A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application
A vertex operator approach for form factors of Belavin's (Z/nZ)-symmetric model is constructed on the basis of bosonization of vertex operators in the An−1⁽¹⁾ model and vertex-face transformation. As simple application for n=2, we obtain expressions for 2m-point form factors related to the σz a...
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| Cite this: | A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application / Y. Quano // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 21 назв. — англ. |
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Quano, Y. 2019-02-11T15:21:26Z 2019-02-11T15:21:26Z 2011 A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application / Y. Quano // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 21 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37K10; 81R12 DOI:10.3842/SIGMA.2011.008 https://nasplib.isofts.kiev.ua/handle/123456789/146795 A vertex operator approach for form factors of Belavin's (Z/nZ)-symmetric model is constructed on the basis of bosonization of vertex operators in the An−1⁽¹⁾ model and vertex-face transformation. As simple application for n=2, we obtain expressions for 2m-point form factors related to the σz and σx operators in the eight-vertex model. This paper is a contribution to the Proceedings of the International Workshop “Recent Advances in Quantum Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2010.html. We would like to thank T. Deguchi, R. Inoue, H. Konno, Y. Takeyama and R. Weston for discussion and their interests in the present work. We would also like to thank S. Lukyanov for useful information. This paper is partly based on a talk given in International Workshop RAQIS’10, Recent Advances in Quantum Integrable Systems, held at LAPTH, Annecy-le-Vieux, France, June 15–18, 2010. We would like to thank L. Frappat and E. Ragoucy for organizing the conference. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application Article published earlier |
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A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application |
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A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application Quano, Y. |
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A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application |
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A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application |
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A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application |
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A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application |
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vertex operator approach for form factors of belavin's (z/nz)-symmetric model and its application |
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Quano, Y. |
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Quano, Y. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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A vertex operator approach for form factors of Belavin's (Z/nZ)-symmetric model is constructed on the basis of bosonization of vertex operators in the An−1⁽¹⁾ model and vertex-face transformation. As simple application for n=2, we obtain expressions for 2m-point form factors related to the σz and σx operators in the eight-vertex model.
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A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application / Y. Quano // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 21 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 008, 16 pages
A Vertex Operator Approach for Form Factors
of Belavin’s (Z/nZ)-Symmetric Model
and Its Application?
Yas-Hiro QUANO
Department of Clinical Engineering, Suzuka University of Medical Science,
Kishioka-cho, Suzuka 510-0293, Japan
E-mail: quanoy@suzuka-u.ac.jp
Received October 22, 2010, in final form January 07, 2011; Published online January 15, 2011
doi:10.3842/SIGMA.2011.008
Abstract. A vertex operator approach for form factors of Belavin’s (Z/nZ)-symmetric
model is constructed on the basis of bosonization of vertex operators in the A
(1)
n−1 model
and vertex-face transformation. As simple application for n = 2, we obtain expressions for
2m-point form factors related to the σz and σx operators in the eight-vertex model.
Key words: vertex operator approach; form factors; Belavin’s (Z/nZ)-symmetric model;
integral formulae
2010 Mathematics Subject Classification: 37K10; 81R12
1 Introduction
In [1] and [2] we derived the integral formulae for correlation functions and form factors, respec-
tively, of Belavin’s (Z/nZ)-symmetric model [3, 4] on the basis of vertex operator approach [5].
Belavin’s (Z/nZ)-symmetric model is an n-state generalization of Baxter’s eight-vertex model [6],
which has (Z/2Z)-symmetries. As for the eight-vertex model, the integral formulae for correla-
tion functions and form factors were derived by Lashkevich and Pugai [7] and by Lashkevich [8],
respectively.
It was found in [7] that the correlation functions of the eight-vertex model can be obtained
by using the free field realization of the vertex operators in the eight-vertex SOS model [9], with
insertion of the nonlocal operator Λ, called ‘the tail operator’. The vertex operator approach for
higher spin generalization of the eight-vertex model was presented in [10]. The vertex operator
approach for higher rank generalization was presented in [1]. The expression of the spontaneous
polarization of the (Z/nZ)-symmetric model [11] was also reproduced in [1], on the basis of
vertex operator approach. Concerning form factors, the bosonization scheme for the eight-vertex
model was constructed in [8]. The higher rank generalization of [8] was presented in [2]. It was
shown in [12, 13] that the elliptic algebra Uq,p(ŝlN ) relevant to the (Z/nZ)-symmetric model
provides the Drinfeld realization of the face type elliptic quantum group Bq,λ(ŝlN ) tensored by
a Heisenberg algebra.
The present paper is organized as follows. In Section 2 we review the basic definitions of
the (Z/nZ)-symmetric model [3], the corresponding dual face model A
(1)
n−1 model [14], and the
vertex-face correspondence. In Section 3 we summarize the vertex operator algebras relevant to
the (Z/nZ)-symmetric model and the A
(1)
n−1 model [1, 2]. In Section 4 we construct the free field
representations of the tail operators, in terms of those of the basic operators for the type I [15]
?This paper is a contribution to the Proceedings of the International Workshop “Recent Advances in Quantum
Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2010.html
mailto:quanoy@suzuka-u.ac.jp
http://dx.doi.org/10.3842/SIGMA.2011.008
http://www.emis.de/journals/SIGMA/RAQIS2010.html
2 Y.-H. Quano
and the type II [16] vertex operators in the A
(1)
n−1 model. Note that in the present paper we
use a different convention from the one used in [1, 2]. In Section 5 we calculate 2m-point form
factors of the σz-operator and σx-operator in the eight-vertex model, as simple application for
n = 2. In Section 6 we give some concluding remarks. Useful operator product expansion (OPE)
formulae and commutation relations for basic bosons are given in Appendix A.
2 Basic definitions
The present section aims to formulate the problem, thereby fixing the notation.
2.1 Theta functions
The Jacobi theta function with two pseudo-periods 1 and τ (Im τ > 0) are defined as follows:
ϑ
[
a
b
]
(v; τ) :=
∑
m∈Z
exp
{
π
√
−1(m+ a) [(m+ a)τ + 2(v + b)]
}
,
for a, b ∈ R. Let n ∈ Z>2 and r ∈ R>1, and also fix the parameter x such that 0 < x < 1. We
will use the abbreviations,
[v] := x
v2
r
−vΘx2r(x
2v), [v]′ := [v]|r 7→r−1, [v]1 := [v]|r 7→1,
{v} := x
v2
r
−vΘx2r(−x2v), {v}′ := {v}|r 7→r−1, {v}1 := {v}|r 7→1,
where
Θq(z) = (z; q)∞
(
qz−1; q
)
∞(q; q)∞ =
∑
m∈Z
qm(m−1)/2(−z)m,
(z; q1, . . . , qm)∞ =
∏
i1,...,im>0
(
1− zqi11 · · · q
im
m
)
.
Note that
ϑ
[
1/2
−1/2
](
v
r
,
π
√
−1
εr
)
=
√
εr
π
exp
(
−εr
4
)
[v],
ϑ
[
0
1/2
](
v
r
,
π
√
−1
εr
)
=
√
εr
π
exp
(
−εr
4
)
{v},
where x = e−ε (ε > 0).
For later conveniences we also introduce the following symbols:
rj(v) = z
r−1
r
n−j
n
gj(z
−1)
gj(z)
, gj(z) =
{x2n+2r−j−1z}{xj+1z}
{x2n−j+1z}{x2r+j−1z}
, (2.1)
r∗j (v) = z
r
r−1
n−j
n
g∗j (z
−1)
g∗j (z)
, g∗j (z) =
{x2n+2r−j−1z}′{xj−1z}′
{x2n−j−1z}′{x2r+j−1z}′
, (2.2)
χj(v) = (−z)−
j(n−j)
n
ρj(z
−1)
ρj(z)
, ρj(z) =
(x2j+1z;x2, x2n)∞(x2n−2j+1z;x2, x2n)∞
(xz;x2, x2n)∞(x2n+1z;x2, x2n)∞
, (2.3)
where z = x2v, 1 6 j 6 n and
{z} = (z;x2r, x2n)∞, {z}′ = (z;x2r−2, x2n)∞.
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 3
In particular we denote χ(v) = χ1(v). These factors will appear in the commutation relations
among the type I and type II vertex operators.
The integral kernel for the type I and the type II vertex operators will be given as the products
of the following elliptic functions:
f(v, w) =
[v + 1
2 − w]
[v − 1
2 ]
, h(v) =
[v − 1]
[v + 1]
,
f∗(v, w) =
[v − 1
2 + w]′
[v + 1
2 ]′
, h∗(v) =
[v + 1]′
[v − 1]′
.
2.2 Belavin’s (Z/nZ)-symmetric model
Let V = Cn and {εµ}06µ6n−1 be the standard orthonormal basis with the inner product
〈εµ, εν〉 = δµν . Belavin’s (Z/nZ)-symmetric model [3] is a vertex model on a two-dimensional
square lattice L such that the state variables take the values of (Z/nZ)-spin. The model is
(Z/nZ)-symmetric in a sense that the R-matrix satisfies the following conditions:
(i) R(v)ikjl = 0, unless i+ k = j + l, mod n,
(ii) R(v)i+pk+p
j+pl+p = R(v)ikjl , ∀ i, j, k, l, p ∈ Z/nZ.
The definition of the R-matrix in the principal regime can be found in [2]. The present R-matrix
has three parameters v, ε and r, which lie in the following region:
ε > 0, r > 1, 0 < v < 1.
2.3 The A
(1)
n−1 model
The dual face model of the (Z/nZ)-symmetric model is called the A
(1)
n−1 model. This is a face
model on a two-dimensional square lattice L∗, the dual lattice of L, such that the state variables
take the values of the dual space of Cartan subalgebra h∗ of A
(1)
n−1:
h∗ =
n−1⊕
µ=0
Cωµ,
where
ωµ :=
µ−1∑
ν=0
ε̄ν , ε̄µ = εµ −
1
n
n−1∑
µ=0
εµ.
The weight lattice P and the root lattice Q of A
(1)
n−1 are usually defined. For a ∈ h∗, we set
aµν = āµ − āν , āµ = 〈a+ ρ, εµ〉 = 〈a+ ρ, ε̄µ〉, ρ =
n−1∑
µ=1
ωµ.
An ordered pair (a, b) ∈ h∗2 is called admissible if b = a+ ε̄µ, for a certain µ (0 6 µ 6 n− 1).
For (a, b, c, d) ∈ h∗4, let W
[
c d
b a
∣∣∣∣ v] be the Boltzmann weight of the A
(1)
n−1 model for the state
configuration
[
c d
b a
]
round a face. Here the four states a, b, c and d are ordered clockwise
4 Y.-H. Quano
from the SE corner. In this model W
[
c d
b a
∣∣∣∣ v] = 0 unless the four pairs (a, b), (a, d), (b, c)
and (d, c) are admissible. Non-zero Boltzmann weights are parametrized in terms of the elliptic
theta function of the spectral parameter v. The explicit expressions of W can be found in [2].
We consider the so-called Regime III in the model, i.e., 0 < v < 1.
2.4 Vertex-face correspondence
Let t(v)aa−ε̄µ be the intertwining vectors in Cn, whose elements are expressed in terms of theta
functions. As for the definitions see [2]. Then t(v)aa−ε̄µ ’s relate the R-matrix of the (Z/nZ)-
symmetric model in the principal regime and Boltzmann weights W of the A
(1)
n−1 model in the
regime III
R(v1 − v2)t(v1)da ⊗ t(v2)cd =
∑
b
t(v1)cb ⊗ t(v2)baW
[
c d
b a
∣∣∣∣ v1 − v2
]
. (2.4)
Let us introduce the dual intertwining vectors satisfying
n−1∑
µ=0
t∗µ(v)a
′
a t
µ(v)aa′′ = δa
′
a′′ ,
n−1∑
ν=0
tµ(v)aa−ε̄ν t
∗
µ′(v)a−ε̄νa = δµµ′ . (2.5)
From (2.4) and (2.5), we have
t∗(v1)bc ⊗ t∗(v2)abR(v1 − v2) =
∑
d
W
[
c d
b a
∣∣∣∣ v1 − v2
]
t∗(v1)ad ⊗ t∗(v2)dc .
For fixed r > 1, let
S(v) = −R(v)|r 7→r−1, W ′
[
c d
b a
∣∣∣∣ v] = −W
[
c d
b a
∣∣∣∣ v]
∣∣∣∣∣
r 7→r−1
,
and t′∗(v)ba is the dual intertwining vector of t′(v)ab . Here,
t′(v)ab := f ′(v)t(v; ε, r − 1)ab ,
with
f ′(v) =
x
− v2
n(r−1)−
(r+n−2)v
n(r−1) −
(n−1)(3r+n−5)
6n(r−1)
n
√
−(x2r−2;x2r−2)∞
× (x2z−1;x2n, x2r−2)∞(x2r+2n−2z;x2n, x2r−2)∞
(z−1;x2n, x2r−2)∞(x2r+2n−4z;x2n, x2r−2)∞
, (2.6)
and z = x2v. Then we have
t′∗(v1)bc ⊗ t′∗(v2)abS(v1 − v2) =
∑
d
W ′
[
c d
b a
∣∣∣∣ v1 − v2
]
t′∗(v1)ad ⊗ t′∗(v2)dc .
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 5
3 Vertex operator algebra
3.1 Vertex operators for the (Z/nZ)-symmetric model
Let H(i) be the C-vector space spanned by the half-infinite pure tensor vectors of the forms
εµ1 ⊗ εµ2 ⊗ εµ3 ⊗ · · · with µj ∈ Z/nZ, µj = i+ 1− j (mod n) for j � 0.
The type I vertex operator Φµ(v) can be defined as a half-infinite transfer matrix. The opera-
tor Φµ(v) is an intertwiner from H(i) to H(i+1), satisfying the following commutation relation:
Φµ(v1)Φν(v2) =
∑
µ′,ν′
R(v1 − v2)µνµ′ν′Φ
ν′(v2)Φµ′(v1).
When we consider an operator related to ‘creation-annihilation’ process, we need another
type of vertex operators, the type II vertex operators that satisfy the following commutation
relations:
Ψ∗ν(v2)Ψ∗µ(v1) =
∑
µ′,ν′
Ψ∗µ′(v1)Ψ∗ν′(v2)S(v1 − v2)µ
′ν′
µν ,
Φµ(v1)Ψ∗ν(v2) = χ(v1 − v2)Ψ∗ν(v2)Φµ(v1).
Let
ρ(i) = x2nHCTM : H(i) → H(i),
where HCTM is the CTM Hamiltonian defined as follows:
HCTM(µ1, µ2, µ3, . . . ) =
1
n
∞∑
j=1
jHv(µj , µj+1),
Hv(µ, ν) =
{
µ− ν − 1 if 0 6 ν < µ 6 n− 1,
n− 1 + µ− ν if 0 6 µ 6 ν 6 n− 1.
(3.1)
Then we have the homogeneity relations
Φµ(v)ρ(i) = ρ(i+1)Φµ(v − n), Ψ∗µ(v)ρ(i) = ρ(i+1)Ψ∗µ(v − n).
3.2 Vertex operators for the A
(1)
n−1 model
For k = a+ρ, l = ξ+ρ and 0 6 i 6 n−1, let H(i)
l,k be the space of admissible paths (a0, a1, a2, . . . )
such that
a0 = a, aj − aj+1 ∈ {ε̄0, ε̄1, . . . , ε̄n−1} for j = 0, 1, 2, 3, . . . ,
aj = ξ + ωi+1−j for j � 0.
The type I vertex operator Φ(v)
a+ε̄µ
a can be defined as a half-infinite transfer matrix. The
operator Φ(v)
a+ε̄µ
a is an intertwiner from H(i)
l,k to H(i+1)
l,k+ε̄µ
, satisfying the following commutation
relation:
Φ(v1)cbΦ(v2)ba =
∑
d
W
[
c d
b a
∣∣∣∣ v1 − v2
]
Φ(v2)cdΦ(v1)da.
The free field realization of Φ(v2)ba was constructed in [15]. See Section 4.2.
6 Y.-H. Quano
The type II vertex operators should satisfy the following commutation relations:
Ψ∗(v2)ξcξdΨ
∗(v1)ξdξa =
∑
ξb
Ψ∗(v1)ξcξbΨ
∗(v2)ξbξaW
′
[
ξc ξd
ξb ξa
∣∣∣∣ v1 − v2
]
,
Φ(v1)a
′
a Ψ∗(v2)ξ
′
ξ = χ(v1 − v2)Ψ∗(v2)ξ
′
ξ Φ(v1)a
′
a .
Let
ρ
(i)
l,k = Gax
2nH
(i)
l,k , Ga =
∏
06µ<ν6n−1
[aµν ],
where H
(i)
l,k is the CTM Hamiltonian of A
(1)
n−1 model in regime III is given as follows:
H
(i)
l,k (a0, a1, a2, . . . ) =
1
n
∞∑
j=1
jHf (aj−1, aj , aj+1),
Hf (a+ ε̄µ + ε̄ν , a+ ε̄µ, a) = Hv(ν, µ),
and Hv(ν, µ) is the same one as (3.1). Then we have the homogeneity relations
Φ(v)a
′
a
ρ
(i)
a+ρ,l
Ga
=
ρ
(i+1)
a′+ρ,l
Ga′
Φ(v − n)a
′
a , Ψ∗(v)ξ
′
ξ ρ
(i)
k,ξ+ρ = ρ
(i+1)
k,ξ′+ρΨ
∗(v − n)ξ
′
ξ .
The free field realization of Ψ∗(v)ξ
′
ξ was constructed in [16]. See Section 4.3.
3.3 Tail operators and commutation relations
In [1] we introduced the intertwining operators between H(i) and H(i)
l,k (k = l + ωi (mod Q)):
T (u)ξa0 =
∞∏
j=0
tµj (−u)
aj
aj+1 : H(i) → H(i)
l,k,
T (u)ξa0 =
∞∏
j=0
t∗µj (−u)
aj+1
aj : H(i)
l,k → H
(i),
which satisfy
ρ(i) =
(
(x2r−2;x2r−2)∞
(x2r;x2r)∞
)(n−1)(n−2)/2
1
G′ξ
∑
k≡l+ωi
(mod Q)
T (u)aξρ
(i)
l,kT (u)aξ. (3.2)
In order to obtain the form factors of the (Z/nZ)-symmetric model, we need the free field
representations of the tail operator which is offdiagonal with respect to the boundary conditions:
Λ(u)ξ
′a′
ξ a = T (u)ξ
′a′T (u)ξ a : H(i)
l,k → H
(i)
l′k′ , (3.3)
where k = a+ ρ, l = ξ + ρ, k′ = a′ + ρ, and l′ = ξ′ + ρ. Let
L
[
a′0 a′1
a0 a1
∣∣∣∣u] :=
n−1∑
µ=0
t∗µ(−u)a1a0t
µ(−u)
a′0
a′1
.
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 7
Then we have
Λ(u)
ξ′a′0
ξ a0
=
∞∏
j=0
L
[
a′j a′j+1
aj aj+1
∣∣∣∣u] .
From the invertibility of the intertwining vector and its dual vector, we have
Λ(u0)ξ
′ a
ξ a = δξ
′
ξ . (3.4)
Note that the tail operator (3.3) satisfies the following intertwining relations [1, 2]:
Λ(u)ξ
′c
ξ bΦ(v)ba =
∑
d
L
[
c d
b a
∣∣∣∣u− v]Φ(v)cdΛ(u)ξ
′d
ξ a , (3.5)
Ψ∗(v)ξcξdΛ(u)ξd a
′
ξa a
=
∑
ξb
L′
[
ξc ξd
ξb ξa
∣∣∣∣u+ ∆u− v
]
Λ(u)ξc a
′
ξb a
Ψ∗(v)ξbξa , (3.6)
where
L′
[
ξc ξd
ξb ξa
∣∣∣∣u] = L
[
ξc ξd
ξb ξa
∣∣∣∣u]∣∣∣∣
r 7→r−1
.
We should find a representation of Λ(u)ξ
′a′
ξ a and fix the constant ∆u that solves (3.5) and (3.6).
4 Free filed realization
4.1 Bosons
In [17, 18] the bosons Bj
m (1 6 j 6 n−1,m ∈ Z\{0}) relevant to elliptic algebra were introduced.
For α, β ∈ h∗ we denote the zero mode operators by Pα, Qβ. Concerning commutation relations
among these operators see [17, 18, 2].
We will deal with the bosonic Fock spaces Fl,k, (l, k ∈ h∗) generated by Bj
−m(m > 0) over
the vacuum vectors |l, k〉 :
Fl,k = C[{Bj
−1, B
j
−2, . . . }16j6n]|l, k〉,
where
|l, k〉 = exp
(√
−1(β1Qk + β2Ql)
)
|0, 0〉,
and
t2 − β0t− 1 = (t− β1)(t− β2), β0 =
1√
r(r − 1)
, β1 < β2.
4.2 Type I vertex operators
Let us define the basic operators for j = 1, . . . , n− 1
U−αj (v) = z
r−1
r : exp
−β1
(√
−1Qαj + Pαj log z)
)
+
∑
m6=0
Bj
m −Bj+1
m
m
(xjz)−m
:,
Uωj (v) = z
r−1
2r
j(n−j)
n : exp
β1
(√
−1Qωj + Pωj log z)
)
−
∑
m 6=0
1
m
j∑
k=1
x(j−2k+1)mBk
mz
−m
:,
8 Y.-H. Quano
where β1 = −
√
r−1
r and z = x2v as usual. The normal product operation places Pα’s to the
right of Qβ’s, as well as Bm’s (m > 0) to the right of B−m’s. For some useful OPE formulae
and commutation relations, see Appendix A.
In what follows we set
πµ =
√
r(r − 1)Pε̄µ , πµν = πµ − πν = rLµν − (r − 1)Kµν .
The operators Kµν , Lµν and πµν act on Fl,k as scalars 〈εµ− εν , k〉, 〈εµ− εν , l〉 and 〈εµ− εν , rl−
(r − 1)k〉, respectively. In what follows we often use the symbols
GK =
∏
06µ<ν6n−1
[Kµν ], G′L =
∏
06µ<ν6n−1
[Lµν ]′.
For 0 6 µ 6 n − 1 the type I vertex operator Φ(v)
a+ε̄µ
a can be expressed in terms of Uωj (v)
and U−αj (v) on the bosonic Fock space Fl,a+ρ. The explicit expression of Φ(v)
a+ε̄µ
a can found
in [15].
4.3 Type II vertex operators
Let us define the basic operators for j = 1, . . . , n− 1
V−αj (v) = (−z)
r
r−1 : exp
−β2
(√
−1Qαj + Pαj log(−z)
)
−
∑
m6=0
Ajm −Aj+1
m
m
(xjz)−m
:,
Vωj (v) = (−z)
r
2(r−1)
j(n−j)
n
× : exp
β2
(√
−1Qωj + Pωj log(−z)
)
+
∑
m6=0
1
m
j∑
k=1
x(j−2k+1)mAkmz
−m
:,
where β2 =
√
r
r−1 and z = x2v, and Ajm = [rm]x
[(r−1)m]x
Bj
m. For some useful OPE formulae and
commutation relations, see Appendix A.
For 0 6 µ 6 n− 1 the type II vertex operator Ψ∗(v)
ξ+ε̄µ
ξ can be expressed in terms of Vωj (v)
and V−αj (v) on the bosonic Fock space Fξ+ρ,k. The explicit expression of Ψ∗(v)
ξ+ε̄µ
ξ can found
in [16].
4.4 Free field realization of tail operators
In order to construct free field realization of the tail operators, we also need another type of
basic operators:
W−αj (v) = ((−1)rz)
1
r(r−1)
× : exp
−β0
(√
−1Qαj + Pαj log(−1)rz)
)
−
∑
m6=0
Ojm −Oj+1
m
m
(xjz)−m
:,
where β0 = β1 + β2 = 1√
r(r−1)
, (−1)r := exp(π
√
−1r) and Ojm = [m]x
[(r−1)m]x
Bj
m. Concerning
useful OPE formulae and commutation relations, see Appendix A.
We cite the results on the free field realization of tail operators. In [1] we obtained the free
field representation of Λ(u)ξ a
′
ξ a satisfying (3.5) for ξ′ = ξ:
Λ(u)
ξa−ε̄µ
ξa−ε̄ν = GK
∮ ν∏
j=µ+1
dzj
2π
√
−1zj
U−αµ+1(vµ+1) · · ·U−αν (vν)
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 9
×
ν−1∏
j=µ
(−1)Ljν−Kjνf(vj+1 − vj , πjν)G−1
K , (4.1)
where vµ = u and µ < ν. In [2] we obtained the free field representation of Λ(v)
ξ+ε̄n−1a+ε̄n−1
ξ+ε̄µa+ε̄n−2
satisfying (3.6) as follows:
Λ(u)
ξ+ε̄n−1a+ε̄n−1
ξ+ε̄µa+ε̄n−2
=
(−1)n−µ[an−2n−1]
(x−1 − x)(x2r;x2r)3
∞
[ξµn−1 − 1]′
[1]′
GKG
′
L
−1
×
∮
C′
n−2∏
j=µ+1
dzj
2π
√
−1zj
W−αn−1
(
u− r−1
2
)
V−αn−2(vn−2) · · ·V−αµ+1(vµ+1)
×
n−2∏
j=µ+1
(−1)Lµj−Kµjf∗(vj − vj+1, πµj)G
−1
K G′L, (4.2)
for 0 6 µ 6 n − 2 with ∆u = −n−1
2 and vn−1 = u. Concerning other types of tail operators
Λ(u)ξa
′
ξa , the expressions of the free field representation can be found in [1, 2].
4.5 Free field realization of CTM Hamiltonian
Let
HF =
∞∑
m=1
[rm]x
[(r − 1)m]x
n−1∑
j=1
j∑
k=1
x(2k−2j−1)mBk
−m(Bj
m −Bj+1
m ) +
1
2
n−1∑
j=1
PωjPαj
=
∞∑
m=1
[rm]x
[(r − 1)m]x
n−1∑
j=1
j∑
k=1
x(2j−2k−1)m(Bj
−m −B
j+1
−m )Bk
m +
1
2
n−1∑
j=1
PωjPαj (4.3)
be the CTM Hamiltonian on the Fock space Fl,k [19]. Then we have the homogeneity relation
φµ(z)qHF = qHF φµ
(
q−1z
)
,
and the trace formula
trFl,k
(
x2nHFGa
)
=
xn|β1k+β2l|2
(x2n;x2n)n−1
∞
Ga.
Let ρ
(i)
l,k = Gax
2nHF . Then the relation (3.2) holds. We thus indentify HF with free field
representations of H
(i)
l,k , the CTM Hamiltonian of A
(1)
n−1 model in regime III.
5 Form factors for n = 2
In this section we would like to find explicit expressions of form factors for n = 2 case, i.e., the
eight-vertex model form factors. Here, we adopt the convention that the components 0 and 1
for n = 2 are denoted by + and −. Form factors of the eight-vertex model are defined as matrix
elements of some local operators. For simplicity, we choose σz as a local operator:
σz = E
(1)
++ − E
(1)
−−,
where E
(j)
µµ′ is the matrix unit on the j-th site. The free field representation of σz is given by
σ̂z =
∑
ε=±
εΦ∗ε(u)Φε(u).
10 Y.-H. Quano
Here, Φ∗ε(u) is the dual type I vertex operator, whose free filed representation can be found
in [1, 2].
The corresponding form factors with 2m ‘charged’ particles are given by
F (i)
m (σz;u1, . . . , u2m)ν1···ν2m =
1
χ(i)
TrH(i)
(
Ψ∗ν1(u1) · · ·Ψ∗ν2m(u2m)σ̂zρ(i)
)
, (5.1)
where
χ(i) = TrH(i)ρ(i) =
(x4;x4)∞
(x2;x2)∞
.
In this section we denote the spectral parameters by zj = x2uj , and denote integral variables by
wa = x2va .
By using the vertex-face transformation, we can rewrite (5.1) as follows:
F (i)
m (σz;u1, . . . , u2m)ν1···ν2m =
1
χ(i)
∑
l1,...,l2m
t′∗ν1
(
u1 − u0 + 1
2
) l1
l · · · t
′∗
ν2m
(
u2m − u0 + 1
2
) l2m
l2m−1
×
∑
k≡l+i(mod 2)
∑
ε=±
ε
∑
k1=k±1
∑
k2=k1±1
t∗ε(u− u0)kk1t
ε(u− u0)k1k2
× TrH(i)
l,k
(
Ψ∗(u1)ll1 · · ·Ψ
∗(u2m)
l2m−1
l2m
Φ∗(u)kk1Φ(u)k1k2Λ(u0)l2mk2l k
[k]x4HF
[l]′
)
,
where HF is the CTM Hamiltonian defined by (4.3).
Let
F (i)
m (σz;u1, . . . , u2m)ll1···l2m =
1
χ(i)
∑
k≡l+i (mod 2)
∑
ε=±
ε
∑
k1=k±1
∑
k2=k1±1
t∗ε(u− u0)kk1t
ε(u− u0)k1k2
× TrH(i)
l,k
(
Ψ∗(u1)ll1 · · ·Ψ
∗(u2m)
l2m−1
l2m
Φ∗(u)kk1Φ(u)k1k2Λ(u0)l2mk2l k
[k]x4HF
[l]′
)
. (5.2)
Then we have
F (i)
m (σz;u1, . . . , u2m)ll1···l2m =
∑
ν1,...,ν2m
F (i)
m (σz;u1, . . . , u2m)ν1···ν2m
× t′ν1
(
u1 − u0 + 1
2
)
l
l1 · · · t
′ν2m (u2m − u0 + 1
2
) l2m−1
l2m
.
For simplicity, let lj = l−j for 1 6 j 6 2m. Then from the relation (3.4), Λ(u0)l2mk2l k vanishes
unless k2 = k − 2. Thus, the sum over k1 and k2 on (5.2) reduces to only one non-vanishing
term. Furthermore, we note the formula∑
ε=±
εt∗ε(u− u0)kk−1t
ε(u− u0)k−1
k−2 = (−1)1−i {0}{u− u0 − 1 + k}
[u− u0][k − 1]
.
Here, we use k − l ≡ i (mod 2). The sum with respect to k for the trace over the zero-modes
parts can be calculated as follows:
∑
k≡l+i (mod 2)
{u− u0 − 1 + k}
2m∏
j=1
(−zj)
rl
2(r−1)
− k
2 (x−1z)−l+
(r−1)k
r
m−1∏
a=1
(−wa)−
rl
r−1
+k
×
(
(−1)rx−r+1z0
)− l
r−1
+ k
r x
rl2
r−1
−2kl+
(r−1)k2
r
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 11
= x
l2
r−1
+l
(
2+ r
r−1
2n∑
j=1
uj−2u− 2r
r−1
m−1∑
a=1
va− 2u0
r−1
)
x
1
r
(u−u0−1)2−(u−u0−1)
∑
k≡l+i (mod 2)
x(k−l)2
×
∑
n∈Z
xrn(n−1)x2(u−u0−1+k)nx
k
(
2
m−1∑
a=1
va+2u−
2m∑
j=1
uj−3
)
=
(−1)1−i
2
x
1
r
(u−u0−1)2− 1
r−1
(
u0+
m−1∑
a=1
va− 1
2
2m∑
j=1
uj
)2
−
(
m−1∑
a=1
va+u− 1
2
2m∑
j=1
uj−1
)2
× Z(i)
m (l, u, u0, uj , va),
where
Z(i)
m (l, u, u0, uj , va) =
l − u0 −
m−1∑
a=1
va + 1
2
2m∑
j=1
uj
′ m−1∑
a=1
va + u− 1
2
2m∑
j=1
uj
1
+ (−1)1−i
l − u0 −
m−1∑
a=1
va + 1
2
2m∑
j=1
uj
′
m−1∑
a=1
va + u− 1
2
2m∑
j=1
uj
1
.
Thus, F
(i)
m (σz;u1, . . . , u2m)ll−1···l−2m can be obtained as follows:
(−1)m−1β−1
m F (i)
m (σz;u1, . . . , u2m)ll−1···l−2m
=
∏
j<j′
(−zj)
r
2(r−1)Fψ∗ψ∗(zj′/zj)
2m∏
j=1
(−zj)−
1
r−1x
(uj−u0+1/2)2
4(r−1)
+
r(uj−u0+1/2)
2(r−1)
+ 1
4
x−1z(xzj/z;x4)∞(x3z/zj ;x4)∞
f ′(uj − u0 + 1
2)
×
∮
C
m−1∏
a=1
dwa
2π
√
−1wa
Z(i)
m (l, u, u0, uj , va)
∏
a<b
(−wb)
2r
r−1 [va − vb]′[va − vb]1x−
r
r−1
(va−vb−1)2
×
m−1∏
a=1
x−2z2x−(va−u)2+va−u[va − u]1(−wa)
2
r−1x−
1
r−1
(u0−va−1)2+u0−va−1[va − u0 + l −m]′
×
m−1∏
a=1
2m∏
j=1
(−zj)−
r
r−1
(x2r−1wa/zj ;x
4, x2r−2)∞(x2r+3zj/wa;x
4, x2r−2)∞
(x−1wa/zj ;x4, x2r−2)∞(x3zj/wa;x4, x2r−2)∞
×(x−1z)
2
r
2
x
− r+2
r
(u0−u)− 1
r
− 1
r−1
(
u0+
m−1∑
a=1
va− 1
2
2m∑
j=1
uj
)2
−
(
m−1∑
a=1
va+u− 1
2
2m∑
j=1
uj−1
)2
, (5.3)
where f ′(v) is defined by (2.6) for n = 2, a scalar function Fψ∗ψ∗(z) and a scalar βm are
Fψ∗ψ∗(z) =
(z;x4, x4, x2r−2)∞(x4z−1;x4, x4, x2r−2)∞
(x2z;x4, x4, x2r−2)∞(x6z−1;x4, x4, x2r−2)∞
× (x2r+2z;x4, x4, x2r−2)∞(x2r+6z−1;x4, x4, x2r−2)∞
(x2rz;x4, x4, x2r−2)∞(x2r+4z−1;x4, x4, x2r−2)∞
,
and
βm =
x−
r−1
4r {0}[m− 1]′!(x−2z)
r−1
2r (x2, x4)2
∞(x2;x2r)∞(x2r+1;x2r−2)∞
(m− 1)![1]′m(x−1 − x)g1(x2)(x2r;x2r)2
∞(x2r+1;x2r)∞
× (x2;x2)m−1
∞ (x2r;x2r−2)m−1
∞
(x4;x4, x4, x2r−2)m∞(x2r+6;x4, x4, x2r−2)m∞
(x6;x4, x4, x2r−2)m∞(x2r+4;x4, x4, x2r−2)m∞
,
12 Y.-H. Quano
with
[m]′! =
m∏
p=1
[p]′.
On (5.3), the integral contour C should be chosen such that all integral variables wa lie in the
convergence domain x3|zj | < |wa| < x|zj |.
Gathering phase factors on (5.3), we have e
−π
√
−1 3mr
2(r−1) . Redefining f ′(v) by a scalar factor,
we thus obtain the equality:
∑
ν1,...,ν2m
F (i)
m (σz;u1, . . . , u2m)ν1···ν2m
2m∏
j=1
ϑ
[
0
bνj
](
uj−u0+ 1
2
+l−j+1
2(r−1) ; π
√
−1
2ε(r−1)
)
= βm
∏
j<j′
z
r
2(r−1)
j Fψ∗ψ∗(zj′/zj)
2m∏
j=1
z
− 1
r−1
j
x−1z(xzj/z;x4)∞(x3z/zj ;x4)∞
×
∮
C
m−1∏
a=1
dwa
2π
√
−1wa
Z(i)
m (l, u, u0, uj , va)
∏
a<b
w
2r
r−1
b [va − vb]′[va − vb]1x−
r
r−1
(va−vb−1)2
×
m−1∏
a=1
x−2z2x−(va−u)2+va−u[va − u]1w
2
r−1
a x−
1
r−1
(u0−va−1)2+u0−va−1[va − u0 + l −m]′
×
m−1∏
a=1
2m∏
j=1
z
− r
r−1
j
(x2r−1wa/zj ;x
4, x2r−2)∞(x2r+3zj/wa;x
4, x2r−2)∞
(x−1wa/zj ;x4, x2r−2)∞(x3zj/wa;x4, x2r−2)∞
×(x−1z)
2
r
2
x
− r+2
r
(u0−u)− 1
r
− 1
r−1
(
u0+
m−1∑
a=1
va− 1
2
2m∑
j=1
uj
)2
−
(
m−1∑
a=1
va+u− 1
2
2m∑
j=1
uj−1
)2
, (5.4)
where
bν =
{
0 (ν = +),
1
2 (ν = −).
By comparing the transformation properties with respect to l for both sides on (5.4), we conclude
that F
(i)
m (σz;u1, . . . , u2m)ν1···ν2m are independent of l, and also that
F
(i)
m (σz;u1, . . . , u2m)ν1···ν2m = 0 unless
1
2
2m∑
j=1
νj ≡ 0 (mod 2),
as expected.
When m = 1, we have
F (i)(σz;u1, u2)ν1ν2 = δν1+ν2,0C
(z)z
r
2(r−1)
1
2∏
j=1
z
− 1
r−1
j
x−1z(xzj/z;x4)∞(x3z/zj ;x4)∞
× (x−1z)
2
r
4
x−
r+2
r
(u0−u)− 1
r
− 1
r−1
(u0−(u1+u2)/2)2−(u−(u1+u2)/2−1)2
× Fψ∗ψ∗(z2/z1)
(
ν1
[u− u1+u2
2 ]1
[u2−u1−1
2 ]′
+ (−1)1−i {u−
u1+u2
2 }1
{u2−u1−1
2 }′
)
, (5.5)
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 13
where C(z) is a constant. This is a same result obtained by Lashkevich in [8], up to a scalar
factor1.
Next, let us choose σx as a local operator:
σz = E
(1)
+− + E
(1)
−+.
Then the relation for F
(i)
m (σx;u1, . . . , u2m)ν1···ν2m reduces to (5.4) with Z
(i)
m (l, u, u0, uj , va) re-
placed by
X(i)
m (l, u, u0, uj , va) =
l − u0 −
m−1∑
a=1
va + 1
2
2m∑
j=1
uj
′
m−1∑
a=1
va + u− 1
2
2m∑
j=1
uj
1
+ (−1)1−i
l − u0 −
m−1∑
a=1
va + 1
2
2m∑
j=1
uj
′ m−1∑
a=1
va + u− 1
2
2m∑
j=1
uj
1
,
and with {0} in βm replaced by [[0]], respectively. Here,
[[v]] := x
v2
r Θx2r(x
2v+r), [[v]]′ := [[v]]|r 7→r−1, [[v]]1 := [[v]]|r 7→1,
{{v}} := x
v2
r Θx2r(−x2v+r), {{v}}′ := {{v}}|r 7→r−1, {{v}}1 := {{v}}|r 7→1.
The transformation properties with respect to l implies that F
(i)
m (σx;u1, . . . , u2m)ν1···ν2m are
independent of l, and also that
F
(i)
m (σx;u1, . . . , u2m)ν1···ν2m = 0 unless
1
2
2m∑
j=1
νj ≡ 1 (mod 2),
as expected. Furthermore, 2-point form factors for σx-operator can be obtained as follows:
F (i)(σx;u1, u2)ν1ν2 = δν1 ν2C
(x)z
r
2(r−1)
1
2∏
j=1
z
− 1
r−1
j
x−1z(xzj/z;x4)∞(x3z/zj ;x4)∞
× (x−1z)
2
r
4
x−
r+2
r
(u0−u)− 1
r
− 1
r−1
(u0−(u1+u2)/2)2−(u−(u1+u2)/2−1)2
× Fψ∗ψ∗(z2/z1)
(
ν1
{{u− u1+u2
2 }}1
[[u2−u1−1
2 ]]′
+ (−1)1−i [[u−
u1+u2
2 ]]1
{{u2−u1−1
2 }}′
)
, (5.6)
where C(x) is a constant. The expressions (5.5) and (5.6) are essentially same as the results
obtained by Lukyanov and Terras [20]2.
6 Concluding remarks
In this paper we present a vertex operator approach for form factors of the (Z/nZ)-symmetric
model. For that purpose we constructed the free field representations of the tail operators Λξ
′a′
ξ a ,
1This scalar factor results from the difference between the present normalization of the type II vertex operators
and that used in [8].
2Strictly speaking, we consider the parameterization of the coupling constants |Jz| > Jx > Jy while Lukyanov
and Terras [20] considered that of Jx > Jy > |Jz|. Thus, the present results (5.5) and (5.6) correspond to their
results of the 2-point form factors for σx-operator and σy-operator, respectively. Furthermore, we note that their
rapidity θj can be obtained from our spectral parameter uj by a constant shift. After such substitution, we claim
that our results (5.5) and (5.6) agree with their corresponding results in [20].
14 Y.-H. Quano
the nonlocal operators which relate the physical quantities of the (Z/nZ)-symmetric model and
the A
(1)
n−1 model. As a result, we can obtain the integral formulae for form factors of the (Z/nZ)-
symmetric model, in principle.
Our approach is based on some assumptions. We assumed that the vertex operator al-
gebra defined by (3.2) and (3.5), (3.6) correctly describes the intertwining relation between the
(Z/nZ)-symmetric model and the A
(1)
n−1 model. We also assumed that the free field representa-
tions (4.1), (4.2) provide relevant representations of the vertex operator algebra.
As a consistency check of our bosonization scheme, we presented the integral formulae for form
factors which are related to the σz-operator and σx-operator in the eight-vertex model, i.e., the
(Z/2Z)-symmetric model. The expressions (5.3) and (5.4) for σz form factors and σx analogues
remind us of the determinant structure of sine-Gordon form factors found by Smirnov [21].
In Smirnov’s approach form factors in integrable models can be obtained by solving matrix
Riemann-Hilbert problems. We wish to find form factors formulae in the eight-vertex model on
the basis of Smirnov’s approach in a separate paper.
A OPE formulae and commutation relations
In this paper we use some different definitions of the basic bosons from the one used in [2].
Accordingly, some formulae listed in Appendix B of [2] should be changed. Here we list such
formulae. Concerning unchanged formulae see [2]. In what follows we denote z = x2v, z′ = x2v′ .
First, useful OPE formulae are:
Vω1(v)Vωj (v
′) = (−z)
r
r−1
n−j
n g∗j (z
′/z) : Vω1(v)Vωj (v
′) :,
Vωj (v)Vω1(v′) = (−z)
r
r−1
n−j
n g∗j (z
′/z) : Vωj (v)Vω1(v′) :,
Vωj (v)V−αj (v
′) = (−z)−
r
r−1
(x2r−1z′/z;x2r−2)∞
(x−1z′/z;x2r−2)∞
: Vωj (v)V−αj (v
′) :,
V−αj (v)Vωj (v
′) = (−z)−
r
r−1
(x2r−1z′/z;x2r−2)∞
(x−1z′/z;x2r−2)∞
: V−αj (v)Vωj (v
′) :,
V−αj (v)V−αj±1(v′) = (−z)−
r
r−1
(x2r−1z′/z;x2r−2)∞
(x−1z′/z;x2r−2)∞
: V−αj (v)V−αj±1(v′) :,
V−αj (v)V−αj (v
′) = (−z)
2r
r−1
(
1− z′
z
)
(x−2z′/z;x2r−2)∞
(x2rz′/z;x2r−2)∞
: V−αj (v)V−αj (v
′) :,
Vωj (v)Uωj (v
′) = (−z)−
j(n−j)
n ρj(z
′/z) : Vω1(v)Uωj (v
′) :,
Uωj (v)Vωj (v
′) = z−
j(n−j)
n ρj(z
′/z) : Uωj (v)Vωj (v
′) :,
Vωj (v)U−αj (v
′) = −z
(
1− z′
z
)
: Vωj (v)U−αj (v
′) := U−αj (v
′)Vωj (v),
Uωj (v)V−αj (v
′) = z
(
1− z′
z
)
: Uωj (v)V−αj (v
′) := V−αj (v
′)Uωj (v),
V−αj (v)U−αj±1(v′) = −z
(
1− z′
z
)
: V−αj (v)U−αj±1(v′) := U−αj±1(v′)V−αj (v),
V−αj (v)U−αj (v
′) =
: V−αj (v)U−αj (v
′) :
z2(1− xz′
z )(1− x−1z′
z )
, (A.1)
U−αj (v)V−αj (v
′) =
: U−αj (v)V−αj (v
′) :
z2(1− xz′
z )(1− x−1z′
z )
, (A.2)
Form Factors of Belavin’s (Z/nZ)-Symmetric Model and Its Application 15
where g∗j (z) and ρj(z) are defined by (2.2) and (2.3), respectively. From (A.1) and (A.2), we
obtain the following commutation relations:
[V−αj (v), U−αj (v
′)] =
δ( z
xz′ )− δ(
z′
xz )
(x− x−1)zz′
: V−αj (v)U−αj (v
′) :,
where the δ-function is defined by the following formal power series
δ(z) =
∑
n∈Z
zn.
Finally, we list the OPE formulae for W−αj (v) and other basic operators:
W−αj (v)V−αj±1(v′) = −(−z)−
1
r−1
(xrz′/z;x2r−2)∞
(xr−2z′/z;x2r−2)∞
: W−αj (v)V−αj±1(v′) :,
V−αj±1(v)W−αj (v
′) = (−z)−
1
r−1
(xrz′/z;x2r−2)∞
(xr−2z′/z;x2r−2)∞
: V−αj±1(v)W−αj (v
′) :,
Vωj (v)W−αj (v
′) = (−z)−
1
r−1
(xrz′/z;x2r−2)∞
(xr−2z′/z;x2r−2)∞
: Vωj (v)W−αj (v
′) :,
W−αj (v)Vωj (v
′) = −(−z)−
1
r−1
(xrz′/z;x2r−2)∞
(xr−2z′/z;x2r−2)∞
: W−αj (v)Vωj (v
′) :,
From these, we obtain
W−αj
(
v + r
2
)
V−αj±1(v) = 0 = V−αj±1(v)W−αj
(
v − r
2
)
,
W−αj
(
v + r
2
)
Vωj (v) = 0 = Vωj (v)W−αj
(
v − r
2
)
.
Acknowledgements
We would like to thank T. Deguchi, R. Inoue, H. Konno, Y. Takeyama and R. Weston for
discussion and their interests in the present work. We would also like to thank S. Lukyanov
for useful information. This paper is partly based on a talk given in International Workshop
RAQIS’10, Recent Advances in Quantum Integrable Systems, held at LAPTH, Annecy-le-Vieux,
France, June 15–18, 2010. We would like to thank L. Frappat and É. Ragoucy for organizing
the conference.
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1 Introduction
2 Basic definitions
2.1 Theta functions
2.2 Belavin's (Z/nZ)-symmetric model
2.3 The A(1)n-1 model
2.4 Vertex-face correspondence
3 Vertex operator algebra
3.1 Vertex operators for the (Z/nZ)-symmetric model
3.2 Vertex operators for the A(1)n-1 model
3.3 Tail operators and commutation relations
4 Free filed realization
4.1 Bosons
4.2 Type I vertex operators
4.3 Type II vertex operators
4.4 Free field realization of tail operators
4.5 Free field realization of CTM Hamiltonian
5 Form factors for n=2
6 Concluding remarks
A OPE formulae and commutation relations
References
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