A Vertex Operator Approach for Form Factors of Belavin’s ( Z /n Z )-Symmetric Model

A Vertex Operator Approach for Form Factors of Belavin’s ( Z /n Z )-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: [email protected]

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 theσ^z andσ^xoperators 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 ann-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 U_q,p(slb_N) relevant to the (Z/nZ)-symmetric model provides the Drinfeld realization of the face type elliptic quantum group B_q,λ(slb_N) 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 Section3we summarize the vertex operator algebras relevant to the (Z/nZ)-symmetric model and theA⁽¹⁾_n−1 model [1,2]. In Section4we 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 athttp://www.emis.de/journals/SIGMA/RAQIS2010.html

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 Section5 we calculate 2m-point form factors of the σ^z-operator and σ^x-operator in the eight-vertex model, as simple application for n= 2. In Section6we give some concluding remarks. Useful operator product expansion (OPE) formulae and commutation relations for basic bosons are given in Appendix A.

2 Basic def initions

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 (Imτ >0) are defined as follows:

ϑ a

b

(v;τ) := X

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^v

2 r−v

Θ_x^2r(x^2v), [v]⁰ := [v]|_r7→r−1, [v]1 := [v]|_r7→1, {v}:=x^v

2

r −vΘ_x^2r(−x^2v), {v}⁰ :={v}|_r7→r−1, {v}₁:={v}|_r7→1, where

Θq(z) = (z;q)∞ qz⁻¹;q

∞(q;q)∞= X

m∈Z

q^m(m−1)/2(−z)^m, (z;q₁, . . . , q_m)∞= Y

i1,...,im>0

1−zqⁱ₁¹· · ·q_m^im . Note that

ϑ 1/2

−1/2 v r,π√

−1 r

= rr

π exp

−r 4

[v], ϑ

0 1/2

v r,π√

−1 r

= rr

π exp

−r 4

{v}, where x=e⁻ ( >0).

For later conveniences we also introduce the following symbols:

r_j(v) =z^{r−1r n−jn} g_j(z⁻¹)

g_j(z) , g_j(z) = {x^{2n+2r−j−1}z}{x^j+1z}

{x^2n−j+1z}{x^2r+j−1z}, (2.1)

r^∗_j(v) =z^{r−1r n−jn} g_j^∗(z⁻¹)

g^∗_j(z) , g_j^∗(z) = {x^{2n+2r−j−1}z}⁰{x^j−1z}⁰

{x^2n−j−1z}⁰{x^2r+j−1z}⁰, (2.2) χ_j(v) = (−z)^−j(n−j)n ρ_j(z⁻¹)

ρj(z) , ρ_j(z) = (x^2j+1z;x², x²ⁿ)∞(x^2n−2j+1z;x², x²ⁿ)∞

(xz;x², x²ⁿ)∞(x²ⁿ⁺¹z;x², x²ⁿ)∞

, (2.3) where z=x^2v, 16j6n and

{z}= (z;x^2r, x²ⁿ)∞, {z}⁰ = (z;x^2r−2, x²ⁿ)∞.

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+¹₂ −w]

[v−¹₂] , h(v) = [v−1]

[v+ 1], f^∗(v, w) = [v−¹₂ +w]⁰

[v+¹₂]⁰ , h^∗(v) = [v+ 1]⁰ [v−1]⁰. 2.2 Belavin’s (Z/nZ)-symmetric model

Let V = Cⁿ and {ε_µ}_06µ6n−1 be the standard orthonormal basis with the inner product hε_µ, ε_νi =δ_µν. 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)^ik_jl = 0, unless i+k=j+l, modn, (ii) R(v)^i+pk+p_j+pl+p =R(v)^ik_jl, ∀i, j, k, l, p∈Z/nZ.

The definition of theR-matrix in the principal regime can be found in [2]. The presentR-matrix has three parameters v,andr, which lie in the following region:

>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 latticeL^∗, the dual lattice ofL, such that the state variables take the values of the dual space of Cartan subalgebra h^∗ ofA⁽¹⁾_n−1:

h^∗=

n−1

M

µ=0

Cωµ,

where ωµ:=

µ−1

X

ν=0

¯

εν, ε¯µ=εµ− 1 n

n−1

X

µ=0

εµ.

The weight lattice P and the root latticeQof A⁽¹⁾_n−1 are usually defined. For a∈h^∗, we set a_µν = ¯a_µ−¯a_ν, ¯a_µ=ha+ρ, ε_µi=ha+ρ,ε¯_µi, ρ=

n−1

X

µ=1

ω_µ.

An ordered pair (a, b)∈h^∗2 is called admissible ifb=a+ ¯ε_µ, for a certainµ(06µ6n−1).

For (a, b, c, d)∈h^∗4, letW

c d b a v

be the Boltzmann weight of theA⁽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

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)^a_a−¯εµ be the intertwining vectors in Cⁿ, whose elements are expressed in terms of theta functions. As for the definitions see [2]. Then t(v)^a_a−¯εµ’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)^d_a⊗t(v2)^c_d=X

b

t(v1)^c_b⊗t(v2)^b_aW

c d b a

v1−v2

. (2.4)

Let us introduce the dual intertwining vectors satisfying

n−1

X

µ=0

t^∗_µ(v)^a_a⁰t^µ(v)^a_a00 =δ_a^a00⁰,

n−1

X

ν=0

t^µ(v)^a_a−¯ενt^∗_µ0(v)^a−¯_a ^εν =δ_µ^µ0. (2.5)

From (2.4) and (2.5), we have t^∗(v₁)^b_c⊗t^∗(v₂)^a_bR(v₁−v₂) =X

d

W

c d b a

v₁−v₂

t^∗(v₁)^a_d⊗t^∗(v₂)^d_c.

For fixedr >1, let

S(v) =−R(v)|r7→r−1, W⁰

c d b a v

=−W

c d b a v

_r7→r−1

,

and t^0∗(v)^b_ais the dual intertwining vector of t⁰(v)^a_b. Here, t⁰(v)^a_b :=f⁰(v)t(v;, r−1)^a_b,

with

f⁰(v) = x⁻

v²

n(r−1)−(r+n−2)v

n(r−1) −(n−1)(3r+n−5) 6n(r−1)

pn

−(x^2r−2;x^2r−2)∞

×(x²z⁻¹;x²ⁿ, x^2r−2)∞(x^2r+2n−2z;x²ⁿ, x^2r−2)∞

(z⁻¹;x²ⁿ, x^2r−2)∞(x^2r+2n−4z;x²ⁿ, x^2r−2)∞

, (2.6)

and z=x^2v. Then we have

t^0∗(v₁)^b_c⊗t^0∗(v₂)^a_bS(v₁−v₂) =X

d

W⁰

c d b a

v₁−v₂

t^0∗(v₁)^a_d⊗t^0∗(v₂)^d_c.

3 Vertex operator algebra

3.1 Vertex operators for the (Z/nZ)-symmetric model

Let H⁽ⁱ⁾ be theC-vector space spanned by the half-infinite pure tensor vectors of the forms εµ1 ⊗εµ2⊗εµ3 ⊗ · · · with µj ∈Z/nZ,µj =i+ 1−j (modn) for j0.

The type I vertex operator Φ^µ(v) can be defined as a half-infinite transfer matrix. The opera- tor Φ^µ(v) is an intertwiner fromH⁽ⁱ⁾ toH⁽ⁱ⁺¹⁾, satisfying the following commutation relation:

Φ^µ(v₁)Φ^ν(v₂) = X

µ⁰,ν⁰

R(v₁−v₂)^µν_µ0ν⁰Φ^ν0(v₂)Φ^µ0(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) =X

µ⁰,ν⁰

Ψ^∗_µ0(v₁)Ψ^∗_ν0(v₂)S(v₁−v₂)^µ_µν^0ν0, Φ^µ(v1)Ψ^∗_ν(v2) =χ(v1−v2)Ψ^∗_ν(v2)Φ^µ(v1).

Let

ρ⁽ⁱ⁾=x^2nHCTM :H⁽ⁱ⁾→ H⁽ⁱ⁾,

where H_CTM is the CTM Hamiltonian defined as follows:

H_CTM(µ₁, µ₂, µ₃, . . .) = 1 n

∞

X

j=1

jH_v(µ_j, µ_j+1), H_v(µ, ν) =

µ−ν−1 if 06ν < µ6n−1,

n−1 +µ−ν if 06µ6ν 6n−1. (3.1)

Then we have the homogeneity relations

Φ^µ(v)ρ⁽ⁱ⁾ =ρ⁽ⁱ⁺¹⁾Φ^µ(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

Fork=a+ρ, l=ξ+ρand 06i6n−1, letH⁽ⁱ⁾_l,kbe the space of admissible paths (a0, a1, a2, . . .) such that

a0=a, aj−aj+1 ∈ {¯ε0,ε¯1, . . . ,ε¯n−1} forj= 0,1,2,3, . . ., aj =ξ+ωi+1−j forj0.

The type I vertex operator Φ(v)^a+¯a ^εµ can be defined as a half-infinite transfer matrix. The operator Φ(v)^a+¯a ^εµ is an intertwiner fromH⁽ⁱ⁾_l,k toH⁽ⁱ⁺¹⁾_l,k+¯ε

µ, satisfying the following commutation relation:

Φ(v1)^c_bΦ(v2)^b_a=X

d

W

c d b a

v1−v2

Φ(v2)^c_dΦ(v1)^d_a.

The free field realization of Φ(v2)^b_a was constructed in [15]. See Section 4.2.

The type II vertex operators should satisfy the following commutation relations:

Ψ^∗(v₂)^ξ_ξ^c

dΨ^∗(v₁)^ξ_ξ^d

a =X

ξb

Ψ^∗(v₁)^ξ_ξ^c

bΨ^∗(v₂)^ξ_ξ^b

aW⁰

ξ_c ξ_d ξ_b ξ_a

v₁−v₂

, Φ(v₁)^a_a⁰Ψ^∗(v₂)^ξ_ξ⁰ =χ(v₁−v₂)Ψ^∗(v₂)^ξ_ξ⁰Φ(v₁)^a_a⁰.

Let

ρ⁽ⁱ⁾_l,k=Gax^2nH

(i)

l,k, Ga= Y

06µ<ν6n−1

[aµν],

where H_l,k⁽ⁱ⁾ is the CTM Hamiltonian ofA⁽¹⁾_n−1 model in regime III is given as follows:

H_l,k⁽ⁱ⁾(a₀, a₁, a₂, . . .) = 1 n

∞

X

j=1

jH_f(aj−1, a_j, a_j+1), H_f(a+ ¯εµ+ ¯εν, a+ ¯εµ, a) =Hv(ν, µ),

and Hv(ν, µ) is the same one as (3.1). Then we have the homogeneity relations Φ(v)^a_a⁰ρ⁽ⁱ⁾_a+ρ,l

G_a = ρ⁽ⁱ⁺¹⁾_a0+ρ,l

G_a⁰ Φ(v−n)^a_a⁰, Ψ^∗(v)^ξ_ξ⁰ρ⁽ⁱ⁾_k,ξ+ρ=ρ⁽ⁱ⁺¹⁾_k,ξ0+ρΨ^∗(v−n)^ξ_ξ⁰. The free field realization of Ψ^∗(v)^ξ_ξ⁰ 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 (modQ)):

T(u)^ξa0 =

∞

Y

j=0

t^µj(−u)^a_a^j_j+1 :H⁽ⁱ⁾→ H_l,k⁽ⁱ⁾, T(u)_ξa0 =

∞

Y

j=0

t^∗_µj(−u)^a_a^j+1_j :H⁽ⁱ⁾_l,k → H⁽ⁱ⁾, which satisfy

ρ⁽ⁱ⁾=

(x^2r−2;x^2r−2)∞

(x^2r;x^2r)∞

(n−1)(n−2)/2

1 G⁰_ξ

X

k≡l+ωi

(modQ)

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_{ξ a}^0a0 =T(u)^ξ0a0T(u)_{ξ a}:H⁽ⁱ⁾_l,k→ H⁽ⁱ⁾_l0k⁰, (3.3) where k=a+ρ,l=ξ+ρ,k⁰ =a⁰+ρ, and l⁰ =ξ⁰+ρ. Let

L

a⁰₀ a⁰₁ a0 a1

u

:=

n−1

X

µ=0

t^∗_µ(−u)^a_a¹₀t^µ(−u)^a_a⁰⁰0 1

.

Then we have Λ(u)^ξ

0a⁰₀ ξ a0 =

∞

Y

j=0

L

a⁰_j a⁰_j+1 aj aj+1

u

.

From the invertibility of the intertwining vector and its dual vector, we have

Λ(u0)^ξ_{ξ a}^0a=δ^ξ_ξ⁰. (3.4)

Note that the tail operator (3.3) satisfies the following intertwining relations [1,2]:

Λ(u)^ξ_{ξ b}^0cΦ(v)^b_a=X

d

L

c d b a

u−v

Φ(v)^c_dΛ(u)^ξ_{ξ a}^0d, (3.5)

Ψ^∗(v)^ξ_ξ^c

dΛ(u)^ξ_ξ^da0

aa =X

ξb

L⁰

ξc ξ_d ξ_b ξ_a

u+ ∆u−v

Λ(u)^ξ_ξ^ca0

baΨ^∗(v)^ξ_ξ^b

a, (3.6)

where L⁰

ξc ξ_d ξ_b ξ_a

u

=L

ξc ξ_d ξ_b ξ_a

u

r7→r−1

.

We should find a representation of Λ(u)^ξ_{ξ a}^0a0 and fix the constant ∆u that solves (3.5) and (3.6).

4 Free f iled realization

4.1 Bosons

In [17,18] the bosonsBm^j (16j6n−1, m∈Z\{0}) relevant to elliptic algebra were introduced.

Forα, β ∈h^∗ we denote the zero mode operators byP_α,Q_β. Concerning commutation relations among these operators see [17,18,2].

We will deal with the bosonic Fock spaces F_l,k, (l, k ∈h^∗) generated by B_−m^j (m > 0) over the vacuum vectors|l, ki:

F_l,k =C[{B₋₁^j , B^j₋₂, . . .}_16j6n]|l, ki, where

|l, ki= exp √

−1(β₁Qk+β2Ql)

|0,0i, and

t²−β₀t−1 = (t−β₁)(t−β₂), β₀= 1

pr(r−1), β₁ < β₂. 4.2 Type I vertex operators

Let us define the basic operators forj= 1, . . . , n−1 U−αj(v) =z^r−1r : exp



−β₁ √

−1Q_αj+Pαjlogz)

+ X

m6=0

Bm^j −Bm^j+1

m (x^jz)^−m



:,

Uωj(v) =z^{r−12r j(n−j)n} : exp



β1

√−1Q_ωj+Pωjlogz)

−X

m6=0

1 m

j

X

k=1

x^(j−2k+1)mB_m^kz^−m



:,

where β1 = − qr−1

r and z = x^2v as usual. The normal product operation places Pα’s to the right of Q_β’s, as well as B_m’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 π_µ=p

r(r−1)P_ε¯µ, π_µν =π_µ−π_ν =rL_µν−(r−1)K_µν.

The operatorsK_µν,L_µν andπ_µν act onF_l,k as scalarshε_µ−ε_ν, ki,hε_µ−ε_ν, li andhε_µ−ε_ν, rl− (r−1)ki, respectively. In what follows we often use the symbols

GK = Y

06µ<ν6n−1

[Kµν], G⁰_L= Y

06µ<ν6n−1

[Lµν]⁰.

For 06µ6n−1 the type I vertex operator Φ(v)^a+¯_a ^εµ can be expressed in terms of U_ωj(v) and U−αj(v) on the bosonic Fock space F_l,a+ρ. The explicit expression of Φ(v)^a+¯a ^εµ can found in [15].

4.3 Type II vertex operators

Let us define the basic operators forj= 1, . . . , n−1 V−α_j(v) = (−z)^r−1r : exp



−β₂ √

−1Q_αj+P_αjlog(−z)

− X

m6=0

A^jm−A^j+1m

m (x^jz)^−m



:, V_ωj(v) = (−z)^2(r−1)r

j(n−j) n

×: exp



β2

√−1Q_ωj+Pωjlog(−z)

+ X

m6=0

1 m

j

X

k=1

x^(j−2k+1)mA^k_mz^−m



:, where β2 = q

r

r−1 and z = x^2v, and A^jm = _[(r−1)m]^[rm]x

xB^jm. For some useful OPE formulae and commutation relations, see AppendixA.

For 06µ6n−1 the type II vertex operator Ψ^∗(v)^ξ+¯_ξ ^εµ can be expressed in terms ofV_ωj(v) and V−α_j(v) on the bosonic Fock space F_ξ+ρ,k. The explicit expression of Ψ^∗(v)^ξ+¯_ξ ^εµ can found in [16].

4.4 Free f ield 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 √

−1Q_αj+Pαjlog(−1)^rz)

−X

m6=0

O^j_m−O_m^j+1

m (x^jz)^−m



:, where β₀ = β₁ +β₂ = √ ¹

r(r−1), (−1)^r := exp(π√

−1r) and O_m^j = _[(r−1)m]^[m]x

xB_m^j . 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}⁰ satisfying (3.5) for ξ⁰=ξ:

Λ(u)^ξa−¯_ξa−¯^ε_ε^µ_ν =GK

I ^ν Y

j=µ+1

dz_j 2π√

−1z_jU−α_µ+1(vµ+1)· · ·U−α_ν(vν)

×

ν−1

Y

j=µ

(−1)^Ljν−Kjνf(v_j+1−v_j, π_jν)G⁻¹_K, (4.1) where vµ =u and µ < ν. In [2] we obtained the free field representation of Λ(v)^ξ+¯_ξ+¯^ε_ε^{n−1a+¯εn−1}

µa+¯εn−2

satisfying (3.6) as follows:

Λ(u)^ξ+¯_ξ+¯^ε_ε^{n−1a+¯εn−1}

µa+¯εn−2 = (−1)^n−µ[an−2n−1] (x⁻¹−x)(x^2r;x^2r)³_∞

[ξµ n−1−1]⁰

[1]⁰ G_KG⁰_L⁻¹

× I

C⁰ n−2

Y

j=µ+1

dz_j 2π√

−1z_jW−αn−1 u−^r−1₂

V−αn−2(vn−2)· · ·V−α_µ+1(v_µ+1)

×

n−2

Y

j=µ+1

(−1)^Lµj−Kµjf^∗(vj−vj+1, πµj)G⁻¹_K G⁰_L, (4.2) for 0 6 µ 6 n−2 with ∆u = −ⁿ⁻¹₂ and vn−1 = u. Concerning other types of tail operators Λ(u)^ξa_ξa⁰, the expressions of the free field representation can be found in [1,2].

4.5 Free f ield realization of CTM Hamiltonian

Let

H_F =

∞

X

m=1

[rm]x

[(r−1)m]_x

n−1

X

j=1 j

X

k=1

x^{(2k−2j−1)m}B_−m^k (B^j_m−B_m^j+1) +1 2

n−1

X

j=1

PωjPαj

=

∞

X

m=1

[rm]_x [(r−1)m]x

n−1

X

j=1 j

X

k=1

x^{(2j−2k−1)m}(B_−m^j −B^j+1_−m)B_m^k +1 2

n−1

X

j=1

P_ωjP_αj (4.3) be the CTM Hamiltonian on the Fock spaceF_l,k [19]. Then we have the homogeneity relation

φ_µ(z)q^HF =q^HFφ_µ q⁻¹z , and the trace formula

trF_l,k x^2nHFG_a

= x^{n|β1k+β2l|2} (x²ⁿ;x²ⁿ)ⁿ⁻¹∞

G_a.

Let ρ⁽ⁱ⁾_l,k = Gax^2nHF. Then the relation (3.2) holds. We thus indentify HF with free field representations ofH_l,k⁽ⁱ⁾, the CTM Hamiltonian ofA⁽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 forn= 2 case, i.e., the eight-vertex model form factors. Here, we adopt the convention that the components 0 and 1 forn= 2 are denoted by + and−. Form factors of the eight-vertex model are defined as matrix elements of some local operators. For simplicity, we chooseσ^z as a local operator:

σ^z =E₊₊⁽¹⁾ −E₋₋⁽¹⁾,

where E_µµ^(j)0 is the matrix unit on thej-th site. The free field representation of σ^z is given by σc^z =X

ε=±

εΦ^∗_ε(u)Φ^ε(u).

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_m⁽ⁱ⁾(σ^z;u1, . . . , u2m)ν1···ν2m = 1

χ⁽ⁱ⁾Tr_H(i)

Ψ^∗_ν1(u1)· · ·Ψ^∗_ν2m(u2m)cσ^zρ⁽ⁱ⁾

, (5.1)

where

χ⁽ⁱ⁾= Tr_H(i)ρ⁽ⁱ⁾= (x⁴;x⁴)∞

(x²;x²)∞

.

In this section we denote the spectral parameters by z_j =x^2uj, and denote integral variables by w_a=x^2va.

By using the vertex-face transformation, we can rewrite (5.1) as follows:

F_m⁽ⁱ⁾(σ^z;u1, . . . , u2m)ν1···ν2m = 1 χ⁽ⁱ⁾

X

l1,...,l2m

t^0∗_ν1 u1−u0+¹_2l1

l · · ·t^0∗_ν2m u2m−u0+¹_2l2m

l2m−1

× X

k≡l+i(mod 2)

X

ε=±

ε X

k1=k±1

X

k2=k1±1

t^∗_ε(u−u₀)^k_k1t^ε(u−u₀)^k_k¹

2

×Tr_H(i) l,k

Ψ^∗(u₁)^l_l1· · ·Ψ^∗(u_2m)^l_l^2m−1

2m Φ^∗(u)^k_k1Φ(u)^k_k¹

2Λ(u₀)^l_{l k}^2mk2[k]x^4HF [l]⁰

, where HF is the CTM Hamiltonian defined by (4.3).

Let

F_m⁽ⁱ⁾(σ^z;u₁, . . . , u_2m)_ll1···l_2m = 1 χ⁽ⁱ⁾

X

k≡l+i(mod 2)

X

ε=±

ε X

k1=k±1

X

k2=k1±1

t^∗_ε(u−u₀)^k_k1t^ε(u−u₀)^k_k¹

2

×Tr_H(i) l,k

Ψ^∗(u1)^l_l1· · ·Ψ^∗(u2m)^l_l^2m−1

2m Φ^∗(u)^k_k1Φ(u)^k_k¹

2Λ(u0)^l_{l k}^2mk2[k]x^4HF [l]⁰

. (5.2)

Then we have

F_m⁽ⁱ⁾(σ^z;u₁, . . . , u_2m)_ll1···l2m = X

ν1,...,ν2m

F_m⁽ⁱ⁾(σ^z;u₁, . . . , u_2m)_ν1···ν2m

×t^0ν1 u1−u0+¹_2l

l1· · ·t^0ν2m u2m−u0+¹₂l2m−1

l2m .

For simplicity, letlj =l−jfor 16j62m. Then from the relation (3.4), Λ(u0)^l_{l k}^2mk2 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

X

ε=±

εt^∗_ε(u−u₀)^k_k−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 tok for the trace over the zero-modes parts can be calculated as follows:

X

k≡l+i(mod 2)

{u−u₀−1 +k}

2m

Y

j=1

(−z_j)^{2(r−1)rl −k2}(x⁻¹z)^{−l+(r−1)kr}

m−1

Y

a=1

(−w_a)^{−r−1rl +k}

× (−1)^rx^−r+1z0

−_r−1^l +^k_r

x^rl

2

r−1−2kl+^(r−1)k_r ²

=x

l2 r−1+l

2+_r−1^r

2n

P

j=1

uj−2u−_r−1^2r

m−1

P

a=1

va−_r−1^2u0

x^{1r(u−u0−1)2−(u−u0−1)} X

k≡l+i(mod 2)

x^(k−l)2

×X

n∈Z

x^rn(n−1)x^{2(u−u0−1+k)n}x

k

2

m−1

P

a=1

va+2u−

2m

P

j=1

uj−3

= (−1)¹⁻ⁱ

2 x

1

r(u−u0−1)²−_r−1¹

u0+

m−1

P

a=1

va−¹₂

2m

P

j=1

uj

²

−

_m−1

P

a=1

va+u−¹₂

2m

P

j=1

uj−1

²

×Z_m⁽ⁱ⁾(l, u, u₀, u_j, v_a), where

Z_m⁽ⁱ⁾(l, u, u₀, u_j, v_a) =



l−u₀−

m−1

X

a=1

v_a+¹₂

2m

X

j=1

u_j





0



m−1

X

a=1

v_a+u−¹₂

2m

X

j=1

u_j





1

+ (−1)¹⁻ⁱ







l−u0−

m−1

X

a=1

va+¹₂

2m

X

j=1

uj







0





m−1

X

a=1

va+u−¹₂

2m

X

j=1

uj







1

.

Thus,Fm⁽ⁱ⁾(σ^z;u1, . . . , u2m)ll−1···l−2m can be obtained as follows:

(−1)^m−1β_m⁻¹F_m⁽ⁱ⁾(σ^z;u1, . . . , u2m)ll−1···l−2m

= Y

j<j⁰

(−z_j)

r

2(r−1)Fψ^∗ψ^∗(z_j⁰/zj)

2m

Y

j=1

(−z_j)^−r−11 x

(uj−u0+1/2)2

4(r−1) +^r(uj_2(r−1)^−u0+1/2)+¹₄

x⁻¹z(xzj/z;x⁴)∞(x³z/zj;x⁴)∞

f⁰(uj−u0+¹₂)

× I

C m−1

Y

a=1

dw_a 2π√

−1w_aZ_m⁽ⁱ⁾(l, u, u0, uj, va)Y

a<b

(−w_b)^r−12r [va−vb]⁰[va−vb]1x^{−r−1r (va−vb−1)2}

×

m−1

Y

a=1

x⁻²z²x^{−(va−u)2+va−u}[va−u]1(−w_a)^r−12 x^{−r−11 (u0−va−1)2+u0−va−1}[va−u0+l−m]⁰

×

m−1

Y

a=1 2m

Y

j=1

(−z_j)^−r−1r (x^2r−1w_a/z_j;x⁴, x^2r−2)∞(x^2r+3z_j/w_a;x⁴, x^2r−2)∞

(x⁻¹wa/zj;x⁴, x^2r−2)∞(x³zj/wa;x⁴, x^2r−2)∞

×(x⁻¹z)^2r

2 x

−^r+2

r (u0−u)−¹

r−_r−1¹

u0+

m−1

P

a=1

va−¹

2 2m

P

j=1

uj

²

−

m−1

P

a=1

va+u−¹

2 2m

P

j=1

uj−1

²

, (5.3)

where f⁰(v) is defined by (2.6) for n= 2, a scalar function F_ψ^∗_ψ^∗(z) and a scalar β_m are F_ψ^∗_ψ^∗(z) = (z;x⁴, x⁴, x^2r−2)∞(x⁴z⁻¹;x⁴, x⁴, x^2r−2)∞

(x²z;x⁴, x⁴, x^2r−2)∞(x⁶z⁻¹;x⁴, x⁴, x^2r−2)∞

×(x^2r+2z;x⁴, x⁴, x^2r−2)∞(x^2r+6z⁻¹;x⁴, x⁴, x^2r−2)∞

(x^2rz;x⁴, x⁴, x^2r−2)∞(x^2r+4z⁻¹;x⁴, x⁴, x^2r−2)∞

, and

βm = x^−r−14r {0}[m−1]⁰!(x⁻²z)^r−12r (x², x⁴)²_∞(x²;x^2r)∞(x^2r+1;x^2r−2)∞

(m−1)![1]^0m(x⁻¹−x)g₁(x²)(x^2r;x^2r)²_∞(x^2r+1;x^2r)∞

×(x²;x²)^m−1_∞ (x^2r;x^2r−2)^m−1_∞ (x⁴;x⁴, x⁴, x^2r−2)^m_∞(x^2r+6;x⁴, x⁴, x^2r−2)^m_∞ (x⁶;x⁴, x⁴, x^2r−2)^m_∞(x^2r+4;x⁴, x⁴, x^2r−2)^m_∞,