν)
(a) as follows
Ĵ
(ν)
(ε)(a) =
∑
|Aµ,i|=ai
1≤i≤r
Ĵ
(ν)
(ε)(µ),
q-Wakimoto Modules and Integral Formulae 9
J
(ν)
(a) =
∑
ε1,...,εN=±
N∏
j=1
εj
∏
1≤a<b≤N
qεbtb − qεata
tb − q−2ta
Ĵ
(ν)
(ε)(a).
Using J
(ν)
(a) , F (ν)(t, z) can be written as
F (ν)(t, z) = (−1)N
(
q − q−1
)−2N
(
r∏
i=1
1
[ni]!
)(
N∏
b=1
t−1
b
)
f (ν)(t, z)Φ(t, z)
∑
(a)
J
(ν)
(a) .
Theorem 1 straightforwardly follows from the following proposition.
Proposition 1. If (a) 6= (n1, n2, . . . , nr), J
(ν)
(a) (t, z) = 0. For (a) = (n1, n2, . . . , nr) we have
J
(ν)
(n1,...,nr)(t, z) = (−1)N
(
1− q−2
)N
q
N(N−L)+
N(N−1)
2
−
( n∑
i=1
li
)
N
×
r∏
s=1
q
( r∑
t=s+1
nt
)
ns−lk(s)ns
[ns]!
ns−1∏
i=0
(1− q2(lk(s)−i))
w(−ν)(t, z).
This proposition is proved by performing integrals in the variables ui,j in the next section.
5 Proof of Proposition 1
We set
[n]! =
n∏
i=1
[i],
[
n
m
]
=
[n]!
[n−m]![m]!
,
for nonnegative integers n, m (n ≥ m). To prove Proposition 1, we have to calculate Ĵ
(ν)
(ε)(a). We
need the following lemmas.
Lemma 1. For n ≥ 1 and n ≥ m ≥ 0, we have
(i)
∑
AtB={1,2,...,n}
|A|=m
∏
i<j
i∈A,j∈B
q2
= qm(n−m)
[
n
m
]
;
(ii)
∑
AtB={1,2,...,n}
|A|=m
µi=1(i∈A), µi=−1(i∈B)
∏
i<j
qµi
= q−
n(n−1)
2
+m(n−1)
[
n
m
]
.
Proof. By the q-binomial theorem
n∏
i=1
(
1 + q−n−1+2ix
)
=
n∑
i=0
[
n
i
]
xi,
we have the equation
∑
1≤i1<···<im≤n
q
2
m∑
j=1
ij
= q(n+1)m
[
n
m
]
.
The assertions (i) and (ii) easily follow from this equation. �
10 K. Kuroki
Lemma 2. Let n ≥ 1, n ≥ m ≥ 0 and 1 ≤ i1 < · · · < im ≤ n. Then we have∑
σ∈Sn
sgn σ tσ(i1)tσ(i2) · · · tσ(im)
∏
1≤a<b≤n
(tσ(b) − q−2tσ(a))
= q
−m(n+1)−n(n−1)
2
+2
m∑
j=1
ij
[m]![n−m]! em(t1, . . . , tn)
∏
1≤a<b≤n
(tb − ta),
where em(t1, . . . , tn) is the m-th elementary symmetric polynomial.
Proof. Set
F (t) =
∑
σ∈Sn
sgn σ tσ(i1)tσ(i2) · · · tσ(im)
∏
1≤a<b≤n
(tσ(b) − q−2tσ(a)).
It is easy to see that F (t) is an antisymmetric polynomial. So we can write
F (t) = S(t)
∏
1≤a<b≤n
(tb − ta),
where S(t) is a symmetric polynomial. Moreover S(t) is a homogeneous polynomial of degree m
and degtiS(t) = 1 for all i ∈ {1, . . . , n}. Hence we have
S(t) = cem(t)
for some constant c.
The number (−1)
m∑
j=1
ij+
n(n−1)
2
−m(m+1)
2
c is equal to the coefficient of
tni1t
n−1
i2
· · · tn−m+1
im
tn−m−1
1 tn−m−2
2 · · · tn−1
in F (t).
We can show
c = q
−2nm+m(m−1)+2
m∑
k=1
ik
(
q−m(m−1)
∑
σ∈Sm
q2`(σ)
)q−(n−m)(n−m−1)
∑
τ∈Sn−m
q2`(τ)
,
where `(σ) is the inversion number of σ.
Using the fact
∑
σ∈Sm
q2`(σ) = q
m(m−1)
2 [m]!, we have the desired result. �
Lemma 3. For 1 ≤ n ≤ l, we have
n∑
s=0
(−1)sq−s(n−1)
[
n
s
] ∑
σ∈Sn
s∏
i=1
(z − qltσ(i))
n∏
i=s+1
(
z − q−ltσ(i)
) ∏
1≤a<b≤n
tσ(b) − q−2tσ(a)
tσ(b) − tσ(a)
= (−1)nq−ln−n(n−1)
2
{
n−1∏
i=0
(
1− q2(l−i)
)}
[n]!t1t2 . . . tn.
Proof. We set
Ln,s =
∑
σ∈Sn
sgn σ
s∏
i=1
(z − qltσ(i))
n∏
j=s+1
(
z − q−ltσ(j)
)∏
i>j
tσ(i) − q−2tσ(j)
ti − tj
,
q-Wakimoto Modules and Integral Formulae 11
Ln =
n∑
s=0
(−1)sq−s(n−1)
[
n
s
]
Ln,s.
Using Lemma 2,
Ln,s =
n∑
k=0
(−1)kzn−kek(t)q−k(n+1)−n(n−1)
2 [k]![n− k]!
k∑
t=0
q2lt−lk
∑
1≤i1<i2<···<it≤s
s<it+1<···<ik≤n
q
k∑
j=1
2ij
=
n∑
k=0
(−1)kzn−kek(t)q−lk−k(n+1)−n(n−1)
2 [k]![n− k]!
×
(
k∑
t=0
q2ltq2s(k−t)+(s+1)t+(n−s+1)(k−t)
[
s
t
] [
n− s
k − t
])
.
Then,
Ln =
n∑
s=0
(−1)sq−s(n−1)
[
n
s
] n∑
k=0
(−1)kzn−kek(t)q−lk−k(n+1)−n(n−1)
2 [k]![n− k]!
×
(
k∑
t=0
q2ltqsk+k+n(k−t)
[
s
t
] [
n− s
k − t
])
= [n]!
n∑
k=0
(−1)kzn−kek(t) q−lk−k(n+1)−n(n−1)
2
×
k∑
t=0
q2ltq(k−t)(n+1)+t
[
k
t
] n−k+t∑
s=t
(−1)sq−s(n−k−1)
[
n− k
s− t
]
= [n]!
n∑
k=0
(−1)kzn−kek(t) q−lk−k(n+1)−n(n−1)
2
×
k∑
t=0
q2ltq(k−t)(n+1)+t
[
k
t
]
(−1)tq−t(n−k−1)δn,k
= [n]!(−1)nq−lnq−
n(n−1)
2
n∑
t=0
(−1)tq2ltq−(n−1)t
[
n
t
]
en(t)
= [n]!(−1)nq−lnq−
n(n−1)
2
{
n−1∏
i=0
(1− q2(l−i))
}
en(t).
Here we have used the q-binomial theorem. �
For a given sequence (mi)r
i=1 (0 ≤ mi ≤ ni), let Mi = {(i, j) | j ≤ mi}. Set
Î
(ν)
(µ)(ε)(m) =
∫
CN
∏
(i1,i2)
dui1,i2
2πiui1,i2
Ĝ
(ν)
(µ)(ε)(m).
Lemma 4. We have
Î
(ν)
(µ)(ε)(m) = q
(L−N)
{ r∑
s=1
(ns−2as)
} ∏
(i1,i2)<j
qµi1,i2
lj
∏
(i1,i2)<(j1,j2)
q−µi1,i2
12 K. Kuroki
×
∑
CitDi=Aµ,i
D′
i=Di∩Mi
1≤i≤r
(
N∏
b=1
q−1−εb
) r∑
i=1
|Ci|
∏
(i1,i2)<(j1,j2)
(i1,i2)∈C1∪···∪Cr
(j1,j2)∈D1∪···∪Dr
q2
×
∑
1≤bi,j≤N
1≤i≤r
1≤j≤|Di|
r∏
i1=1
|Di1
|∏
i2=1
(1− q
−1−εbi1,i2 )
∏
b6=bi1,i2
tbi1,i2
− q−1−εbtb
tbi1,i2
− tb
×
∏
(i1,i2)<(j1,j2)
tbi1,i2
− tbj1,j2
tbi1,i2
− q−2tbj1,j2
k(i1)−1∏
j=1
zj − q−lj tbi1,i2
zj − qlj tbi1,i2
|Di1
|∏
i2=|D′
i1
|+1
zk(i1) − q−lk(i1)tbi1,i2
zk(i1) − qlk(i1)tbi1,i2
.
Proof. We integrate with respect the variables ui,j , (i, j) ∈ A+
µ in the order u`+1,1
, . . . , u`+
1,a+
1
,
u`+2,1
, . . . , u`+
r,a+
r
. With respect to u`+1,1
the only singularity outside C`+1,1
is ∞. Then the integral
in u`+1,1
is calculated by taking the residue at ∞. After this integration the integrand as a
function of u`+1,2
has a similar structure. Then the integral with respect to u`+1,2
is calculated by
taking residue at ∞ and so on. Finally we get
Î
(ν)
(ε)(µ)(m) = (−1)
r∑
i=1
a+
i
Res
u
`+
r,a+
r
=∞
· · · Res
u
`+r,1
=∞
· · · Res
u
`+
1,a+
1
=∞
· · · Res
u
`+1,1
=∞
Ĝν
(ε)(µ)(m)(t, z|u)
=
∏
(i1,i2)
q(L−N)µi1,i2
∏
(i1,i2)<j
qµi1,i2
lj
∏
(i1,i2)<(j1,j2)
q−µi1,i2
×
∫
C
N−
r∑
i=1
a+
i
∏
(i1,i2)∈Aµ
dui1,i2
2πiui1,i2
∏
j<(i1,i2)
(i1,i2)∈Aµ
zj − q−lj−k−2ui1,i2
zj − qlj−k−2ui1,i2
×
∏
(i1,i2)∈Aµ
1≤b≤N
ui1,i2 − qk+1−εbtb
ui1,i2 − qk+2tb
∏
(i1,i2)<(j1,j2)
(i1,i2),(j1,j2)∈Aµ
ui1,i2 − uj1,j2
ui1,i2 − q−2uj1,j2
,
where C
N−
r∑
i=1
a+
i
is the resulting contour for (u`1,1 , . . . , u`r,ar
). We set
I
(ν)+
(ε)(µ)(m)(t, z) =
∏
(i1,i2)∈Aµ
1
ui1,i2
∏
j<(i1,i2)
(i1,i2)∈Aµ
zj − q−lj−k−2ui1,i2
zj − qlj−k−2ui1,i2
×
∏
(i1,i2)∈Aµ
1≤b≤N
ui1,i2 − qk+1−εbtb
ui1,i2 − qk+2tb
∏
(i1,i2)<(j1,j2)
(i1,i2),(j1,j2)∈Aµ
ui1,i2 − uj1,j2
ui1,i2 − q−2uj1,j2
.
q-Wakimoto Modules and Integral Formulae 13
Next we perform integrations with respect to the remaining variables ui,j , (i, j) ∈ Aµ in the
order u`r,ar
, . . . , u`r,1 , u`r−1,ar−1
, . . . , u`1,1 . The poles of the integrand inside C`r,ar
are 0 and
qk+2tb, b = 1, . . . , N . Thus we have∫
C`r,ar
du`r,ar
2πi
I
(ν)+
(ε)(µ)(m)(t, z)
=
∏
i≤b≤N
q−1−εb
∏
(i1,i2)∈Aµ
(i1,i2) 6=`r,ar
1
ui1,i2
∏
j<(i1,i2)
(i1,i2)∈Aµ−{`r,ar}
zj − q−lj−k−2ui1,i2
zj − qlj−k−2ui1,i2
×
∏
(i1,i2)∈Aµ−{`r,ar}
1≤b≤N
ui1,i2 − qk+1−εbtb
ui1,i2 − qk+2tb
∏
(i1,i2)<(j1,j2)<`r,ar
(i1,i2),(j1,j2)∈Aµ
ui1,i2 − uj1,j2
ui1,i2 − q−2uj1,j2
+
∑
1≤b`r,ar
≤N
(1− q
−1−εb`r,ar )
∏
j<`r,ar
zj − q−lj tb`r,ar
zj − qlj−k−2tb`r,ar
∏
1≤b≤N
b6=b`r,ar
tb`r,ar
− q−1−εbtb
tb`r,ar
− tb
×
∏
(i1,i2)<`r,ar
ui1,i2 − qk+2tb`r,ar
ui1,i2 − qktb`r,ar
∏
(i1,i2)∈Aµ
(i1,i2) 6=`r,ar
1
ui1,i2
∏
j<(i1,i2)
(i1,i2)∈Aµ−{`r,ar}
zj − q−lj−k−2ui1,i2
zj − qlj−k−2ui1,i2
×
∏
(i1,i2)∈Aµ−{`r,ar}
1≤b≤N
ui1,i2 − qk+1−εbtb
ui1,i2 − qk+2tb
∏
(i1,i2)<(j1,j2)<`r,ar
(i1,i2),(j1,j2)∈Aµ
ui1,i2 − uj1,j2
ui1,i2 − q−2uj1,j2
.
The integrand in u`r,ar−1 has the poles at 0 and qk+2tb inside C`r,ar−1 and so on. Finally we get
Î
(ν)
(ε)(µ) =
∏
(i1,i2)
q(L−N)µi1,i2
∏
(i1,i2)<j
qµi1,i2
lj
∏
(i1,i2)<(j1,j2)
q−µi1,i2
×
∑
w`i1,i2
∈{0}∪(T−Wi1,i2
)
(i1,i2)∈Aµ
Res
u`1,1
=w`1,1
· · · Res
u`r,ar
=w`r,ar
I
(ν)+
(ε)(µ),
where T = {t1, t2, . . . , tN}, Wi1,i2 = ∪
`i1,i2
<`j1,j2
{w`j1,j2
}.
Set Ci = {`i,j |w`i,j
= 0}, Di = Aµ,i − Ci. Then we have the desired result. �
Now we can calculate Ĵ
(ν)
(ε)(a).
Proposition 2. We have
Ĵ
(ν)
(ε)(a) = (−1)
r∑
i=1
ai
q
r∑
s=1
( n∑
t=k(s)+1
lt
)
(ns−2as)
(q
(L−N)
{ r∑
s=1
(ns−2as)
})
14 K. Kuroki
×
q
−
r∑
s=1
r∑
t=s+1
ns(nt−2at)
∑
1≤bi1,i2
≤N
1≤i1≤r
1≤i2≤ai1
∏
i1<j1
tbi1,i2
− tbi1,i2
tbi1,i2
− q−2tbi1,i2
×
r∏
i1=1
ai1∑
si1
=0
qai1
(ni1
−si1
−1)−
ni1
(ni1
−1)
2
[ni1 ]!
[si1 ]![ai1 − si1 ]!
×
ni1
−ai1∑
i=0
(−1)i+si1 qi(2lk(i1)−ni1
−ai1
+1)+si1
1
[i]![ni1 − ai1 − i]!
×
ai1∏
i2=1
(1− q
−1−εbi1,i2
) ∏
b6=bi1,i2
tbi1,i2
− q−1−εbtb
tbi1,i2
− tb
∏
i2<j2
tbi1,i2
− tbi1,j2
tbi1,i2
− q−2tbi1,j2
×
k(i1)−1∏
j=1
zj − q−lj tbi1,i2
zj − qlj tbi1,i2
ai1∏
i2=si1
+1
zk(i1) − q−lk(i1)tbi1,i2
zk(i1) − qlk(i1)tbi1,i2
.
Proof. Using Lemma 4 we have
Ĵ
(ν)
(ε)(a) = (−1)
r∑
i=1
ai ∑
|Aµ,i|=ai
1≤i≤r
∑
0≤mi≤ni
1≤i≤r
(−1)
r∑
i=1
mi
{
r∏
i=1
qmilk(i)q−mi(ni−1)
[
ni
mi
]}
×
(
r∏
i=1
q(L−N)(ni−2ai)
) ∏
(i1,i2)<j
qµi1,i2
lj
∏
(i1,i2)<(j1,j2)
q−µi1,i2
×
∑
CitDi=Aµ,i
D′
i=Di∩Mi
1≤i≤r
(
N∏
b=1
q−1−εb)
r∑
i=1
|Ci|
∏
(i1,i2)<(j1,j2)
(i1,i2)∈C1∪···∪Cr
(j1,j2)∈D1∪···∪Dr
q2
×
∑
1≤bi,j≤N
1≤i≤r
1≤j≤|Di|
r∏
i1=1
|Di1
|∏
i2=1
(1− q
−1−εbi1,i2
) ∏
b6=bi1,i2
tbi1,i2
− q−1−εbtb
tbi1,i2
− tb
×
∏
(i1,i2)<(j1,j2)
tbi1,i2
− tbj1,j2
tbi1,i2
− q−2tbj1,j2
k(i1)−1∏
j=1
zj − q−lj tbi1,i2
zj − qlj tbi1,i2
×
|Di1
|∏
i2=|D′
i1
|+1
zk(i1) − q−lk(i1)tbi1,i2
zk(i1) − qlk(i1)tbi1,i2
. (4)
Set λi = |Aµ,i ∩Mi|, γi = |Di|, si = |D′
i|, 1 ≤ i ≤ r. Then the right hand side of (4) is equal to∑
0≤mi≤ni
1≤i≤r
(−1)
r∑
i=1
mi
{
r∏
i=1
qmilk(i)q−mi(ni−1)
[
ni
mi
]}
×
∑
0≤γj≤aj
1≤j≤r
∑
0≤sj≤γj
1≤j≤r
∑
0≤λj≤mj
1≤j≤r
C(a)(γ)
{
q
r∑
s=1
lk(s)(ms−2λs)
}
q-Wakimoto Modules and Integral Formulae 15
×
r∏
i1=1
∑
|Aµ,i1
|=ai1
|Aµ,i1
∩Mi1
|=λi1
∏
i2<j2
i1=j1
q−µi1,i2
×
(
r∏
i1=1
q
λi1
γi1
+ai1
γi1
−ai1
si1
−γ2
i1
[
λi1
si1
] [
ai1 − λi1
γi1 − si1
])
×
∑
1≤bi1,i2
≤N
1≤i1≤r
1≤i2≤γi1
r∏
i1=1
γi1∏
i2=1
(1− q
−1−εbi1,i2 )
∏
b6=bi1,i2
tbi1,i2
− q−1−εbtb
tbi1,i2
− tb
×
∏
i2<j2
tbi1,i2
− tbi1,j2
tbi1,i2
− q−2tbi1,j2
k(i1)−1∏
j=1
zj − q−lj tbi1,i2
zj − qlj tbi1,i2
γi1∏
i2=si1
+1
zk(i1) − q−lk(i1)tbi1,i2
zk(i1) − qlk(i1)tbi1,i2
×
∏
i1<j1
tbi1,i2
− tbj1,j2
tbi1,i2
− q−2tbj1,j2
,
where
C(a)(γ) = (−1)
r∑
i=1
ai
q
r∑
s=1
( n∑
t=k(s)+1
lt
)
(ns−2as)
(q
(L−N)
{ r∑
s=1
(ns−2as)
})
×
q
−
r∑
s=1
(ns−2as)
( r∑
t=s+1
nt
)(q
2
r∑
s=1
∑
s<t
γt(as−γs)
) ∏
1≤b≤N
q−1−εb
r∑
s=1
(as−γs)
.
Here we have used Lemma 1 (i).
By (ii) of Lemma 1 we have
Ĵ
(ν)
(ε)(a) =
r∑
j=1
∑
0≤γj≤aj
1≤j≤r
C(a)(γ)
∑
1≤bi1,i2
≤N
1≤i1≤r
1≤i2≤γi2
∏
i1<j1
tbi1,i2
− tbj1,j2
tbi1,i2
− q−2tbj1,j2
×
r∏
i1=1
γi1∑
si1
=0
aj−γi1
+si1∑
λi1
=si1
ni1∑
mi1
=0
(−1)mi1 q−mi1
(ni1
−1)q2lk(i1)(mi1
−λi1
)
[
ni1
mi1
]
×
∑
|Aµ,i1
|=ai1
|Aµ,i1
∩Mi1
|=λi1
∏
i2<j2
i1=j1
q−µi1,i2
(
q
λi1
γi1
+ai1
γi1
−ai1
si1
−γ2
i1
[
λi1
si1
] [
ai1 − λi1
γi1 − si1
])
×
γi1∏
i2=1
(1− q
−1−εbi1,i2 )
∏
b6=bi1,i2
tbi1,i2
− q−1−εbtb
tbi1,i2
− tb
∏
i2<j2
tbi1,i2
− tbi1,j2
tbi1,i2
− q−2tbi1,j2
×
k(i1)−1∏
j=1
zj − q−lj tbi1,i2
zj − qlj tbi1,i2
γi1∏
i2=si1
+1
zk(i1) − q−lk(i1)tbi1,i2
zk(i1) − qlk(i1)tbi1,i2
16 K. Kuroki
=
∑
0≤γj≤aj
1≤j≤r
C(a)(γ)
∑
1≤bi1,i2
≤N
1≤i1≤r
1≤i2≤γi2
∏
i1<j1
tbi1,i2
− tbj1,j2
tbi1,i2
− q−2tbj1,j2
×
r∏
i1=1
γi1∑
si1
=0
aj−γi1
+si1∑
λi1
=si1
ni1∑
mi1
=0
(−1)mi1 q−mi1
(ni1
−1)q2lk(i1)(mi1
−λi1
)
[
ni1
mi1
]
×
(
qni1
λi1
+ai1
ni1
−ai1
mi1
−ai1
−
ni1
(ni1
−1)
2
[
mi1
λi1
] [
ni1 −mi1
ai1 − λi1
])
×
(
q
λi1
γi1
+ai1
γi1
−ai1
si1
−γ2
i1
[
λi1
si1
] [
ai1 − λi1
γi1 − si1
])
×
γi1∏
i2=1
(1− q
−1−εbi1,i2 )
∏
b6=bi1,i2
tbi1,i2
− q−1−εbtb
tbi1,i2
− tb
∏
i2<j2
tbi1,i2
− tbi1,j2
tbi1,i2
− q−2tbi1,j2
×
k(i1)−1∏
j=1
zj − q−lj tbi1,i2
zj − qlj tbi1,i2
γi1∏
i2=si1
+1
zk(i1) − q−lk(i1)tbi1,i2
zk(i1) − qlk(i1)tbi1,i2
.
It is easy to show
a−γ+s∑
λ=s
n∑
m=0
(−1)mq−m(n−1)
[
n
m
]
q2l(m−λ)
(
qnλ+an−am−a−n(n−1)
2
[
m
λ
] [
n−m
a− λ
])
×
(
qλγ+aγ−as−γ2
[
λ
s
] [
a− λ
γ − s
])
= (−1)sqa(n−s−1)+sq−
n(n−1)
2
[n]!
[s]![a− s]!
n−a∑
i=0
(−1)iqi(2l−n−a+1) 1
[i]![n− a− i]!
δa,γ ,
for 0 ≤ s ≤ γ ≤ a ≤ n.
Hence
Ĵ
(ν)
(ε)(a) = (−1)
r∑
i=1
ai
q
r∑
s=1
( n∑
t=k(s)+1
lt
)
(ns−2as)
(q
(L−N)
{ r∑
s=1
(ns−2as)
})
×
q
−
r∑
s=1
(ns−2as)
( r∑
t=s+1
nt
) ∑
1≤bi1,i2
≤N
1≤i1≤r
1≤i2≤ai1
∏
i1<j1
tbi1,i2
− tbj1,j2
tbi1,i2
− q−2tbj1,j2
×
r∏
i1=1
ai1∑
si1
=0
(−1)si1 qai1
(ni1
−si1
−1)+si1 q−
ni1
(ni1
−1)
2
[ni1 ]!
[si1 ]![ai1 − si1 ]!
×
ni1
−ai1∑
i=0
(−1)iqi(2lk(i1)−ni1
−ai1
+1) 1
[i]![ni1 − ai1 − i]!
×
ai1∏
i2=1
(1− q
−1−εbi1,i2 )
∏
b6=bi1,i2
tbi1,i2
− q−1−εbtb
tbi1,i2
− tb
∏
i2<j2
tbi1,i2
− tbi1,j2
tbi1,i2
− q−2tbi1,j2
×
k(i1)−1∏
j=1
zj − q−lj tbi1,i2
zj − qlj tbi1,i2
ai1∏
i2=si1
+1
zk(i1) − q−lk(i1)tbi1,i2
zk(i1) − qlk(i1)tbi1,i2
. �
q-Wakimoto Modules and Integral Formulae 17
Lemma 5. If ai 6= ni for some i,
∑
εi=±
1≤i≤N
N∏
j=1
εj
(∏
a<b
qεbtb − qεata
tb − q−2ta
)
Ĵ
(ν)
(ε)(a) = 0.
Proof. It is enough to show the following equation. For 1 ≤ bi1,i2 ≤ N (1 ≤ i1 ≤ r, 1 ≤ i2 ≤
ai1), bi1,i2 6= bj1,j2 ((i1, i2) 6= (j1, j2)),
∑
εi=±
1≤i≤N
(
N∏
i=1
εi
)∏
a<b
(qεbtb − qεata)
∏
1≤i1≤r
1≤i2≤ai1
(1− q
−1−εbi1,i2 )
∏
b6=bi1,i2
(tbi1,i2
− q−1−εbtb)
= (1− q−2)Nq
N(N−1)
2
(
r∏
s=1
δas,ns
){∏
a<b
(tb − ta)
} ∏
b6=bi1,i2
(tbi1,i2
− q−2tb)
. (5)
For a set {b1,1, . . . , br,ar} = {b1, . . . , bα}, let{c1, . . . , cN−α} be defined by
{b1, . . . , bα} t {c1, . . . , cN−α} = {1, . . . , N},
where α =
r∑
i=1
ai.
Then the left hand side of (5) is equal to
(1− q−2)α
∏
1≤i≤α
δεbi
,+
∏
i<j
q(tbj
− tbi
)
∏
1≤i,j≤α
i6=j
(tbi
− q−2tbj
)
×
∏
bi<cj
(−q)
∏
ci<bj
q
∑
εci=±
1≤i≤N−α
(
N−α∏
i=1
εci
)∏
i<j
(qεcj tcj − qεci tci)
×
∏
1≤i≤α
∏
1≤j≤N−α
(tbi
− qεcj−1tcj )
∏
1≤i≤α
∏
1≤j≤N−α
(tbi
− q−1−εcj tcj )
.
Using
(tbj
− qεci−1tci)
(
tbi
− q−1−εcj tcj
)
= (tbj
− tci)
(
tbi
− q−2tcj
)
,
we have
∑
ε
(
N∏
i=1
εi
){∏
a<b
(
qεbtb − qεata
)}
∏
1≤i1≤r
1≤i2≤ai1
(
1− q
−1−εbi1,i2
)
∏
b6=bi1,i2
(
tbi1,i2
− q−1−εbtb
)
= (1− q−2)α
∏
1≤i≤α
δεbi
,+
∏
i<j
q(tbj
− tbi
)
∏
1≤i,j≤α
i6=j
(tbi
− q−2tbj
)
×
∏
1≤i≤α
∏
1≤j≤N−α
(tbi
− tcj )
(
tbi
− q−2tcj
)
18 K. Kuroki
×
∏
bi<cj
(−q)
∏
ci<bj
q
∑
εci=±
1≤i≤N−α
(
N−α∏
i=1
εci
)∏
i<j
(
qεcj tcj − qεci tci
) .
Let α 6= N and ai(ε) =t (1, qεti, (qεti)2, . . . , (qεti)N−α−1). Then
∑
εi=±
1≤i≤N−α
(
N−α∏
i=1
εi
)∏
i<j
(qεj tj − qεiti)
=
∑
εi=±
1≤i≤N−α
(
N−α∏
i=1
εi
)
det(a1(ε1),a2(ε2), . . . ,aN−α(εN−α)). (6)
Since ∑
εi=±
εiai(ε) =t (0, (q − q−1)ti, . . . , (qN−α−1 − q−(N−α−1))tN−α−1
i ),
the right hand side of (6) is equal to 0. �
If ai = ni for all i, then
∑
εi=±
1≤i≤N
(
N∏
i=1
εi
) ∏
1≤a<b≤N
qεbtb − qεata
tb − q−2ta
Ĵ
(ν)
(ε)(a)
= C1
(∏
a<b
tb − ta
tb − q−2ta
) ∑
Γ1t···tΓr={1,...,N}
|Γs|=ns (s=1,...,r)
∑
bi1,i2
∈Γi1
1≤i1≤r
1≤i2≤ni1
∏
i1>j1
tbi1,i2
− q−2tbj1,j2
tbi1,i2
− tbj1,j2
×
r∏
i1=1
ni1∑
si1
=0
(−1)si1 q−(ni1
−1)si1
[
ni1
si1
] si1∏
i2=1
(zk(i1) − qlk(i1)tbi1,i2
)
×
ni1∏
i2=si1
+1
(zk(i1) − q−lk(i1)tbi1,i2
)
∏
i2>j2
tbi1,i2
− q−2tbi1,j2
tbi1,i2
− tbi1,j2
×
ni1∏
i2=1
1
zk(i1) − qlk(i1)tbi1,i2
k(i1)−1∏
j=1
zj − q−lj tbi1,i2
zj − qlj tbi1,i2
, (7)
where
C1 = (−1)N
(
1− q−2
)N
qN2−LNq
N(N−1)
2 q
r∑
i=1
ni(ni−1)
2
×
q
−
r∑
s=1
( n∑
t=k(s)+1
lt
)
ns
q
r∑
s=1
( r∑
t=s+1
nt
)
ns
.
By Lemma 3 the right hand side of (7) becomes
C1
r∏
s=1
{
(−1)ns [ns]!q−lk(s)nsq−
ns(ns−1)
2
{
ns−1∏
i=0
(
1− q2(lk(s)−i)
)}}
q-Wakimoto Modules and Integral Formulae 19
×
(∏
a<b
tb − ta
tb − q−2ta
) ∑
Γ1t···tΓr={1,...,N}
|Γs|=ns (s=1,...,r)
∏
1≤i<j≤r
a∈Γi,b∈Γj
tb − q−2ta
tb − ta
×
r∏
s=1
∏
b∈Γs
tb
zk(s) − qlk(s)tb
k(s)−1∏
i=1
zi − q−litb
zi − qlitb
.
This completes the proof of Proposition 1. �
A The representation V (l)
z
Let qhi , ei, fi (i = 0, 1) and qd be the generators of Uq(ŝl2). (See [4] for more details.) The
actions of the generators of Uq(ŝl2) on V
(l)
z are given as follows.
For 0 ≤ i ≤ l and n ∈ Z,
e0v
(l)
j ⊗ zn = [l − i]v(l)
i+1 ⊗ zn+1, e1v
(l)
j ⊗ zn = [i]v(l)
i−1 ⊗ zn,
f0v
(l)
j ⊗ zn = [i]v(l)
i−1 ⊗ zn−1, f1v
(l)
j ⊗ zn = [l − i]v(l)
i+1 ⊗ zn,
qh0v
(l)
i ⊗ zn = q−(l−2i)v
(l)
i ⊗ zn, qh1v
(l)
i ⊗ zn = ql−2iv
(l)
i ⊗ zn,
qdv
(l)
j ⊗ zn = qnv
(l)
j ⊗ zn.
B R-matrix
We give examples of explicit forms of R-matrix in the case of l1 = 1 or l2 = 1. They are taken
from [4]. If we write
R1,l2(z)
(
v(1)
ε ⊗ v
(l2)
j
)
=
∑
ε′=0,1
v
(1)
ε′ ⊗ r1l2
ε′ε (z)v(l2)
j ,
Rl1,1(z)
(
v
(l1)
j ⊗ v(1)
ε
)
=
∑
ε′=0,1
rl11
ε′ε (z)v(l1)
j ⊗ v
(1)
ε′ ,
then we have(
r1l2
00 (z) r1l2
01 (z)
r1l2
10 (z) r1l2
11 (z)
)
=
1
q1+l2/2 − z−1q−l2/2
(
q1+h/2 − z−1q−h/2 (q − q−1)z−1fqh/2
(q − q−1)eq−h/2 q1−h/2 − z−1qh/2
)
,(
rl11
00 (z) rl11
01 (z)
rl11
10 (z) rl11
11 (z)
)
=
1
zql1/2 − q−1−l1/2
(
zqh/2 − q−1−h/2 (q − q−1)zqh/2f
(q − q−1)q−h/2e zq−h/2 − q−1+h/2
)
,
h = h1, e = e1 and f = f1.
C Free field representations
The following formulae are given in [6]. 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.
20 K. Kuroki
The normal ordering is defined by specifying N+, ã0, b̃0, c̃0 as annihilation operators, N−, Qa,
Qb, Qc as creation operators.
Define operators
J−(z) : Fr,s → Fr,s+1, S(z) : Fr,s → Fr−2,s−1, φ(l)
m (z) : Fr,s → Fr+l,s+l−m,
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∏
i=1
[i], [X, Y ]q = XY − qY X,
and the integral in φ
(l)
l−r(z) signifies to take the coefficient of (u1 · · ·ur)−1.
D List of OPE’s
The following formulae are given in [6]
φl1(z1)φl2(z2) = (qkz1)
l1l2
2(k+2)
(
ql1+l2+2k+6 z2
z1
; q4, q2(k+2)
)
∞
(
q−l1−l2+2k+6 z2
z1
; q4, q2(k+2)
)
∞(
ql1−l2+2k+6 z2
z1
; q4, q2(k+2)
)
∞
(
q−l1+l2+2k+6 z2
z1
; q4, q2(k+2)
)
∞
× : φl1(z1)φl2(z2) :, |q−l1−l2+2k+6z2| < |z1|,
φl(z)J−µ (u) =
z − qµl−k−2u
z − ql−k−2u
: φl(z)J−µ (u) :, |q−l−k−2u| < |z|,
J−µ (u)φl(z) = qµl u− q−µl+k+2z
u− ql+k+2z
: φl(z)J−µ (u) :, |q−l+k+2u| < |z|,
φl(z)Sε(t) =
(
ql t
z ; p
)
∞(
q−l t
z ; p
)
∞
(qkz)−
l
k+2 : φl(z)Sε(t) :, |z| > |q−lt|,
J−µ (u)Sε(t) = q−µ u− q−µ(k+1)−εt
u− q−µ(k+2)t
: J−µ (u)Sε(t) :, |u| > |q−k−2t|,
q-Wakimoto Modules and Integral Formulae 21
J−µ1
(u1)J−µ2
(u2) =
q−µ1u1 − q−µ2u2
u1 − q−2u2
: J−µ1
(u1)J−µ2
(u2) :, |u1| > |q−2u2|,
Sε1(t1)Sε2(t2) = (q−2t1)
2
k+2
qε1t1 − qε2t2
t1 − q−2t2
(
q−2 t2
t1
; p
)
∞(
q2 t2
t1
; p
)
∞
: Sε1(t1)Sε2(t2) :, |t1| > |q−2t2|.
Acknowledgements
After completing the paper we know that similar results to the present paper are obtained by
H. Awata, S. Odake and J. Shiraishi [12]. I would like to thank all of them for kind correspon-
dences and permitting me to write the results in a single authored paper. I am also grateful
to Professor H. Konno for useful comments. I would like to thank Professor A. Nakayashiki for
constant encouragements.
References
[1] Abada A., Bougourzi A.H., El Gradechi M.A., Deformation of the Wakimoto construction, Modern Phys.
Lett. A 8 (1993), 715–724, hep-th/9209009.
[2] Bougourzi A.H., Weston R.A., Matrix elements of Uq(su(2)k) vertex operators via bosonization, Internat.
J. Modern Phys. A 9 (1994), 4431–4447, hep-th/9305127.
[3] Frenkel I.B., Reshetikhin N.Yu., Quantum affine algebras and holonomic difference equations, Comm. Math.
Phys. 146 (1992), 1–60.
[4] Idzumi M., Tokihiro T., Iohara K., Jimbo M., Miwa T., Nakashima T., Quantum affine symmetry in vertex
models, Internat. J. Modern Phys. A 8 (1993), 1479–1511, hep-th/9208066.
[5] Jimbo M., Miwa T., Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in
Mathematics, Vol. 85, American Math. Soc., Providence, RI, 1995.
[6] Kato A., Quano Y.-H., Shiraishi J., Free boson representation of q-vertex operators and their correlation
functions, Comm. Math. Phys. 157 (1993), 119–137, hep-th/9209015.
[7] Konno H., Free-field representation of the quantum affine algebra Uq(ŝl2) and form factors in the higher-spin
XXZ model, Nuclear Phys. B 432 (1994), 457–486, hep-th/9407122.
[8] Kuroki K., Nakayashiki A., Free field approach to solutions of the quantum Knizhnik–Zamolodchikov equa-
tions, SIGMA 4 (2008), 049, 13 pages, arXiv:0802.1776.
[9] Matsuo A., Quantum algebra structure of certain Jackson integrals, Comm. Math. Phys. 157 (1993), 479–
498.
[10] Matsuo A., A q-deformation of Wakimoto modules, primary fields and screening operators, Comm. Math.
Phys. 160 (1994), 33–48, hep-th/9212040.
[11] Shiraishi J., Free boson representation of quantum affine algebra, Phys. Lett. A 171 (1992), 243–248.
[12] Shiraishi J., Free boson realization of quantum affine algebras, PhD thesis, University of Tokyo, 1995.
[13] Tarasov V., Varchenko A., Geometry of q-hypergeometric functions, quantum affine algebras and elliptic
quantum groups, Astérisque 246 (1997), 1–135.
http://arxiv.org/abs/hep-th/9209009
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/9407122
http://arxiv.org/abs/0802.1776
http://arxiv.org/abs/hep-th/9212040
1 Introduction
2 Tarasov-Varchenko's formulae
3 Free field realizations
4 Integral formulae
5 Proof of Proposition 1
A The representation V^{(l)}_z
B R-matrix
C Free field representations
D List of OPE's
References
|