Integrable structure of W_{3} Conformal Field Theory, Quantum Boussinesq Theory and

Boundary Affine Toda Theory.

Vladimir V. Bazhanov^{1,2,}^{†}, Anthony N. Hibberd^{1,}^{‡} and Sergey M. Khoroshkin^{3,4}^{§}

1 Department of Theoretical Physics, Research School of Physical Sciences and Engineering,

Institute of Advanced Studies, Australian National University, Canberra, ACT, 0200, Australia

2Centre for Mathematics and its Applications, Institute of Advanced Studies, Australian National University, Canberra, ACT, 0200, Australia

3Institute of Theoretical and Experimental Physics Moscow, 117259, Russia

4Max-Planck Institut fur Mathemetik, Vivatsgasse 7, 53111, Bonn, Germany

†e-mail: vladimir.bazhanov@rsphysse.anu.edu.au ‡e-mail: anthony.hibberd@rsphysse.anu.edu.au §e-mail: khor@heron.itep.ru

Abstract

In this paper we study the Yang-Baxter integrable structure of Conformal Field Theories with extended conformal symmetry generated by the W_{3} algebra. We explicitly construct various T and Q-operators which act in the irreducible highest weight modules of the W_{3} algebra. These operators can be viewed as continuous field theory analogues of the commuting transfer matrices and Q-matrices of the integrable lattice systems associated with the quantum algebra U_{q}(ŝl(3)). We formulate several conjectures detailing certain analytic characteristics of the Q-operators and propose exact asymptotic expansions of the T and Q-operators at large values of the spectral parameter. We show, in particular, that the asymptotic expansion of the T-operators generates an infinite set of local integrals of motion of the W_{3} CFT which in the classical limit reproduces an infinite set of conserved Hamiltonians associated with the classical Boussinesq equation. We further study the vacuum eigenvalues of the Q-operators (corresponding to the highest weight vector of the W_{3} module) and show that they are simply related to the expectation values of the boundary exponential fields in the non-equilibrium boundary affine Toda field theory with zero bulk mass.

Contents

1 Introduction 1

2 Quantum Boussinesq theory 5 2.1 Extended W_{3} conformal symmetry and local integrals of motion . . . . . . 5 2.2 Relation to classical Boussinesq theory . . . . . . . . . . . . . . . . . . . . 8

3 Quantum Yang-Baxter equation 12 3.1 Universal R-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Free field representation and the vertex operators . . . . . . . . . . . . . . 15 3.3 The universal L-operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Quantum T- and Q-operators 19 4.1 Symmetry relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 The T-operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 The Q-operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.4 Vacuum eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Functional Relations 29

6 Conjectures: Exact Asymptotic Expansions 32 6.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.3.1 The case g = 1/2, c = −10 . . . . . . . . . . . . . . . . . . . . . . . 40 6.3.2 The case g = 2/3, p = 1/6 . . . . . . . . . . . . . . . . . . . . . . . 41 6.3.3 The case g = 5/6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

7 Applications 42 7.1 Boundary Affine Toda Theory . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.2 Relation to third order differential equations . . . . . . . . . . . . . . . . . 50

8 Discussion 52 8.1 Truncation of the functional relations . . . . . . . . . . . . . . . . . . . . . 53 8.2 Connection with U_{q}(A_{2}^{(2)}) related IQFT . . . . . . . . . . . . . . . . . . . . 53 8.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

1 Introduction

The Yang-Baxter equation and its associated integrability structures play an important role in the understanding of Integrable Quantum Field Theory (IQFT) [1]. Recently, the authors of [2–5] developed a continuous version of the Quantum Inverse Problem Method (QIPM) [6] or Baxter’s commuting transfer matrix method [7] to study IQFT, especially the finite volume IQFT. This new method inherits all the analytic power of the theory of integrable lattice models but does not require intermediate lattice regularization and allows from the very beginning to work in terms of the continuous IQFT. The method has been most successfully applied [2–4] to the c < 1 Conformal Field Theory (CFT) which is a simplest nontrivial example of IQFT. The fundamental problem addressed in these papers is the simultaneous diagonalization of an infinite set of mutually commuting local integrals of motion (IM) in CFT.

In this paper we extend the results of [2–4] to CFT’s with extended conformal symmetry generated by the simplest W -algebra W_{3} [8]. Such a CFT involves additional local spin-3 fields W (u) and \overline{W} (\overline{u}) besides the usual left and right components T (u) and \overline{T} (\overline{u}) of the stress-energy tensor. In what follows we will concentrate on the left chiral component W_{3} of the full symmetry algebra W_{3} × \overline{W}_{3}. The algebra W_{3} is generated by the fields T (u) and W (u) along with all composite fields built from products of T (u) and W (u) and their derivatives. We will interpret the variable u as a complex coordinate on a 2D cylinder of circumference 2π and impose the periodic boundary conditions

T (u + 2π) = T (u),  W (u + 2π) = W (u).  (1.1)

The algebra W_{3} possesses an infinite-dimensional Abelian subalgebra [9,10] generated by the local integrals of motion (IM), I_{k} ∈ W_{3}

I_{k} = ∫_{0}^{2π} \frac{du}{2π} T_{k+1}(u),  k ∈ Z_{+},  k ≠ 0 (mod 3)  (1.2)

where the densities T_{k}(u) are appropriately regularized polynomials in T (u) and W (u) and their derivatives. The first two densities read

T_{2}(u) = T (u),  T_{3}(u) = W (u),  (1.3)

while the higher densities are determined by the requirement of commutativity

[I_{k}, I_{l}] = 0,  (1.4)

and by the “spin” assignment [9] (see Section 2 for the details).

We explicitly construct operator valued functions 𝕋(t) and \overline{𝕋}(t) where t is a complex parameter. These operators act invariantly in each irreducible highest weight module of the W_{3} algebra

𝕋(t), \overline{𝕋}(t) :  V_{Δ_{2},Δ_{3}} → V_{Δ_{2},Δ_{3}}  (1.5)

where Δ_{2}, Δ_{3} are the highest weights and they commute among themselves for any values of t, i.e.,

[𝕋(t), 𝕋(t′)] = [𝕋(t), \overline{𝕋}(t′)] = [\overline{𝕋}(t), \overline{𝕋}(t′)] = 0.  (1.6)

These operators are defined in terms of certain monodromy matrices associated with the three-dimensional representation of the quantum algebra U_{q}(ŝl(3)) where

q = e^{iπg},  (1.7)

and g is related to the central charge c of the W_{3} algebra as

c = 50 − 24(g + g^{−1}),  (1.8)

Here we mostly restrict our considerations to the “quasiclassical domain” [2]

−∞ < c < −2,  (1.9)

however, the results are valid in a wider region −∞ < c < 2, corresponding to 0 < g < 1.

Obviously, the operators 𝕋(t) and \overline{𝕋}(t) are W_{3} analogues of Baxter’s commuting transfer matrices from lattice theory. They enjoy remarkable analyticity properties, namely, they are entire functions of t with an essential singularity at t = ∞ and their asymptotic expansion near this point is described in terms of the local IM (1.2). For instance,

log 𝕋(t) ≃ m t^{\frac{1}{3(1−g)}} + ∑_{k=1}^{∞} C′_{k} t^{− \frac{k}{3(1−g)}} I_{k},  (1.10)

where m and C′_{k} are some explicitly known coefficients which depend on the central charge c only and C′_{k} ≡ 0 for k = 0 (mod 3). A similar expansion holds for \overline{𝕋}(t). Therefore the operators 𝕋(t) and \overline{𝕋}(t) can be thought of as generating functions of the local IM since their eigenvalues contain all the information about the eigenvalues of the local IM.

Another powerful method known in the lattice theory is based on the so-called Qoperators. This method was introduced by Baxter in his original solution of the eightvertex model of statistical mechanics [11]. Here we construct appropriate W_{3} versions of Baxter’s Q-operators (actually, there are six such operators Q_{i}(t) and \overline{Q}_{i}(t), i = 1, 2, 3). They are defined again as traces of certain monodromy matrices this time associated with infinite-dimensional representations of the so-called q-oscillator algebra. The Q-operators satisfy the following relations with the operators 𝕋(t) and \overline{𝕋}(t) discussed above

Q_{i}(tq^{3}) − 𝕋(tq^{\frac{1}{2}} ) Q_{i}(tq) + \overline{𝕋}(tq^{− \frac{1}{2}} ) Q_{i}(tq^{−1}) − Q_{i}(tq^{−3}) = 0,

\overline{Q}_{i}(tq^{3}) − \overline{𝕋}(tq^{\frac{1}{2}} ) \overline{Q}_{i}(tq) + 𝕋(tq^{− \frac{1}{2}} ) \overline{Q}_{i}(tq^{−1}) − \overline{Q}_{i}(tq^{−3}) = 0,

(1.11)

where i = 1, 2, 3. These relations are obviously the U_{q}(ŝl(3))-analogues [12,13] of Baxter’s famous T-Q relation. The relations (1.11) are just the first representatives of a rich family of the functional relations obeyed by the Q-operators in the U_{q}(ŝl(3)) related integrable systems. In particular, in Section 4 we define more general T-operators T_{μ}(t) and \overline{T}_{μ}(t) corresponding to finite-dimensional irreducible representations π_{μ} of U_{q}(gl(3)). Here μ = (μ_{1}, μ_{2}, μ_{3}) is the highest weight of the representation π_{μ} such that μ_{1} − μ_{2} and μ_{2} − μ_{3} are non-negative integers^{1}. All these T-operators can be expressed through Q_{i}(t) and \overline{Q}_{i}(t) by the means of only two elegant relations

T_{(μ_{1},μ_{2},μ_{3})}(t) = z_{0}^{−1} det ∥ Q_{i}(t q^{+2μ′_{j}})∥_{i,j=1,2,3}

−\overline{T}_{(μ_{1},μ_{2},μ_{3})}(t) = z_{0}^{−1} det ∥ \overline{Q}_{i}(t q^{−2μ′_{j}})∥_{i,j=1,2,3}

(1.12)

where μ′_{1} = μ_{1} + 1, μ′_{2} = μ_{2}, μ′_{3} = μ_{3} − 1, and where z_{0} is a simple operator acting diagonally in each W_{3} module which can be interpreted as the “quantum Wronskian” [3] of the third order difference equations (1.11)

z_{0} = det ∥Q_{i}(tq^{−2j})∥_{i,j=1,2,3} = − det ∥\overline{Q}_{i}(tq^{2j})∥_{i,j=1,2,3}  (1.13)

We prove the relations (1.12) algebraically by using decomposition properties of direct products of certain “q-oscillator” representations of the Borel subalgebra of U_{q}(ŝl(3))^{2}. These relations provide a very simply tool for derivation of various “fusion relations” for the T-operators [16, 17]. In particular, the so-called RSOS-reductions [16] of the Toperators simply follow from (1.12) when two columns of the three-by-three matrices therein become proportional.

The eigenvalues of the Q-operators can be studied by various methods. In particular, they satisfy the Destri-de-Vega non-linear integral equations [18–20]. We solve these equations exactly in a particular case when Δ_{2}, Δ_{3} → ∞, where Δ_{2}, Δ_{3} are the highest weight in (1.5). On the basis of this result in Section 6 we conjecture exact asymptotic asymptotic expansions of the eigenvalues of the Q-operators at large values of t. These expansions exhibit remarkable duality properties of the Q-operators with respect to the substitution g → g^{−1} similar to the case of c < 1 CFT [3]. We support our conjectures by consideration of certain exactly solvable cases and numerical calculation of the zeroes of eigenvalues of the Q-operators.

1The existence of the two different sets of operators T_{μ}(t) and \overline{T}_{μ}(t) is related to the fact that there are two non-equivalent “evaluation maps” from U_{q}(ŝl(3)) to U_{q}(gl(3)).

2It should noted that some particular cases of (1.12) were previously obtained [13,14] from the properties of the corresponding eigenvalues determined by the so-called “nested Bethe Ansatz” solutions [15].

As shown in [2] the T and Q-operators define integrable boundary conditions [21] for CFT in the geometry of the half-infinite cylinder. In particular, the vacuum eigenvalues of these operators can be interpreted [3] as finite-temperature partition functions of certain integrable QFT models with a non-trivial boundary interaction. These models are of interest in many branches of physics. For example, the integrable boundary sine-Gordon model [21] (with zero bulk mass) finds important application in dissipative quantum mechanics [22–25]; as explained in [26,27] it also describes the universal current through the point contact in the quantum Hall system [3, 28–30]. Here we consider a simplest generalisation of this model, namely, the boundary A_{2}^{(1)} affine Toda theory with zero bulk mass. Its Hamiltonian has the form

H = \frac{1}{4πg} ∫_{−∞}^{0} dx ( Π^{2} + (∂_{x}Φ)^{2} ) − \frac{κ}{2g} ∑_{j=1}^{3} y_{j} e^{i(e_{j},Φ(0))} ,  (1.14)

where g, κ are parameters and Φ(x) = (Φ_{1}(x), Φ_{2}(x)), Π(x) = (Π_{1}(x), Π_{2}(x)) are field operators obeying canonical commutation relations

[Π_{a}(x) , Φ_{b}(x′)] = −2πig δ(x − x′)δ_{ab},  a, b = 1, 2.  (1.15)

and e_{j}, j = 1, 2, 3 are the simple roots of the algebra A_{2}^{(1)}. The “amplitudes” y_{1}, y_{2}, y_{3} in (1.14) are not c-numbers, but quantum operators describing some additional boundary degrees of freedom. They satisfy the commutation relations

y_{1}y_{2} = qy_{2}y_{1},  y_{2}y_{3} = qy_{3}y_{2},  y_{3}y_{1} = qy_{1} y_{3},  y_{1}y_{2}y_{3} = 1,  (1.16)

where q = e^{iπg}. It should be noted that the integrability of the boundary affine Toda systems has been studied previously both on classical [31] and quantum levels [32]. In [31] it was shown that (with the exception of the boundary sine-Gordon model) the integrability in the boundary affine Toda theory imposes severe restrictions on the amplitudes of the boundary interaction. Namely, the boundary interaction should vanish when the bulk mass is zero. This conclusion was then confirmed [32] with the perturbation theory calculations in the quantum case. We would like to stress here that the model (1.14) does not fall under this “no-go theorem” since (1.14) involves additional boundary interaction, which was not considered in [31, 32]. We claim here that the theory (1.14) is integrable, but postpone detailed considerations of this question to a separate publication. Actually, the sufficient condition for the integrability of (1.14) is that operators y_{j} obey the Serre relations of quantum affine algebra U_{q}(A_{2}^{(1)})

y_{i}^{2} y_{j} − (q + q^{−1}) y_{i} y_{j} y_{i} + y_{j} y_{i}^{2} = 0,  i ≠ j.  (1.17)

and not necessarily restricted to the case (1.16). A Similar statement holds for generalisations of (1.14) to other affine algebras. In Section 7.1 we consider a non-equilibrium variant of (1.14) with a particular time-dependent boundary interaction. We show that expectation values of boundary exponential fields over a stationary non-equilibrium state in this theory are simply related to the vacuum eigenvalues of the Q-operators of the W_{3} CFT. This calculation generalises the corresponding result of [30] for the boundary sine-Gordon model.

2 Quantum Boussinesq theory

2.1 Extended W_{3} conformal symmetry and local integrals of motion

In this paper we consider CFT with extended conformal symmetry generated by the simplest W -algebra W_{3} (sometimes called W A_{2}) [8, 33]. Such a CFT involves additional local spin-3 fields W (u) and \overline{W} (\overline{u}) besides the usual left and right components T (u) and \overline{T} (\overline{u}) of the stress-energy tensor. The left chiral component W_{3} of the full symmetry algebra W_{3} ×\overline{W}_{3} is generated by the fields T (u) and W (u) along with all composite fields built from products of T (u) and W (u) and their derivatives. We will interpret the variable u as a complex coordinate on a 2D cylinder of circumference 2π and impose the periodic boundary conditions (1.1). The Fourier mode expansions of T (u) and W (u)

T (u) = −\frac{c}{24} + ∑_{−∞}^{+∞} L_{−n} e^{inz},  W (u) = ∑_{−∞}^{+∞} W_{−n} e^{inz},  (2.1)

are expressed in terms of the operators L_{n} and W_{n} satisfying the commutation relations [8]

[L_{n}, L_{m}] = (n − m) L_{n+m} + \frac{c}{12} (n^{3} − n) δ_{n+m,0},
[L_{n}, W_{m}] = (2n − m) W_{n+m},  (2.2)
[W_{n}, W_{m}] = \frac{1}{3}(n − m)Λ_{n+m} + \frac{(n − m)}{3b^{2}} [\frac{1}{15}(n + m + 2)(n + m + 3)  (2.3)

−\frac{1}{6}(n + 2)(m + 2)] L_{n+m} + \frac{c}{1080 b^{2}} (n^{2} − 4)(n^{2} − 1)nδ_{n+m,0},

where c is the central charge,

b^{2} = 16/(22 + 5c),

and the operators Λ_{n} are expressed through L_{n} as

Λ_{n} = ∑_{k=−∞}^{+∞} : L_{k}L_{n−k} : +\frac{1}{5} x_{n}L_{n},  (2.4)

with

x_{2l} = (1 + l)(1 − l), x_{2l+1} = (2 + l)(1 − l).

The symbol : : in (2.4) denotes the normal ordering such that operators L_{n} with bigger n are placed to the right.

The chiral space of state H_{chiral} is built up from a suitable collection of highest weight modules

H_{chiral} = ⊕_{a} V_{(Δ_{2}^{(a)},Δ_{3}^{(a)})},  (2.5)

where the parameters Δ_{2} and Δ_{3} are the highest weights, and the associated highest weight vectors |Δ_{2}, Δ_{3}⟩ ∈ V_{(Δ_{2},Δ_{3})} satisfy the equations

L_{n}|Δ_{2}, Δ_{3}⟩ = 0,  W_{n}|Δ_{2}, Δ_{3}⟩ = 0,  n > 0;  (2.6)
L_{0}|Δ_{2}, Δ_{3}⟩ = Δ_{2}|Δ_{2}, Δ_{3}⟩,  W_{0}|Δ_{2}, Δ_{3}⟩ = Δ_{3}|Δ_{2}, Δ_{3}⟩.  (2.7)

The algebra W_{3} possesses an infinite Abelian subalgebra [9,10] generated by the local integrals of motion (IM), I_{k} ∈ W_{3}

I_{k} = ∫_{0}^{2π} \frac{du}{2π} T_{k+1}(u),  k ∈ Z_{+},  k ≠ 0 (mod 3)  (2.8)

where the densities T_{k}(u) are appropriately regularized polynomials in T (u) and W (u) and their derivatives. The integer k + 1 in (2.8) denotes the spin of the local field T_{k+1}

∮_{C} \frac{dw}{2πi} (w − u)T (T (w)T_{k+1}(u)) = (k + 1) T_{k+1}(u),  (2.9)

where T stands for the chronological ordering

T (A(w)B(u)) = \{ \begin{matrix}A(w)B(u) , & if ℑm u > ℑm w ; \\ B(u)A(w) , & if ℑm w > ℑm u . \end{matrix}  (2.10)

The set of possible values of k in (2.8) is determined by the allowed values of spins of the densities T_{k+1}(u) [9]

spin[T_{k+1}(u)] = k + 1,  k ∈ ℤ_{+},  k ≠ 3 (mod 3)  (2.11)

The first few densities T_{k}(u) can be written as [10]^{3}

T_{2}(u) = T (u),  T_{3}(u) = W (u),  T_{5}(u) =: T (u)W (u) :,  (2.12)
T_{6}(u) =: T (u)^{3} : +9 : W (u)^{2} : +\frac{c − 10}{32} : T′(u)^{2} :,  (2.13)

where the prime denotes the derivative and : : denotes the regularized product of the fields, such that the product : A(u)B(u) : is defined as the regular term (of order O((u − v)^{0}) in the operator product expansion of A(u)B(v).^{4} The densities (2.13) are defined up to total derivatives which, obviously, have no effect in (2.8). A general expression for the densities (2.13) is not known, however, once the first two densities in (2.13) are fixed all other higher densities can be uniquely determined (up to a normalisation) by the requirement of commutativity

[I_{k}, I_{l}] = 0,  (2.14)

and the spin assignment (2.11). Alternatively one can formulate the last requirement as the “degree assignment”, namely, if one assigns degrees 2 and 3 to the fields T (u) and W (u), respectively, and the degree 1 to each derivative then T_{k+1}(u) is a homogeneous polynomial of the degree k + 1. We normalise the densities T_{k+1}(u) by fixing coefficients to the terms containing highest power of T (u),

T_{2n}(u) = (T (u))^{n} + …,  T_{2n+1}(u) = (T (u))^{n−1} W (u) + …,  n = 1, 2, 3, … (2.15)

The integrals (2.8) define operators I_{k}: V_{Δ_{2},Δ_{3}} → V_{Δ_{2},Δ_{3}} which can be expressed in terms of the generators L_{n} and W_{n}, for example

3A misprint in the expression for T_{6}(u) in [10] is corrected here.

4We assume that these regularized product are calculated in the cylinder geometry, as they are not invariant under conformal transformations.

I_{1} = L_{0} − \frac{c}{24} ,
I_{2} = W_{0} ,
I_{4} = ∑_{n=1}^{+∞}(L_{−n}W_{n} + W_{−n}L_{n}) + W_{0}L_{0} − \frac{c + 6}{24}W_{0} ,  (2.16)
I_{5} = ∑_{n_{1}+n_{2}+n_{3}=0} : L_{n_{1}}L_{n_{2}}L_{n_{3}} : + ∑_{n=1}^{∞} (\frac{14 + c}{16}n^{2} − 2 − \frac{c}{4}) L_{−n}L_{n} + \frac{3}{2} ∑_{r=1}^{∞} L_{1−2r}L_{2r−1}

+18 ∑_{n=1}^{∞} W_{−n}W_{n} + 9W_{0}^{2} − \frac{c + 8}{8}L_{0}^{2} + \frac{(c + 15)(c + 2)}{192}L_{0} − \frac{c(c + 23)(7c + 30)}{96768}.

Here the symbol : : denotes the same normal ordering as in (2.4) (where L_{n} with bigger n are placed to the right). Note that although these operators are not polynomials in L_{n} and W_{n} their action in V_{Δ_{2},Δ_{3}} is well defined.

2.2 Relation to classical Boussinesq theory

In this paper we will study the problem of simultaneous diagonalization of the infinite set of the operators I_{k} in V_{(Δ_{2},Δ_{3})}. This problem can be regarded as the quantum version of the related theory of the classical Boussinesq equation [34], which is a well known example of a completely integrable Hamiltonian system possessing an infinite set of local IM [35]. As is known [9] the W-algebras can regarded as the quantization of the Gelfand-Dickey Poisson bracket algebras [36]. The classical limit of (2.2) corresponds to c → −∞. In this limit the substitution

T (u) → \frac{c}{24} U(u),  W (u) → i( − \frac{c}{24})^{\frac{3}{2}} V (u),  [ , ] → \frac{48π}{ic} \{ , \},  (2.17)

where

U(u + 2π) = U(u), V (u + 2π) = V (u), (2.18)

reduces the algebra (2.2) to the following Poisson bracket algebra [37]

{U(u), U(v)} = −(U(u) + U(v)) δ′(u − v) + 2 δ′′′(u − v),
{U(u), V (v)} = −(V (u) + 2V (v)) δ′(u − v),  (2.19)
\{V (u), V (v)\} = −[\frac{1}{4}(U′′(u) + U′′(v)) + \frac{1}{3}(U^{2}(u) + U^{2}(v))] δ′(u − v)
+ \frac{5}{12}(U(u) + U(v))δ′′′(u − v) − \frac{1}{6}δ^{(5)}(u − v),

which is a particular example of the Gelfand-Dickey bracket algebra. The substitution (2.17) brings the local IM (2.8) to their classical counterparts

I_{k} → i^{−k−1}( − \frac{c}{24})^{\frac{k+1}{2}} I_{k}^{(cl)},  c → −∞  (2.20)

which form a commutative Poisson bracket algebra {I_{k}^{(cl)}, I_{l}^{(cl)}} = 0 . For example, the first few classical IM I_{k}^{(cl)} read

I_{1}^{(cl)} = ∫_{0}^{2π} \frac{du}{2π} U(u) ,  (2.21)
I_{2}^{(cl)} = ∫_{0}^{2π} \frac{du}{2π} V (u),  (2.22)
I_{4}^{(cl)} = ∫_{0}^{2π} \frac{du}{2π} U(u)V (u),  (2.23)
I_{4}^{(cl)} = ∫_{0}^{2π} \frac{du}{2π} (U(u)^{3} + 9V (u)^{2} + \frac{3}{4}U′(u)^{2}).  (2.24)

The Poisson bracket algebra (2.19) is known to describe the second Hamiltonian structure of the Boussinesq equation [37,38] provided one of the infinite set of the classical IM I_{k}^{(cl)} is taken as a Hamiltonian.

The evolution equations^{5}

∂_{t_{k}}U = {I_{k}^{(cl)}, U},  ∂_{t_{k}}V = {I_{k}^{(cl)}, V },  t_{1} = u,  k ≠ 0 (mod 3),  (2.25)

describe isospectral deformations of the third-order differential operator

L = ∂_{u}^{3} − U(u) ∂_{u} + V (u) − \frac{1}{2}U′(u) − λ^{3},  (2.26)

where λ is the spectral parameter. In particular, if one defines the monodromy matrix M(λ), which is an element of the group SL(3), as

(ψ_{1}(u + 2π), ψ_{2}(u + 2π), ψ_{3}(u + 2π)) = (ψ_{1}(u), ψ_{2}(u), ψ_{3}(u))M(λ),  (2.27)

where ψ_{1}(u), ψ_{2}(u) and ψ_{3}(u) are three linearly independent solutions of the equation Lψ = 0, then the eigenvalues of M(λ) are in involution among themselves and with all local IM (2.24) (with respect to the Poisson bracket algebra (2.19)). The trace of the monodromy matrix

T^{(cl)}(λ) = Tr M(λ),  (2.28)

can be regarded as a generating function for the local IM (2.24) as it expands in the asymptotic series [38]

\frac{1}{2π} log T^{(cl)}(λ) = λ + ∑_{k=1}^{∞}c_{k} I_{k}^{(cl)}λ^{−k},  λ → +∞  (2.29)

∂_{t} U = −2∂_{u} V, ∂_{t} V = \frac{1}{3} ∂_{u} [\frac{1}{2}∂_{u}^{2} U − U^{2}],

5The original Boussinesq equation of [34] corresponds to k = 2 in (2.25)

where c_{k} (c_{k} = 0 for k = 0 (mod 3)) are some numerical coefficients. The simplest way to calculate these coefficients is to consider the case when both U(u) and V (u) are constant (i.e., independent of u) and take real values. In this case (2.29) reduces to the λ → +∞ expansion of the positive root of the cubic equation x^{3} + 2Ux + V − λ^{3} = 0 which can be easily calculated. Similarly to (2.15) we normalise the classical IM I_{k}^{(cl)} by the term containing the highest power of U(u)

I_{2n−1}^{(cl)} = ∫_{0}^{2π} \frac{du}{2π}(U(u))^{n} + …,  I_{2n}^{(cl)} = ∫_{0}^{2π} \frac{du}{2π}(U(u))^{n−1}V (u) + …,  n = 1, 2, 3, …

(2.30)

Note that this normalisation is consistent with (2.15), (2.17) and (2.20). It follows then that

c_{2n−1} = \frac{4π cos (\frac{(n−2)π}{3})Γ(\frac{2n−1}{3})}{3 n!Γ(\frac{2−n}{3})},  c_{2n+2} = \frac{4π cos (\frac{(n+1)π}{3})Γ(\frac{2(n+1)}{3})}{n! Γ( − \frac{n+1}{3})},  (2.31)

where n = 0, 1, 2, . . .. The required calculations are given in Appendix C; the above expressions are simple corollaries of (C.9).

For our purposes it will be more convenient to work with a first order differential operator instead of (2.26). To define it we need some additional notation. The algebra gl(3) is generated by the elements H_{i}, i = 1, 2, 3 and E_{ij}, (i, j) = (1, 2), (2, 1), (2, 3), (3, 2), for which we use also the notation

E_{α} = E_{12}, E_{β} = E_{23}, F_{α} = E_{21}, F_{β} = E_{32},

and

H_{α} = H_{1} − H_{2},  H_{β} = H_{2} − H_{3},  H_{α+β} = H_{1} − H_{3}.

They satisfy the relations

[H_{i}, H_{j}] = 0,  [H_{i}, E_{kl}] = (δ_{ik} − δ_{il})E_{kl},  [E_{α_{i}}, F_{α_{j}}] = δ_{α_{i},α_{j}} H_{α_{i}},  (2.32)
\begin{aligned}E_{α_{i}}^{2} E_{α_{j}} − 2 E_{α_{i}} E_{α_{j}} E_{α_{i}} + E_{α_{j}} E_{α_{i}}^{2} & = & 0, \\ F_{α_{i}}^{2} F_{α_{j}} − 2 F_{α_{i}} F_{α_{j}} F_{α_{i}} + F_{α_{j}} F_{α_{i}}^{2} & = & 0, \end{aligned}  α_{i} ≠ α_{j}.  (2.33)

where in the last three relations greek indices α_{i} and α_{j} take two values α or β. Introduce two-dimensional vectors

\begin{aligned}e_{1} & = & (\frac{1}{2}, −\frac{\sqrt{3}}{2}), & e_{2} & = & (−1, 0), & e_{3} & = & ( \frac{1}{2}, \frac{\sqrt{3}}{2}), \\ w_{1} & = & (1, \frac{1}{\sqrt{3}}), & w_{2} & = & ( 0, −\frac{2}{\sqrt{3}}), & w_{3} & = & (−1, \frac{1}{\sqrt{3}}), \end{aligned}  (2.34)

such that

e_{1} + e_{2} + e_{3} = 0,  w_{1} + w_{2} + w_{3} = 0,  (2.35)

and

e_{i} = \frac{1}{2}(w_{j} − w_{k}),  w_{i} = −\frac{2}{3}(e_{j} − e_{k}),  (2.36)

where (i, j, k) is a cyclic permutation of (1, 2, 3). Moreover,

e_{i}e_{j} = \frac{1}{2}a_{ij},  w_{i}w_{j} = \frac{2}{3}a_{ij},  e_{i}w_{j} = ϵ_{ij},  i, j = 1, 2, 3,  (2.37)

where ab denotes the (Euclidean) scalar product, a_{ij}, i, j = 1, 2, 3, is the Cartan matrix of the algebra A_{2}^{(1)}

a_{ii} = 2,  a_{ij} = −1,  i ≠ j,  i, j = 1, 2, 3,  (2.38)

and

ϵ_{11} = ϵ_{22} = ϵ_{33} = 0,  ϵ_{12} = ϵ_{23} = ϵ_{31} = 1,  ϵ_{21} = ϵ_{32} = ϵ_{13} = −1.  (2.39)

It is also useful to define

E_{31} = [F_{β}, F_{α}],  H = w_{1}H_{1} + w_{2}H_{2} + w_{3}H_{3}.  (2.40)

For any μ = (μ_{1}, μ_{2}, μ_{3}), such that μ_{α} = μ_{1} − μ_{2} and μ_{β} = μ_{2} − μ_{3} are non-negative integers, denote by π_{μ} the finite dimensional irreducible representation of gl(3) with the highest weight μ and highest weight vector | 0⟩ defined as

E_{α}| 0⟩ = E_{β}| 0⟩ = 0,  H_{i}| 0⟩ = μ_{i}| 0⟩,  i = 1, 2, 3.  (2.41)

The dimension of such a representation is dim[π_{μ}] = \frac{1}{2}(μ_{α} + 1)(μ_{β} + 1)(μ_{α} + μ_{β} + 2). Consider the following differential operator

L = ∂_{u} − ϕ′(u) H − λ(E_{23} + E_{31} + E_{12}),  (2.42)

where the components of the vector ϕ(u) = (ϕ_{1}(u), ϕ_{2}(u)) are the canonical variables with Poisson bracket

\{ϕ_{i}(u), ϕ_{j}(v)\} = \frac{1}{2}δ_{ij}ϵ(u − v),  ϵ(u) = \frac{u}{2π} + \frac{1}{π} ∑_{n=1}^{∞} \frac{sin(nu)}{n},  0 ≤ u, v < 2π; (2.43)

which are related to U(u) and V (u) by the generalised Miura transform [39]

U(u) = ϕ′_{1}(u)^{2} + ϕ′_{2}(u)^{2} + 2ϕ′′_{1}(u),
\sqrt{3}V (u) = \frac{2}{3}ϕ′_{2}(u)^{3} − 2ϕ′_{1}(u)^{2}ϕ′_{2}(u) − 3ϕ′_{1}(u)ϕ′′_{2}(u) − ϕ′′_{1}(u)ϕ′_{2}(u) − ϕ′′′_{2}(u). (2.44)

For the periodic potentials U(u) and V (u), (2.18) the field ϕ(u) is, in general, quasiperiodic

ϕ(u + 2π) = ϕ(u) + 2πip.  (2.45)

The monodromy matrix associated with the equation π_{μ}[L]Ψ(u) = 0 (where π_{μ} denotes a representation of gl(3)) is given by

M_{μ}(λ) = π_{μ} [e^{2πipH} P exp (λ ∫_{0}^{2π}(V_{1}E_{23} + V_{2}(u)E_{31} + V_{3}(u)E_{12})du)] ,  (2.46)

where the symbol P denotes the “path ordered” exponential and

V_{i}(u) = e^{−2 e_{i} ϕ(u)} ,  i = 1, 2, 3.  (2.47)

The associated L-operator

L_{μ}^{(cl)}(λ) = π_{μ}[e^{−πipH} ]M_{μ}(λ),  (2.48)

satisfies the “r-matrix” Poisson bracket algebra [40,41]

{L_{μ}^{(cl)}(λ) ⊗_{,} L_{μ′}^{(cl)}(λ′)} = [r_{μμ′}(λ/λ′), L_{μ}^{(cl)}(λ) ⊗ L_{μ′}^{(cl)}(λ′)] ,  (2.49)

The quantity r_{μμ′}(λ) = π_{μ} ⊗ π_{μ′}[ r ] is the “classical r-matrix” given by

r(λ) = \frac{λ^{\frac{3}{2}} + λ^{− \frac{3}{2}}}{λ^{\frac{3}{2}} − λ^{− \frac{3}{2}}} ∑_{i=1}^{3} H_{i} ⊗ H_{i} + \frac{2}{λ^{\frac{3}{2}} − λ^{− \frac{3}{2}}} ∑_{(ij)} (λ^{− \frac{1}{2}} E_{ij} ⊗ E_{ji} + λ^{\frac{1}{2}} E_{ji} ⊗ E_{ij}) , (2.50)

where the second sum is taken over the values (ij) = (12), (23), (31).

It follows from (2.49) that the quantities

T_{μ}^{(cl)}(λ) = Tr_{π_{μ}} M_{μ}(λ),  (2.51)

are in involution with respect to the Poisson brackets (2.19),(2.43)

{T_{μ}^{(cl)}(λ), T_{μ′}^{(cl)}(λ′)} = 0.  (2.52)

Note, in particular, that T_{(1,0,0)}^{(cl)}(λ) coincides with (2.28).

3 Quantum Yang-Baxter equation

3.1 Universal R-matrix

The quantum Kac-Moody algebra A = U_{q}(ŝl(3)) is generated by the elements x_{i}, y_{i} and h_{i}, i = 1, 2, 3, and the grading element d, satisfying the commutation relations

[h_{i}, h_{j}] = 0 ,  [h_{i}, x_{j}] = −a_{ij}x_{j} ,  [h_{i}, y_{j}] = a_{ij}y_{j} ,  (3.1)

[d, y_{i}] = δ_{i,3}y_{i} ,  [d, x_{i}] = −δ_{i,3}x_{i} ,  [y_{i}, x_{j}] = δ_{ij} \frac{q^{h_{i}} − q^{−h_{i}}}{q − q^{−1}} ,  (3.2)

and the cubic Serre relations

\begin{aligned}x_{i}^{2} x_{j} − [2]_{q} x_{i} x_{j} x_{i} + x_{j} x_{i}^{2} & = & 0, \\ y_{i}^{2} y_{j} − [2]_{q} y_{i} y_{j} y_{i} + y_{j} y_{i}^{2} & = & 0, \end{aligned}  i ≠ j.  (3.3)

Here the indices i, j take three values i, j = 1, 2, 3; the Cartan matrix a_{ij} is the same as (2.38) and the “q-numbers” [n]_{q} are defined as

[n]_{q} = \frac{q^{n} − q^{−n}}{q − q^{−1}} .  (3.4)

The element k = h_{1} + h_{2} + h_{3} is the central element of the algebra A. In the following we will always work with the representations of the subalgebra of A with suppressed grading element d on which the central element k acts by zero:

h_{1} + h_{2} + h_{3} = 0.  (3.5)

The algebra A = U_{q}(ŝl(3)) is a Hopf algebra with the co-multiplication

δ : A ⟶ A ⊗ A,

defined as

\begin{aligned}δ(x_{i}) & = & x_{i} ⊗ 1 + q^{−h_{i}} ⊗ x_{i} , \\ δ(y_{i}) & = & y_{i} ⊗ q^{h_{i}} + 1 ⊗ y_{i} , \\ δ(h_{i}) & = & h_{i} ⊗ 1 + 1 ⊗ h_{i} , \\ δ(d) & = & d ⊗ 1 + 1 ⊗ d , \end{aligned}  (3.6)

where i = 1, 2, 3. As usual, we introduce the “twisted” co-multiplication

δ′ = σ ∘ δ ,  σ ∘ (a ⊗ b) = b ⊗ a ( ∀a, b ∈ A) .  (3.7)

Define also two Borel subalgebras B_{−} ⊂ A and B_{+} ⊂ A generated by h_{i}, x_{i}, i = 1, 2, 3, and h_{i}, y_{i}, i = 1, 2, 3, respectively. Let B̂_{±} be their extensions by d, that is, the subalgebras of A, generated by B_{±} and d .

There exists a unique element [42,43]

R ∈ B̂_{+} ⊗ B̂_{−} ,  (3.8)

satisfying the following relations

δ′(a) R = R δ(a) (∀ a ∈ A) ,  (3.9)
(δ ⊗ 1) R = R^{13} R^{23} ,  (3.10)
(1 ⊗ δ) R = R^{13} R^{12} ,  (3.11)

where R^{12}, R^{13}, R^{23} ∈ A ⊗ A ⊗ A and R^{12} = R ⊗ 1, R^{23} = 1 ⊗ R, R^{13} = (σ ⊗ 1) R^{23}. The element R is called the universal R-matrix. It satisfies the Yang-Baxter equation

R^{12}R^{13}R^{23} = R^{23}R^{13}R^{12} ,  (3.12)

which is a simple corollary of the definitions (3.9)-(3.11). The universal R-matrix is understood as a formal series in generators in B̂_{+} ⊗ B̂_{−}. Its dependence on the Cartan elements can be isolated as a simple factor

R = q^{t} \overline{R},  (3.13)

following where t is an invariant tensor in the tensor square of the extended Cartan subalgebra. It can be chosen to be [44,45] t = h_{1} ⊗ h^{1} + h_{2} ⊗ h^{2} + k ⊗ d + d ⊗ k,

h^{1} = \frac{2}{3}h_{1} + \frac{1}{3}h_{2}, h^{2} = \frac{2}{3}h_{2} + \frac{1}{3}h_{1},

and q^{t} satisfies the following defining properties:

q^{t}(y_{i} ⊗ 1) = (y_{i} ⊗ q^{h_{i}})q^{t},  q^{t}(x_{i} ⊗ 1) = (x_{i} ⊗ q^{−h_{i}})q^{t},
q^{t}(1 ⊗ y_{i}) = (q^{h_{i}} ⊗ y_{i})q^{t},  q^{t}(1 ⊗ x_{i}) = (q^{−h_{i}} ⊗ x_{i})q^{t}.

(3.14)

The factor \overline{R} is the “reduced” universal R-matrix \overline{R} ⊂ B_{+} ⊗ B_{−} which is a formal power series in y_{i} ⊗ 1 and 1 ⊗ x_{i} only

(with no dependence on the Cartan elements)

\overline{R} = 1 + (q − q^{−1})(y_{1} ⊗ x_{1} + y_{2} ⊗ x_{2} + y_{3} ⊗ x_{3}) + …  (3.15)

Below we will use the following important property of the reduced universal R-matrix. Proposition 1 Suppose elements I, A_{i} and B_{i}, i = 1, 2, 3, satisfy the relations

[I, x_{i}] = \frac{A_{i} − B_{i}}{q − q^{−1}} ,  (3.16)
A_{i} x_{j} = q^{a_{ij}} x_{j} A_{i},  B_{i} x_{j} = q^{−a_{ij}} x_{j} B_{i},  (3.17)

where i, j = 1, 2, 3. Then

[1 ⊗ I, \overline{R}] = ( ∑_{i=1}^{3} y_{i} ⊗ A_{i}) \overline{R} − \overline{R} ( ∑_{i=1}^{3} y_{i} ⊗ B_{i}).  (3.18)

The proof of this statement is given in Appendix A.

3.2 Free field representation and the vertex operators

The quantum version of the Miura transformation (2.44) is the free field representation [46] of the algebra W_{3},

−gT (u) = : ϕ′_{1}(u)^{2} : + : ϕ′_{2}(u)^{2} : +2(1 − g)ϕ′′_{1}(u) + \frac{g}{12},
−\frac{(3g)^{\frac{3}{2}} i}{2}W (u) = : ϕ′_{2}(u)^{3} : −3 : ϕ′_{1}(u)^{2}ϕ′_{2}(u) : +\frac{3}{2}(g − 1) : ϕ′′_{1}(u)ϕ′_{2}(u) : (3.19)
+ \frac{9}{2}(g − 1) : ϕ′_{1}(u)ϕ′′_{2}(u) : −\frac{3}{2}(g − 1)^{2}ϕ′′′_{2}(u),

in terms of a two-component free chiral Bose field φ(u) = (φ_{1}(u), φ_{2}(u)),

φ(u) = iX + iP u + ∑_{n≠0} \frac{b_{−n}}{n} e^{inu}.  (3.20)

The mode operators X = (X_{1}, X_{2}), P = (P_{1}, P_{2}) and b_{n} = (b_{1,n}, b_{2,n}) (n = ±1, ±2, …) satisfy the commutation relations of the Heisenberg algebra

[X_{i}, P_{j}] = \frac{ig}{2} δ_{i,j},  [b_{i,n}, b_{j,m}] = \frac{ng}{2} δ_{n+m,0} δ_{i,j},  i, j = 1, 2  (3.21)

where the parameter g is related to the central charge c as

g = \frac{50 − c − \sqrt{(2 − c)(98 − c)}}{48},  c = 50 − 24(g + g^{−1}).  (3.22)

Let F_{p} be the highest weight module over the algebra (3.21), with highest weight vector | p⟩, p = (p_{1}, p_{2}) defined as

P_{i}| p⟩ = p_{i}| p⟩,  a_{i,n}| p⟩ = 0,  n > 0,  i = 1, 2.  (3.23)

For generic c and p, the Fock space F_{p} is isomorphic to the space V_{Δ_{2},Δ_{3}} with the highest weights

Δ_{2} = \frac{p_{1}^{2} + p_{2}^{2}}{g} + \frac{c − 2}{24},  Δ_{3} = \frac{2p_{2} (p_{2}^{2} − 3p_{1}^{2})}{(3g)^{3/2}}.  (3.24)

The space

F̂_{p} = ⊕_{n_{1},n_{2}=−∞}^{∞} F_{p+g(n_{1}e_{1}+n_{2}e_{2})},  (3.25)

supports the action of the vertex operators

V_{i}(u) = q^{\frac{1}{2}} : e^{−2 e_{i} φ(u)} : ,  i = 1, 2, 3,  (3.26)

defined as

: e^{−2 e_{i}φ(u)} :≡ exp ( − 2 ∑_{n=1}^{∞} \frac{e_{i} b_{−n}}{n} e^{inu}) exp ( − 2ie_{i}(X + Pu)) exp ( + 2 ∑_{n=1}^{∞} \frac{e_{i} b_{n}}{n} e^{−inu}),

(3.27) where the root vectors e_{i} are given in (2.34) and

q = e^{iπg} .  (3.28)

Using (3.21) it is easy to check that

\begin{aligned}V_{i}(u_{1})V_{j}(u_{2}) & = & q^{a_{ij}}V_{j}(u_{2})V_{i}(u_{1}),  u_{1} > u_{2}, \\ [P , V_{j}(u)] & = & −g e_{j} V_{j}(u), \end{aligned}  (3.29)

where the a_{ij} are given by (2.38) and

V_{j}(u + 2π) = q^{−2}e^{−4iπPe_{j}}V_{j}(u).  (3.30)

It will sometimes be convenient to use the operators

z_{j} = e^{2πiw_{j}P} ,  z_{1} z_{2} z_{3} = 1,  j = 1, 2, 3,  (3.31)

where the vectors w_{j} are defined in (2.34). For example, with these operators the relations (3.29),(3.30) take the form

z_{i}V_{j}(u) = q^{2ϵ_{ij}}V_{j}(u)z_{i},  (3.32)

where ϵ_{ij} is defined in (2.39) and

V_{i}(u + 2π) = q^{−2} z_{j}^{−1} z_{k} V_{i}(u),  (3.33)

where (i, j, k) is a cyclic permutation of (1, 2, 3).

3.3 The universal L-operator

Consider the following operator [47,48]

L = e^{iπPh} P exp (∫_{0}^{2π} K(u) du) ,  (3.34)

where

h = \frac{2}{3}(e_{1} h_{1} + e_{2} h_{2} + e_{3} h_{3}),  (3.35)

with the vectors e_{1}, e_{2} and e_{3} defined in (2.34) and

K(u) = V_{1}(u)y_{1} + V_{2}(u)y_{2} + V_{3}(u)y_{3}.  (3.36)

Here h_{i}, y_{i}, i = 1, 2, 3, h_{1} + h_{2} + h_{3} = 0, are the generators of the Borel subalgebra B_{+} ⊂ U_{q}(ŝl(3)) defined above. The ordered exponential in (3.34) (the symbol P denotes the path ordering) is defined as a series

L = e^{iπPh} ∑_{n=0}^{∞} ∫_{2π≥u_{1}≥u_{2}≥...≥u_{n}≥0} K(u_{1})K(u_{2}) ⋯ K(u_{n})du_{1}du_{2} … du_{n}.  (3.37)

The integrals in (3.37) converge when

−∞ < c < −2,  (3.38)

(which corresponds to the range 0 < g < 2/3). For the values of c outside this quasiclassical domain, (3.38), the integrals in (3.37) can be defined via analytic continuation in c [3]. The operator (3.34) is an element of the algebra B_{+} whose coefficients are operators acting in the quantum space (3.25). Following the arguments of [4] it is not difficult to show that this L-operator satisfies the Yang-Baxter equation

R ( L ⊗ 1 ) ( 1 ⊗ L) = ( 1 ⊗ L) ( L ⊗ 1 ) R .  (3.39)

where R is the universal R-matrix for the algebra U_{q}(ŝl(3)) defined above. The only property of the universal R-matrix required for (3.39) to hold is relation (3.9).

An important remark is in order. As the reader might expect, the L-operator (3.34) can be obtained as a specialization of the universal R-matrix and the equation (3.39) is, in fact, a special case of the Yang-Baxter equation (3.12). The exact correspondence (first established in [4] for the case of U_{q}(ŝl(2))) is formulated as follows. Each side of the equation (3.12) is an element of the direct product B_{+} ⊗ A ⊗ B_{−}. Therefore to reduce (3.12) to (3.39) it is enough to find an appropriate realization of the Borel subalgebra B_{−} (appearing in the third factor of the product) in the quantum space (3.25). Let us identify the generators x_{i} ∈ B_{−}, i = 1, 2, 3, of this Borel subalgebra with the integrals of the vertex operators

x_{i} = \frac{1}{q − q^{−1}} ∫_{0}^{2π} V_{i}(u)du,  i = 1, 2, 3.  (3.40)

One can check that these generators satisfy [49] the Serre relations (3.3). Substituting (3.40) into the reduced universal R-matrix, (3.15), one gets an element of B_{+} whose co-ordinates are operators acting in the quantum space (3.25). As expected, it exactly coincides with the P-ordered exponent from (3.34).

Proposition 2 The specialization of the “reduced” universal R-matrix \overline{R} ∈ B_{+} ⊗ B_{−} for the case when x_{i} ∈ B_{−}, i = 1, 2, 3, are realized as in (3.40) coincide with the P-exponent

\overline{R} |_{x_{i}=(3.40)} = P exp ( ∫_{0}^{2π} K(u) du) ≡ \overline{L},  (3.41)

where K(u) is defined in (3.36).

This is precisely the conjecture stated in ref. [4]. It was formulated there for U_{q}(ŝl(2)), but the statement is rather general and (with obvious modifications) holds for any quantized Kac-Moody algebra. A proof of the conjecture of [4] based on combinatorial properties of the representations of quantum groups has been obtained by E.Frenkel and F.A.Smirnov [50]. Here we present an alternative (and, in fact, rather elementary) proof of (3.41) which is based on the defining properties of the universal R-matrix.

The reduced universal R-matrix (3.15) is a formal power series in the generators \overline{y}_{i} = y_{j} ⊗ 1 and \overline{x}_{i} = 1 ⊗ x_{i}, i = 1, 2, 3, such that \overline{x}_{i}\overline{y}_{j} = \overline{y}_{j}\overline{x}_{i}, therefore we can write it as \overline{R} = \overline{R}(\overline{y}_{i}, \overline{x}_{i}). Then, substituting (3.13) into (3.11) one obtains

\overline{R}(\overline{y}_{j}, x′_{i} + x′′_{i}) = \overline{R}(\overline{y}_{j}, x′_{i}) \overline{R}(\overline{y}_{j}, x′′_{i})  (3.42)

where \overline{y}_{i} = y_{i} ⊗ 1 ⊗ 1, x′_{i} = 1 ⊗ q^{−h_{i}} ⊗ x_{i}, x′′_{i} = 1 ⊗ x_{i} ⊗ 1. Both sets x′_{i} and x′′_{i} satisfy the Serre relations (3.3) of B_{−}, commute with \overline{y}_{j}

x′_{i}\overline{y}_{j} = \overline{y}_{j}x′_{i},  x′′_{i}\overline{y}_{j} = \overline{y}_{j}x′′_{i}  (3.43)

and also obey the relations

x′_{i}x′′_{j} = q^{a_{ij}}x′′_{j}x′_{i}  (3.44)

Conversely, we can regard (3.42) as an identity for \overline{R}(\overline{y}_{i}, \overline{x}_{i}) which should be valid once all the arguments involved there are subjected to the above relations (namely, (3.3), (3.43) and (3.44)).

Consider integrals more general than (3.40) which also satisfy the Serre relations (3.3)

\overline{x}_{i} = \frac{1}{q − q^{−1}} ∫_{u_{2}}^{u_{1}} V_{i}(u)du,  i = 1, 2, 3,  (3.45)

where 2π ≥ u_{1} > u_{2} ≥ 0. Denote by \overline{L}(u_{1}, u_{2}) the result of substituting these generators \overline{x}_{i} ∈ B_{−} into the reduced universal R-matrix

\overline{L}(u_{1}, u_{2}) = \overline{R}(\overline{y_{i}}, \overline{x}_{i}) |_{\overline{x}_{i}=(3.45)}  (3.46)

Note that from (3.15) it follows that for (u_{1} − u_{2}) → 0

\overline{L}(u_{1}, u_{2}) = 1 + ∫_{u_{2}}^{u_{1}} K(u) du + O((u_{1} − u_{2})^{2})  (3.47)

where K(u) is defined in (3.36).

Now, let 2π > u_{1} ≥ u_{2} ≥ u_{3} ≥ 0, then the identity (3.42) implies that

\overline{L}(u_{1}, u_{3}) = \overline{L}(u_{1}, u_{2})\overline{L}(u_{2}, u_{3}),  (3.48)

since two sets of quantities

x′_{i} = \frac{1}{q − q^{−1}} ∫_{u_{2}}^{u_{1}} V_{i}(u)du,

x′′_{i} = \frac{1}{q − q^{−1}} ∫_{u_{3}}^{u_{2}} V_{i}(u)du,

satisfy all the required relations (3.3), (3.43) and (3.44). In particular, the relation (3.44) readily follows from (3.29). With an account of (3.47) the functional equation (3.48) has a unique solution

\overline{L}(u_{1}, u_{2}) = P exp (∫_{u_{2}}^{u_{1}} K(u) du) .  (3.49)

Finally, setting u_{2} = 0 and u_{1} = 2π one arrives at the statement of Proposition 2.

4 Quantum T- and Q-operators

4.1 Symmetry relations

Consider the symmetry properties of the L-operator (3.34). Any linear orthogonal operator U in two-dimensional Euclidean space induces an automorphism U of the twocomponent Heisenberg algebra (3.21) by the rule U[φ(u)] = Uφ(u). In particular, the Weyl reflections associated with the vectors w_{i}, i = 1, 2, 3, generate the symmetric group S_{3} permuting the vertex operators

σ_{ij}[V_{i}(u)] = V_{j}(u),  σ_{ij}[V_{j}(u)] = V_{i}(u),  σ_{ij}[V_{k}(u)] = V_{k}(u),  (4.1)
σ_{ij}[z_{i}] = z_{j}^{−1},  σ_{ij}[z_{j}] = z_{i}^{−1},  σ_{ij}[z_{k}] = z_{k}^{−1}.  (4.2)

where (i, j, k) is any permutation of (1, 2, 3). Define also the cyclic automorphism τ = σ_{12}σ_{23} corresponding to the rotation of the Dynkin diagram of A_{2}^{(1)}

τ[V_{i}(u)] = V_{i+1 (mod 3)}(u).  (4.3)

The weights (3.24) can be written in a symmetric form

Δ_{2} = −\frac{r_{1}r_{2} + r_{1}r_{3} + r_{2}r_{3}}{g} + \frac{c − 2}{24},  Δ_{3} = −\frac{r_{1}r_{2}r_{3}}{g^{3/2}} ,  (4.4)

where

r_{i} = w_{i}p,  i = 1, 2, 3,  (4.5)

with w_{i} given in (2.34). From the symmetry properties

τ[r_{i}] = r_{i+1 (mod 3)},  σ_{ij}[r_{i}] = −r_{j},  σ_{ij}[r_{j}] = −r_{i},  σ_{ij}[r_{k}] = −r_{k},  (4.6)

it is obvious that the cyclic automorphism τ transforms the Fock module F_{p} to an isomorphic module

τ[F_{p}] ∼ F_{p},  (4.7)

while the Weyl reflections σ_{ij} negate p and the weight Δ_{3}

σ_{ij}[F_{p}] ∼ F_{−p} ∼ V_{Δ_{2},−Δ_{3}}.  (4.8)

Further, the algebra of outer automorphisms of B_{+} ⊂ U_{q}(ŝl(3)) also contains the symmetric group S_{3}: σ[y_{i}] = y_{σ(i)}, σ[h_{i}] = h_{σ(i)}. Again by τ we denote the automorphism corresponding to rotation of the Dynkin diagram of A_{2}^{(1)}

τ[y_{i}] = y_{i+1 (mod 3)},  τ[h_{i}] = h_{i+1 (mod 3)} ,  (4.9)

and by σ_{ij} the transpositions

\begin{aligned}σ_{ij}[y_{i}] & = & y_{j}, & σ_{ij}[y_{j}] & = & y_{i}, & σ_{ij}[y_{k}] & = & y_{k}, \\ σ_{ij}[h_{i}] & = & h_{j}, & σ_{ij}[y_{h}] & = & h_{i}, & σ_{ij}[h_{k}] & = & h_{k}. \end{aligned}  (4.10)

The L-operator (3.34) is invariant with respect to the diagonal action of the automorphisms

(σ ⊗ σ)[L] = L,  σ ∈ S_{3}.  (4.11)

4.2 The T-operators

We will need several specializations of the universal L-operator (3.34) corresponding to various matrix representations of the Borel subalgebra B_{+} ⊂ U_{q}(ŝl(3)). First consider the so-called evaluation maps from B_{+} to the finite dimensional algebra U_{q}(gl(3)). This latter algebra is generated by the elements H_{i}, i = 1, 2, 3 and E_{ij}, where (i, j) = (1, 2), (2, 1), (2, 3), (3, 2), for which we use also the notations

E_{α} = E_{12}, E_{β} = E_{23}, F_{α} = E_{21}, F_{β} = E_{32},

and

H_{α} = H_{1} − H_{2}, H_{β} = H_{2} − H_{3}, H_{α+β} = H_{1} − H_{3}.

They satisfy the relations

[H_{i}, H_{j}] = 0, [H_{i}, E_{kl}] = (δ_{ik} − δ_{il})E_{kl},
[E_{α_{i}}, F_{α_{j}}] = δ_{α_{i},α_{j}} \frac{q^{H_{α_{i}}} − q^{−H_{α_{i}}}}{q − q^{−1}} ,  (4.12)

\begin{aligned}E_{α_{i}}^{2} E_{α_{j}} − [2]_{q} E_{α_{i}} E_{α_{j}} E_{α_{i}} + E_{α_{j}} E_{α_{i}}^{2} & = & 0, \\ F_{α_{i}}^{2} F_{α_{j}} − [2]_{q} F_{α_{i}} F_{α_{j}} F_{α_{i}} + F_{α_{j}} F_{α_{i}}^{2} & = & 0, \end{aligned} α_{i} ≠ α_{j},

where [n]_{q} = (q^{n}−q^{−n})/(q−q^{−1}) and the greek indices α_{i} and α_{j} in the last three relations take two values α or β. The algebra U_{q}(gl(3)) has an outer automorphism σ_{13}, induced by the automorphism of its Dynkin diagram:

\begin{matrix}σ_{13}[E_{α}] = E_{β},  σ_{13}[E_{β}] = E_{α},  σ_{13}[F_{α}] = F_{β},  σ_{13}[F_{β}] = F_{α}, \\ σ_{13}[H_{1}] = −H_{3},  σ_{13}[H_{2}] = −H_{2},  σ_{13}[H_{3}] = −H_{1}. \end{matrix}  (4.13)

Let us introduce the following root vectors:

F_{α+β} = q^{\frac{1}{2}} F_{β}F_{α} − q^{− \frac{1}{2}} F_{α}F_{β}, \overline{F}_{α+β} = q^{− \frac{1}{2}} F_{β}F_{α} − q^{\frac{1}{2}} F_{α}F_{β},

and

E_{31} = F_{α+β}q^{H_{1}+H_{3}}, \overline{E}_{31} = \overline{F}_{α+β}q^{−H_{1}−H_{3}}.

There are two non-equivalent evaluation maps B_{+} → U_{q}(gl(3)). The first one is

Ev_{t}(y_{1}) = E_{23},  Ev_{t}(y_{2}) = tE_{31},  Ev_{t}(y_{3}) = E_{12},  (4.14)
Ev_{t}(h_{1}) = H_{2} − H_{3},  Ev_{t}(h_{2}) = H_{3} − H_{1},  Ev_{t}(h_{3}) = H_{1} − H_{2}.  (4.15)

where t is a spectral parameter. The second evaluation map \overline{Ev}_{t} : B_{+} → U_{q}(gl(3))[t] has the form

\overline{Ev}_{t} = σ_{13} · Ev_{−t} · σ_{13},  (4.16)

where the automorphisms σ_{13} entering this formula are defined in (4.10), (4.13). Explicitly one has

\overline{Ev}_{t}(y_{1}) = E_{23},  \overline{Ev}_{t}(y_{2}) = t\overline{E}_{31},  \overline{Ev}_{t}(y_{3}) = E_{12},  (4.17)
\overline{Ev}_{t}(h_{1}) = H_{2} − H_{3},  \overline{Ev}_{t}(h_{2}) = H_{3} − H_{1},  \overline{Ev}_{t}(h_{3}) = H_{1} − H_{2}.  (4.18)

One can easily check that both these maps are algebra homomorphisms as all the defining relations (3.3), (3.1) of the algebra B_{+} become simple corollaries of (4.12).

For any μ = (μ_{1}, μ_{2}, μ_{3}), such that μ_{α} = μ_{1} − μ_{2} and μ_{β} = μ_{2} − μ_{3} are nonnegative integers denote by π_{μ} the finite dimensional representation of U_{q}(gl(3)) with the highest weight μ and highest weight vector | 0⟩ defined as

E_{12}| 0⟩ = E_{23}| 0⟩ = 0,  H_{i}| 0⟩ = μ_{i}| 0⟩,  i = 1, 2, 3.  (4.19)

Further let π_{μ}(t) be the representation of B_{+} given by the composition of π_{μ} with the evaluation homomorphism Ev_{t}. Evidently, the operator-valued matrices

L_{μ}(t) = π_{μ}(t)[L],  (4.20)

(whose matrix elements are operators acting in (3.25)) satisfy suitable specializations of the quantum Yang-Baxter equation (3.39). These operators (4.20) are quantum analogues of the classical L-operators (2.48) whereas the operators

T_{μ}(t) = Tr_{π_{μ}(t)} [e^{iπP h}L],  (4.21)

are the quantum analogues of T^{(cl)}(λ) in (2.51) (the spectral parameter t is related to λ in (2.48) and (2.51) as t = λ^{3}). ¿From the standard arguments based on the Yang-Baxter equation it follows that

[T_{μ}(t), T_{μ′}(t′)] = 0.  (4.22)

Let us show that they also commute with all local IM (2.8),

[T_{μ}(t), I_{k}] = 0.  (4.23)

The characteristic property of the local IM is that their commutators with vertex operators (3.26) reduce to total derivatives [FF]

[ I_{k} , V_{i}(u) ] = ∂_{u}{ : O_{k}^{(i)}(u)V_{i}(u) : } ≡ ∂_{u}Θ_{k}^{(i)}(u),  (4.24)

where O_{k}^{(i)}(u) are some polynomials with respect to the field ∂_{u}ϕ(u) and its derivatives. Obviously, the operators Θ_{k}^{(i)}(u) in (4.24) obey the same periodicity properties as the vertex operators V_{i}(u), (3.30),

Θ_{k}^{(j)}(u + 2π) = q^{−2}e^{−4iπPe_{j}}Θ_{k}^{(j)}(u),  (4.25)

and satisfy the commutation relations

\begin{aligned}Θ_{k}^{(i)}(u_{1})V_{j}(u_{2}) & = & q^{a_{ij}}V_{j}(u_{2})Θ_{k}^{(i)}(u_{1}),  u_{1} > u_{2}, \\ [P , Θ_{k}^{(j)}(u)] & = & −g e_{j} Θ_{k}^{(j)}(u) , \end{aligned}  (4.26)

where a_{ij} is defined in (2.38). It is easy to check that the triple

I = I_{k},  A_{i} = Θ_{k}^{(i)}(2π),  B_{i} = Θ_{k}^{(i)}(0),  (4.27)

satisfies the conditions (3.16),(3.17) of Proposition 1 with the generators x_{i} ∈ B_{+} realized as in (3.40) Therefore taking into account (3.41) and applying (3.18) one obtains

[I_{k}, \overline{L}] = Θ_{k}(2π) \overline{L} − \overline{L} Θ_{k}(0),  (4.28)

where

Θ_{k}(u) = ∑_{i=1}^{3} y_{i}Θ_{k}^{(i)}(u).  (4.29)

Now substitute (4.21) into the RHS of (4.23). Using the relation (4.28), the commutation relations (3.1), (4.26) and the cyclic property of the trace it is not difficult to show that the commutator (4.23) actually vanishes.

The most important T-operators correspond to the three-dimensional (fundamental) representations of π_{ω_{α}}, and π_{ω_{β}} of U_{q}(gl(3)) with the highest weights ω_{α} = (1, 0, 0) and ω_{β} = (1, 1, 0) (The corresponding representation matrices are given explicitly in Appendix D). We will introduce special notation for these T-operators (adjusting the normalisation of the spectral parameter for later convenience)

𝕋(t) ≡ T_{(1,0,0)}(tq^{− \frac{3}{2}} ),  \overline{𝕋}(t) ≡ T_{(1,1,0)}(tq^{− \frac{1}{2}} ).  (4.30)

Computing the trace in (4.21) in this case one obtains

𝕋(t) = ∑_{n=0}^{∞} t^{n} G_{n} ,  \overline{𝕋}(t) = ∑_{n=0}^{∞} t^{n} \overline{G}_{n}.  (4.31)

Here

G_{0} = z_{1} + z_{2} + z_{3}, \overline{G}_{0} = z_{1}^{−1} + z_{2}^{−1} + z_{3}^{−1} , (4.32)

with z_{1}, z_{2} and z_{3} defined in (3.31) and for n ≥ 1

\begin{aligned}G_{n} & = & z_{1} J_{312}^{(n)} + z_{2} J_{123}^{(n)} + z_{3} J_{231}^{(n)}, \\ (−1)^{n}\overline{G}_{n} & = & z_{1}^{−1}J_{213}^{(n)} + z_{2}^{−1}J_{321}^{(n)} + z_{3}^{−1}J_{132}^{(n)}, \end{aligned}  (4.33)

J_{ijk}^{(n)} = ∫_{2π≥u_{1}≥u_{2}...≥u_{3n}} (V_{i}(u_{1})V_{j}(u_{2})V_{k}(u_{3}) V_{i}(u_{4})V_{j}(u_{5})V_{k}(u_{6}) ⋯  (4.34)
⋯ V_{i}(u_{3n−2})V_{j}(u_{3n−1})V_{k}(u_{3n})) du_{1} du_{2} … du_{3n},

where (i, j, k) is a permutation of (1, 2, 3).

It follows from (4.22) and (4.23) that the operators G_{n} and \overline{G}_{n} can be regarded as the “nonlocal” IM which commute among themselves and with all the local IM (2.8)

[G_{k}, G_{l}] = [G_{k}, \overline{G}_{l}] = [\overline{G}_{k}, \overline{G}_{l}] = [G_{k}, I_{l}] = [\overline{G}_{k}, I_{l}] = 0.  (4.35)

Definitions similar to those in (4.21) but with the map (4.18) instead of (4.15) lead to a new set of T-operators

\overline{T}_{μ}(t) = Tr_{\overline{π}_{μ}(t)} [e^{iπP h}L],  (4.36)

where \overline{π}_{μ}(t) denotes the representation of B_{+} obtained by the composition of the representation π_{μ} of U_{q}(gl(3)) with the evaluation map (4.18). The operators \overline{T}_{μ}(t) with different values of t commute among themselves and with all T_{μ}(t),

[\overline{T}_{μ}(t), \overline{T}_{μ′}(t′)] = [\overline{T}_{μ}(t), T_{μ′}(t′)] = 0.  (4.37)

In general for the same π_{μ} the formulae (4.21) and (4.36) define different operators. However, when π_{μ} is restricted to symmetric powers of a fundamental representation π_{ω_{α}} and π_{ω_{β}} then the two sets of T-operators coincide (up to a rescaling of spectral parameter). For future references it will be convenient to define

𝕋_{m}(t) ≡ T_{(m,0,0)}(tq^{−m− \frac{1}{2}} ) = \overline{T}_{(m,0,0)}(tq^{m+ \frac{1}{2}} ),  (4.38)

and

\overline{𝕋}_{m}(t) ≡ T_{(m,m,0)}(tq^{−m+ \frac{1}{2}} ) = \overline{T}_{(m,m,0)}(tq^{m− \frac{1}{2}} ),  (4.39)

where m = 0, 1, 2, …, ∞. In particular, 𝕋_{0}(t) = \overline{𝕋}_{0}(t) ≡ 1 while 𝕋_{1}(t) ≡ 𝕋(t), \overline{𝕋}_{1}(t) ≡ \overline{𝕋}(t) are exactly the same as in (4.31).

4.3 The Q-operators

The operators (4.21) and (4.36) are obviously the CFT versions of Baxter’s transfermatrices of the lattice models related to the U_{q}(ŝl(3)) algebra. It is well known that the theory of solvable lattice models involves other important objects, namely Baxter’s Q-matrix. Similarly to the standard CFT case [3] appropriate W_{3} CFT versions of Qoperators can be obtained as certain specializations of the universal L-operator (3.34).

The generators y_{i}, h_{i}, i = 1, 2, 3, h_{1} + h_{2} + h_{3} = 0, of the Borel subalgebra B_{+} ⊂ U_{q}(ŝl(3)) satisfy the commutation relations (3.1)

[h_{i}, h_{j}] = 0,  [h_{i}, y_{i}] = a_{ij},  (4.40)

and the Serre relations (3.3). The later can be written as

y_{i}y_{ij} = q^{−1}y_{i,j}y_{i},  i, j = 1, 2, 3,  i ≠ j,  (4.41)

where y_{ij} stands for the q-commutator of y_{i} and y_{j}:

y_{ij} = y_{i}y_{j} − qy_{j}y_{i}.  (4.42)

Let a_{i}^{+}, a_{i}^{−}, H_{i}, i = 1, 2 be the generators of two independent q-oscillator algebras H_{q},

[H_{i}, a_{i}^{±}] = ±a_{i}^{±},  qa_{i}^{+}a_{i}^{−} − q^{−1}a_{i}^{−}a_{i}^{+} = \frac{1}{q − q^{−1}} .  (4.43)

Define the following homomorphism ρ_{t} : B_{+} → H_{q} ⊗ H_{q}

\begin{aligned}ρ_{t}(y_{1}) & = & q^{\frac{1}{2}} t(q − q^{−1}) q^{−H_{2}}a_{1}^{−}a_{2}^{+}, & ρ_{t}(y_{2}) & = & a_{2}^{−}, & ρ_{t}(y_{3}) & = & a_{1}^{+}q^{H_{2}}, \\ ρ_{t}(h_{1}) & = & −H_{1} + H_{2}, & ρ_{t}(h_{2}) & = & −H_{1} − 2H_{2}, & ρ_{t}(h_{3}) & = & 2H_{1} + H_{2}, \end{aligned} (4.44)

where t is a spectral parameter. Note that this map subjects the generators y_{i}, i = 1, 2, 3 to some additional relations which are not implied by (4.40) and the Serre relations (4.41).

ρ_{t} (qy_{3}y_{21} − q^{−1}y_{21}y_{3}) = ρ_{t} (qy_{13}y_{2} − q^{−1}y_{2}y_{13}) = −\frac{q^{\frac{1}{2}} t}{q − q^{−1}} ,  ρ_{t} (y_{23}) = 0, (4.45)

where y_{ij} is defined in (4.42). Define also the maps ρ_{i}(t) and \overline{ρ}_{i}(t) : B_{+} → H_{q} ⊗ H_{q}, i = 1, 2, 3 obtained as compositions of (4.44) with the automorphisms (4.9),(4.10) of the algebra B_{+}

ρ_{i}(t) = ρ_{t} · τ^{−i+1},  \overline{ρ}_{i}(t) = ρ_{−t} · τ^{−i+1} · σ_{jk}  i = 1, 2, 3,  (4.46)

where ρ_{t} is given by (4.44) and ρ_{−t} is obtained from (4.44) by the replacement t → −t. Let π_{i} and \overline{π}_{i} be any representations of H_{q} ⊗ H_{q} such that the traces

Z_{i} = Tr_{π_{i}ρ_{i}(t)}[e^{2iπp h}] and \overline{Z}_{i} = Tr_{\overline{π}_{i}\overline{ρ}_{i}(t)}[e^{2iπp h}],  (4.47)

converge and do not vanish for

ℑm e_{1}p < 0,  ℑm e_{3}p < 0,  (4.48)

where p = (p_{1}, p_{2}) ∈ ℂ^{2}.

Define the operators

A_{i}(t) = Z_{i}^{−1} Tr_{π_{i}ρ_{i}^{+}(t)}[e^{iπP h}L] and \overline{A}_{i}(t) = \overline{Z}_{i}^{−1} Tr_{\overline{π}_{i}\overline{ρ}_{i}^{+}(t)}[e^{iπP h}L],  (4.49)

where L is given in (3.34). Since we are interested in the action of these operators in F_{p} the operator P in (4.49) can be substituted by its eigenvalue p = (p_{1}, p_{2}). The definitions (4.49) obviously apply to the domain (4.48). However they can be extended for all complex p ∈ ℂ^{2} (except for some set of singular points for real values p_{1} and p_{2}) by an analytic continuation in p.

Obviously, the operators (4.49) can be written as a power series. For example,

A_{i}(t) = 1 + ∑_{n=1}^{∞} ∑_{{σ_{i}=1,2,3}} a_{3n}^{(i)}(σ_{1}, σ_{2}, …, σ_{3n})𝕁_{3n}(σ_{1}, σ_{2}, …, σ_{3n}),  (4.50)

where

𝕁_{3n}(σ_{1}, σ_{2}, …, σ_{3n}) = ∫_{2π≥u_{1}≥···≥u_{3n}} V_{σ_{1}}(u_{1})V_{σ_{2}}(u_{2}) ⋯ V_{σ_{3n}}(u_{3n}),  (4.51)

and

a_{3n}^{(i)}(σ_{1}, σ_{2}, …, σ_{3n}) = Z_{i}^{−1}Tr_{π_{i}ρ_{i}(t)}(e^{2πiPh}y_{σ_{1}}y_{σ_{2}} ⋯ y_{σ_{3n}}).  (4.52)

Note that these coefficients vanish unless the “total charge” of the product of vertex operators in (4.51)

e_{σ_{1}} + e_{σ_{2}} + ⋯ + e_{σ_{3n}} = 0,  (4.53)

is zero. The spectral parameter t is absorbed in the coefficients a_{3n}^{(i)} ∼ O(t^{n}) so that (4.50) is a power series in t.

A remarkable feature of the definitions (4.49) is that the coefficients (4.52) are completely determined by the commutation relations (4.40), (4.41), (4.45) and the cyclic property of the trace, so the specific choice of the representations π_{i} and \overline{π}_{i} is not significant as long as the above convergence property is maintained. The operator coefficients in the power series (4.50) can be expressed in terms of basic nonlocal IM (4.33). For example, calculating the first nontrivial coefficient in (4.50) one gets

\begin{aligned}A_{i}(t) & = & 1 + \frac{q^{\frac{5}{2}} (qz_{i}G_{1} − \overline{G}_{1})}{(q^{2} − 1)(z_{j} − q^{2}z_{i})(z_{k} − q^{2}z_{i})} t + O(t^{2}), \\ \overline{A}_{i}(t) & = & 1 − \frac{q^{\frac{5}{2}} (z_{i}G_{1} − q\overline{G}_{1})}{(q^{2} − 1)(q^{2}z_{j} − z_{i})(q^{2}z_{k} − z_{i})} t + O(t^{2}), \end{aligned}  (4.54)

where z_{i}, i = 1, 2, 3, defined in (3.31). For further references introduce an alternative set of nonlocal IM, H_{i}^{(n)}, \overline{H}_{i}^{(n)}, i = 1, 2, 3, n = 1, 2 … ∞ defined as coefficients in the expansions

log A_{j}(t) = − ∑_{n=1}^{∞} s^{n} H_{n}^{(j)},  log \overline{A}_{j}(t) = − ∑_{n=1}^{∞} s^{n} \overline{H}_{n}^{(j)},  (4.55)

where

s = i (g^{−1}Γ(1 − g))^{3} t.  (4.56)

The operators A_{i}(t) and \overline{A}_{i}(t) simplify considerably when acting in the space F_{p} with values of 2p belonging to the weight lattice of the algebra sl_{3}

2p = n_{1}w_{1} + n_{2}w_{2},  n_{1}, n_{2} ∈ ℤ.  (4.57)

In this case the coefficients (4.52) can be written as

a_{3n}(σ_{1}, σ_{2}, …, σ_{3n}) = Tr_{u}(u_{σ_{1}}u_{σ_{2}} ⋯ u_{σ_{3n}}),  (4.58)

where u_{1}, u_{2} are generators of the Weyl algebra

u_{1}u_{2} = qu_{2}u_{1},  (4.59)

with the trace defined as [51]

Tr_{u}(u_{1}^{n}u_{2}^{m}) = δ_{n,0}δ_{m,0},  n, m ∈ ℤ  (4.60)

and

u_{3} = \frac{q^{\frac{1}{2}} t}{(q − q^{−1})^{3}} u_{2}^{−1}u_{1}^{−1} .  (4.61)

Obviously, the expression (4.58) is invariant with respect to the cyclic automorphism τ in (4.9) therefore all three operators A_{i}(t), i = 1, 2, 3 coincide. With the help of (4.50), (4.58) they can be written as

A_{i}(t)|_{2p=(4.57)} = Tr_{u}[P exp ∫_{0}^{2π} (u_{1}V_{1}(v)+u_{2}V_{2}(v)+u_{3}V_{3}(v))dv],  i = 1, 2, 3. (4.62)

Exactly the same expression but with u_{1} and u_{2} interchanged (which is equivalent to the replacement q → q^{−1} in (4.59)) holds for the operators \overline{A}_{i}(t).

The Q-operators (the CFT analogues of Baxter’s Q-matrix for U_{q}(ŝl(3) related lattice models) are defined as

Q_{i}(t) = t^{\frac{w_{i}P}{g}} A_{i}(t),  \overline{Q}_{i}(t) = t^{− \frac{w_{i}P}{g}} \overline{A}_{i}(t),  (4.63)

where w_{i}, i = 1, 2, 3, are defined in (2.34).

Evidently, the operators Q_{i}(t) and \overline{Q}_{i}(t) with different spectral parameters commute among themselves and with all the operators T_{μ}(t) and \overline{T}_{μ}(t) constructed above. They also commute with all the local IM defined in (2.8)

[Q_{i}(t), I_{k}] = 0,  [\overline{Q}_{i}(t), I_{k}] = 0.  (4.64)

Indeed, the proof of the similar statement (4.23) for the T-operators given above does not depend on a particular choice of the representation of B_{+} and relies only on the commutation relations (4.40), (4.41) and the cyclic property of the trace. Therefore this proof applies to (4.64) as well.

As is known [52–54] the local IM I_{k} do not change under the transformation g → g^{−1}, provided one makes simultaneous substitution ϕ(u) → g^{−1}ϕ(u)

I_{k}{g, ϕ(u)} = I_{k}{g^{−1}, g^{−1}ϕ(u)}.  (4.65)

Obviously, the nonlocal IM G_{n} and \overline{G}_{n} (as well as H_{i}^{(n)} and \overline{H}_{i}^{(n)}) do change and there exists an infinite set of dual nonlocal IM [3] obtained from (2.8) by the above substitution. For instance,

G̃_{n} = z1^{\frac{1}{g}} J̃_{312}^{(n)} + z2^{\frac{1}{g}} J̃_{123}^{(n)} + z3^{\frac{1}{g}} J̃_{231}^{(n)},  (4.66)
J̃_{ijk}^{(n)} = ∫_{2π≥u_{1}≥u_{2}...≥u_{3n}} (Ṽ_{i}(u_{1})Ṽ_{j}(u_{2})Ṽ_{k}(u_{3}) Ṽ_{i}(u_{4})Ṽ_{j}(u_{5})Ṽ_{k}(u_{6}) ⋯  (4.67)
⋯ Ṽ_{i}(u_{3n−2})Ṽ_{j}(u_{3n−1})Ṽ_{k}(u_{3n})) du_{1} du_{2} … du_{3n},

where (i, j, k) is a permutation of (1, 2, 3) and

Ṽ_{i}(u) = q̃^{\frac{1}{2}} e^{−2e_{i}ϕ(u)/g},  q̃ = e^{iπ/g}.  (4.68)

Similarly, define

˜\overline{G}_{n}\{g, ϕ(u)\} = \overline{G}_{n}\{g^{−1}, g^{−1}ϕ(u)\},  (4.69)

and

H̃_{i}^{(n)}\{g, ϕ(u)\} = H_{i}^{(i)}\{g^{−1}, g^{−1}ϕ(u)\},  ˜  \overline{H}_{i}^{(n)}\{g, ϕ(u)\} = \overline{H}_{i}^{(n)}\{g^{−1}, g^{−1}ϕ(u)\}. (4.70)

For g < 3/2 the above definitions require analytic continuation in g as described in [3].

4.4 Vacuum eigenvalues

The Fock space (3.25) naturally splits into level subspaces F_{p}^{(ℓ)}, ℓ = 0, 1, 2, …, defined as

F_{p} = ∑_{ℓ=0}^{∞} F_{p}^{(ℓ)},  L_{0}F_{p}^{(ℓ)} = (Δ_{2} + ℓ)F_{p}^{(ℓ)}.  (4.71)

All the IM (both local and nonlocal) act invariantly in each level subspace F_{p}^{(ℓ)}. In particular the vacuum state is an eigenstate of all these operators,

I_{k} |p⟩ = I_{k}^{(vac)} |p⟩,  G_{n} |p⟩ = G_{n}^{(vac)} |p⟩,  \overline{G}_{n} |p⟩ = \overline{G}_{n}^{(vac)} |p⟩.  (4.72)

Below we list a few vacuum eigenvalues I_{k}^{(vac)} which readily follow from (2.16)

I_{1}^{(vac)} = Δ_{2} − \frac{c}{24} ,
I_{2}^{(vac)} = Δ_{3} ,
I_{4}^{(vac)} = (Δ_{2} − \frac{c + 6}{24}) Δ_{3} ,  (4.73)
I_{5}^{(vac)} = Δ_{2}^{3} + \frac{4}{3}Δ_{3}^{2} − \frac{c + 8}{8}Δ_{2}^{2} + \frac{(c + 15)(c + 2)}{192}Δ_{2} − \frac{c(c + 23)(7c + 30)}{96768}.

Using (4.33) one can express the vacuum eigenvalues of the nonlocal IM as Coulombtype integrals in a circle. In general they cannot be evaluated in a closed form. Even the simplest eigenvalues G_{1}^{(vac)} and \overline{G}_{1}^{(vac)} reduce to the generalised hypergeometric integral

J(x, y) = ∫_{0}^{1} \frac{dω_{1}}{ω_{1}} ∫_{0}^{1} \frac{dω_{2}}{ω_{2}} \frac{ω_{1}^{x+g}ω_{2}^{y+g}}{(1 − ω_{1})^{g}(1 − ω_{2})^{g}(1 − ω_{1}ω_{2})^{g}}
= \frac{Γ(g + x)Γ(g + y)Γ^{2}(1 − g)}{Γ(1 + x)Γ(1 + y)} _{3}F_{2} ({g, g + x, g + y \atop 1 + x, 1 + y} ; 1) .  (4.74)

For example,

G_{1}^{(vac)} = z_{2} ∫_{0}^{2π} du_{3} ∫_{u_{3}}^{u_{3}+2π} du_{1} ∫_{u_{3}}^{u_{1}} du_{2} \frac{e^{−2ip(e_{1}u_{1}+e_{2}u_{2}+e_{3}u_{3})}}{[(2 sin \frac{u_{1} − u_{2}}{2} )(2 sin \frac{u_{1} − u_{3}}{2} )(2 sin \frac{u_{2} − u_{3}}{2} )]^{g}} ,

(4.75)

can be expressed through (4.74) in the following alternative forms

G_{1}^{(vac)} = −2πq^{− \frac{3}{2}} [q(q^{2}z_{1} − z_{2})J_{32} − (q^{2}z_{2} − z_{3})J_{12}],
= −2πq^{− \frac{3}{2}} [q(q^{2}z_{2} − z_{3})J_{13} − (q^{2}z_{3} − z_{1})J_{23}],  (4.76)
= −2πq^{− \frac{3}{2}} [q(q^{2}z_{3} − z_{1})J_{21} − (q^{2}z_{1} − z_{2})J_{31}],

where

J_{ij} = J(2e_{i}p, −2e_{j}p),  (4.77)

with J(x, y) defined in (4.74) and z_{1}, z_{2} and z_{3} denote the vacuum eigenvalues of the corresponding operators in (3.31). The vacuum eigenvalue of \overline{G}_{1}^{(vac)} is obtained from G_{1}^{(vac)} by the negation of the overall sign in (4.75), (4.76) and the replacement

z_{1} → z_{3}^{−1},  z_{2} → z_{2}^{−1},  z_{3} → z_{1}^{−1},  e_{1} ↔ e_{3},  e_{2} → e_{2},  (4.78)

induced by the automorphism σ_{13},

\overline{G}_{1}^{(vac)} = 2πq^{− \frac{3}{2}} [q(q^{2}z_{3}^{−1} − z_{2}^{−1})J_{12} − (q^{2}z_{2}^{−1} − z_{1}^{−1})J_{32}],
= 2πq^{− \frac{3}{2}} [q(q^{2}z_{2}^{−1} − z_{1}^{−1})J_{31} − (q^{2}z_{1}^{−1} − z_{3}^{−1})J_{21}],  (4.79)
= 2πq^{− \frac{3}{2}} [q(q^{2}z_{1}^{−1} − z_{3}^{−1})J_{23} − (q^{2}z_{3}^{−1} − z_{2}^{−1})J_{13}].  (4.80)

Using (4.54), (4.76) and (4.79) one can obtain the vacuum eigenvalues of the nonlocal IM H_{i}^{(1)} and \overline{H}_{i}^{(1)} defined in (4.55),

H_{1}^{(i)(vac)} = −\frac{π g^{3} J_{kj}}{Γ^{3}(1 − g) sin(πg)},  \overline{H}_{1}^{(i)(vac)} = \frac{π g^{3} J_{jk}}{Γ^{3}(1 − g) sin(πg)},  (4.81)

where (i, j, k) is a cyclic permutation of (1, 2, 3).

It will be useful to consider the special choice of vacuum parameter p = (p, 0). In this case, we have

(e_{1}p) = \frac{p}{2},  (e_{2}p) = −p,  (e_{3}p) = \frac{p}{2},  (4.82)

and

z_{1} = z_{3}^{−1} = e^{2πip},  z_{2} = 1.  (4.83)

Both G_{1}^{(vac)} and H_{1}^{(1)(vac)} simplify considerably in this case,

G_{1}^{(vac)}(p, 0) = \frac{4π^{3}Γ^{2}(1 − g)Γ(1 − \frac{3g}{2})}{Γ(1 − \frac{g}{2})Γ(1 − g + p)Γ(1 − g − p)Γ(1 − g/2 + p)Γ(1 − g/2 − p)}, (4.84)
H_{1}^{(1)(vac)}(p, 0) = −\frac{π}{2 sin(πg)} \frac{g^{3}Γ (1 − \frac{3g}{2})}{Γ (1 − \frac{g}{2}) Γ (1 − g)} \frac{Γ(g + p)Γ(g/2 + p)}{Γ(1 − g + p)Γ(1 − g/2 + p)}, (4.85)

where we have used eq. (4.4(5)) of [55] to simplify the hypergeometric functions J_{ij}.

5 Functional Relations

The T- and Q-operators introduced in the previous Section enjoy a variety of important functional relations (FR). In fact, many such functional relations have been previously obtained in the context of the solvable lattice models associated with U_{q}(ŝl(3)) algebra (see, e.g. [12–14, 16, 17]). It turns out that the explicit construction of the Q-operators given in the previous Section, (which defines them as special quantum transfer matrices associated with the q-oscillator algebra), can be used to considerably simplify the derivation of these FR. In fact, we found it rather remarkable that all known FR relevant here (as well as some new ones) can be obtained as elementary corollaries of just two basic relations which express the operators T_{μ}(t), (4.21), and \overline{T}_{μ}(t), (4.36), corresponding to finite dimensional representations π_{μ} of U_{q}(gl(3)) (i.e. those with integral non-negative weights μ_{α} = μ_{1} − μ_{2} ≥ 0 and μ_{β} = μ_{2} − μ_{3} ≥ 0) in terms of the operators Q_{i}(t) and \overline{Q}_{i}(t)

T_{(μ_{1},μ_{2},μ_{3})}(t) = z_{0}^{−1} det ∥ Q_{i}(t q^{+2μ′_{j}})∥_{i,j=1,2,3}  (5.1)
−\overline{T}_{(μ_{1},μ_{2},μ_{3})}(t) = z_{0}^{−1} det ∥ \overline{Q}_{i}(t q^{−2μ′_{j}})∥_{i,j=1,2,3}  (5.2)

where μ′_{1} = μ_{1} + 1, μ′_{2} = μ_{2}, μ′_{3} = μ_{3} − 1, and

z_{0} = (z_{1} − z_{2})(z_{1} − z_{3})(z_{2} − z_{3}),  (5.3)

with z_{1}, z_{2} and z_{3} defined in (3.31). The operators Q_{i}(t), \overline{Q}_{i}(t) are not functionally independent. They satisfy the relations

c_{i} \overline{Q}_{i}(t) = Q_{j}(tq)Q_{k}(tq^{−1}) − Q_{k}(tq)Q_{j}(tq^{−1}),  (5.4)
−c_{i} Q_{i}(t) = \overline{Q}_{j}(tq)\overline{Q}_{k}(tq^{−1}) − \overline{Q}_{k}(tq)\overline{Q}_{j}(tq^{−1}),  (5.5)

where (i, j, k) = cycle(1, 2, 3), and

c_{i} = zi^{\frac{1}{2}} (z_{j} − z_{k}).  (5.6)

The proof of these relations is given in Appendix B (see also the discussion at the end of this Section). Below we present some simple corollaries of (5.1), (5.2), (5.4), and (5.5).

The operators 𝕋(t) and \overline{𝕋}(t) defined in (4.31) corresponding to the three-dimensional fundamental representations of U_{q}(gl(3)) satisfy the following relations with the Q-opera-tors

Q_{i}(tq^{3}) − 𝕋(tq^{\frac{1}{2}} ) Q_{i}(tq) + \overline{𝕋}(tq^{− \frac{1}{2}} ) Q_{i}(tq^{−1}) − Q_{i}(tq^{−3}) = 0,  (5.7)
\overline{Q}_{i}(tq^{3}) − \overline{𝕋}(tq^{\frac{1}{2}} ) \overline{Q}_{i}(tq) + 𝕋(tq^{− \frac{1}{2}} ) \overline{Q}_{i}(tq^{−1}) − \overline{Q}_{i}(tq^{−3}) = 0,  (5.8)

where i = 1, 2, 3, These relations are obviously the U_{q}(ŝl(3))-analogues [12] of Baxter’s famous T-Q relation [11]. Given 𝕋(t) and \overline{𝕋}(t) each of the equations (5.7) is a third order difference equation for Q’s which should have three linearly independent solutions. Since 𝕋(t) and \overline{𝕋}(t) are single-valued functions of t, (i.e. periodic functions of log t) the operators Q_{i}(t) and \overline{Q}_{i}(t) can be interpreted as the “Bloch wave” solutions of (5.7) which satisfy the “quantum Wronskian” condition

z_{0} = det ∥Q_{i}(tq^{−2j})∥_{i,j=1,2,3} = − det ∥\overline{Q}_{i}(tq^{2j})∥_{i,j=1,2,3}  (5.9)

which is just a particular case of (5.1), (5.2) for μ_{i} = 0.

The equations (5.7) and (5.8) imply more familiar expressions

𝕋(t) = \frac{Q(t q^{\frac{5}{2}} )}{Q(t q^{\frac{1}{2}} )} + \frac{Q(t q^{− \frac{3}{2}} ) \overline{Q} (t q^{\frac{3}{2}} )}{Q(t q^{\frac{1}{2}} ) \overline{Q} (t q^{− \frac{1}{2}} )} + \frac{\overline{Q} (t q^{− \frac{5}{2}} )}{\overline{Q} (t q^{− \frac{1}{2}} )},  (5.10)

\overline{𝕋}(t) = \frac{\overline{Q}(t q^{\frac{5}{2}} )}{\overline{Q}(t q^{\frac{1}{2}} )} + \frac{\overline{Q}(t q^{− \frac{3}{2}} ) Q (t q^{\frac{3}{2}} )}{\overline{Q}(t q^{\frac{1}{2}} ) Q (t q^{− \frac{1}{2}} )} + \frac{Q (t q^{− \frac{5}{2}} )}{Q (t q^{− \frac{1}{2}} )},  (5.11)

where \{Q(t), \overline{Q}(t)\} is any of the six pairs \{Q_{i}(t), \overline{Q}_{j}(t)\} with i ≠ j. More general expressions [16] of this type for the T-operators with arbitrary weight μ can be obtained, for example, from (5.10), (5.11) and the relation (5.16) given below.

Further, using (5.4) and (5.5) in (5.1) and (5.2) one can easily deduce that

𝕋_{m}(t) = z_{0}^{−1} ∑_{i=1}^{3} c_{i} Q_{i}(tq^{m+ \frac{3}{2}} )\overline{Q}_{i}(tq^{−m− \frac{3}{2}} ),  (5.12)
\overline{𝕋}_{m}(t) = z_{0}^{−1} ∑_{i=1}^{3} c_{i} Q_{i}(tq^{−m− \frac{3}{2}} )\overline{Q}_{i}(tq^{m+ \frac{3}{2}} ),  (5.13)

Using these two results, the well known fusion relations, [17], can be deduced,

𝕋_{m}(tq)𝕋_{m}(tq^{−1}) = \overline{𝕋}_{m}(t) + 𝕋_{m−1}(t)𝕋_{m+1}(t),  (5.14)
\overline{𝕋}_{m}(tq)\overline{𝕋}_{m}(tq^{−1}) = 𝕋_{m}(t) + \overline{𝕋}_{m−1}(t)\overline{𝕋}_{m+1}(t),  (5.15)

Note that the formulae (5.1) and (5.2) strongly resemble the first Weyl formula for characters of gl(3). Note also, that the corresponding analogue of the second Weyl formula for the transfer matrices (sometimes called the quantum Jacobi-Trudi identity [17]) has been known for some time [16]

T_{(μ_{1},μ_{2},0)}(t) = det ∥ τ^{(μ_{i}^{T} −i+j)}(t q^{2(j−1)}) ∥_{1≤i,j≤μ_{1}}  (5.16)

where

τ^{(0)}(t) = τ^{(3)}(t) ≡ 1,  τ^{(1)}(t) = T_{(1,0,0)}(t),  τ^{(2)}(t) = T_{(1,1,0)}(t),  (5.17)

and τ^{(k)}(t) ≡ 0, for k < 0 and k > 3. The integers μ_{i}^{T}, i = 1, …, μ_{1}, read

μ_{i}^{T} = 2, 1 ≤ i ≤ μ_{2}; μ_{i}^{T} = 1, μ_{2} < i ≤ μ_{1}.

Again, the formula (5.16) easily follows from (5.1). Also, a similar expression for \overline{T}_{μ}(t) is obtained from (5.16) by the replacement q → q^{−1}.

The proof of the basic relations (5.1) and (5.2) given in Appendix B is a straightforward (but rather more technically complicated) generalisation of the corresponding U_{q}(ŝl(2)) results of [4]. We consider more general T-operators which correspond to the infinite dimensional highest weight representations of U_{q}(gl(3)). These new T-operators are defined by the same formulae as (4.21)

T_{μ}^{+}(t) = Tr_{π_{μ}^{+}} [e^{iπP h}L_{μ}^{+}(t)],  L_{μ}^{+}(t) = π_{μ}^{+}(t)[L],  (5.18)

except the trace is now taken over the infinite dimensional representations π_{μ}^{+} of U_{q}(gl(3)) with arbitrary highest weights μ = (μ_{1}, μ_{2}, μ_{3}) the highest weight vector | 0⟩ defined by (2.41). Similar modifications to (4.36) allow us to define the operators \overline{T}_{μ}^{+}(t) related to the second evaluation map (4.18). Then, by considering the tensor product of three q-oscillator algebras corresponding to the product of three Q_{i}(t) operators, we show that

T_{μ_{1},μ_{2},μ_{3}}^{+}(t) = z_{0}^{−1}Q_{1}(tq^{2μ_{1}+2})Q_{2}(tq^{2μ_{2}})Q_{3}(tq^{2μ_{3}−2}),  (5.19)

and similarly

\overline{T}_{μ_{1},μ_{2},μ_{3}}^{+}(t) = −z_{0}^{−1}\overline{Q}_{1}(tq^{−2μ_{1}−2})\overline{Q}_{2}(tq^{−2μ_{2}})\overline{Q}_{3}(q^{−2μ_{3}+2}).  (5.20)

Then, we make use of the Bernstein-Gel’fand-Gel’fand (BGG) resolution of the finitedimensional modules. The BGG result allows one to express the finite dimensional highest weight modules in terms of a direct sum of infinite dimensional highest weight modules. This implies that the T-operators for a finite dimensional module can be written in terms of (5.19), (5.20) as

T_{μ}(t) = ∑_{σ∈S_{3}}(−1)^{l(σ)}T_{σ(μ+ρ)−ρ}^{+}(t),  (5.21)

where ρ = (1, 0, −1), the summation is over all permutations σ ∈ S_{3} of three elements, and l(σ) is the parity of the permutation. There an analogous result for the operators \overline{T}_{μ_{1},μ_{2},μ_{3}}(t). The determinant formulas (5.1) and (5.2) follow immediately.

6 Conjectures: Exact Asymptotic Expansions

6.1 Assumptions

We will make the following assumptions

i) Analyticity. The operators A_{i}(t) and \overline{A}_{i}(t) (and hence all the T-operators introduced above) are entire functions of the complex variable t in the sense that all their matrix elements and eigenvalues are entire functions of this variable. Their leading asymptotics at large t are given by

log A_{i}(t) ∼ M (+it)^{\frac{1}{3(1−g)}} ,  log \overline{A}_{i}(t) ∼ M (−it)^{\frac{1}{3(1−g)}} ,  |t| → ∞  (6.1)

where M is some c-number constant depending on the central charge c only.

ii) Location of zeroes. For real p and large t the zeros of the eigenvalues of A_{i}(t) and \overline{A}_{i}(t) accumulate along the positive imaginary and negative imaginary axes of t respectively. The asymptotics (6.1) are valid everywhere except in the vicinity of the lines of zeroes (i.e. when | arg(it)| < π for A_{i}(t) and when | arg(−it)| < π for \overline{A}_{i}(t)).

As is known the zeroes of the eigenvalues of Q-operators satisfy the Bethe Ansatz equations. In our case there are six sets of these equations involving different pairs of eigenvalues \{A_{i}(t), \overline{A}_{j}(t)\} with i ≠ j. To simplify the notation let us choose a particular pair of eigenvalues, say \{A_{1}(t), \overline{A}_{3}(t)\}, and denote by \{t_{k}\}|_{k=1}^{∞} and \{\overline{t}_{k}\}|_{k=1}^{∞} their sets of zeroes

A_{1}(t_{k}) = 0, \overline{A}_{3}(\overline{t}_{k}) = 0, (6.2)

ordered as |t_{1}| ≤ |t_{2}| ≤ … and similarly \overline{t}_{k}. The equations (5.10), (5.11) (together with the standard arguments based on the analyticity of T-operators) imply the following Bethe-Ansatz equations

\frac{1}{2πi} log [\frac{A_{1}(t_{k}q^{2})}{A_{1}(t_{k}q^{−2})} \frac{\overline{A}_{3}(t_{k}q^{−1})}{\overline{A}_{3}(t_{k}q)}] = −2e_{3}p − n_{k}^{(1)} + \frac{1}{2},
\frac{1}{2πi} log [\frac{\overline{A}_{3}(\overline{t}_{k}q^{2})}{\overline{A}_{3}(\overline{t}_{k}q^{−2})} \frac{A_{1}(\overline{t}_{k}q^{−1})}{A_{1}(\overline{t}_{k}q)}] = −2e_{1}p − n_{k}^{(3)} + \frac{1}{2},

(6.3)

where n_{k}^{(1)}, n_{k}^{(3)} ∈ ℤ. Different eigenvalues of the operators A_{1}(t) and \overline{A}_{3}(t) correspond to different sets of integer phases {n_{k}^{(1)}, n_{k}^{(3)}}_{k=1}^{∞}.

The following assumption concerns the structure of the vacuum eigenvalues A_{1}^{(vac)}(t) and \overline{A}_{3}^{(vac)}(t) of operators A_{1}(t) and \overline{A}_{3}(t) acting in F_{p}.

iii). For real values of 2(w_{1}p) < g and 2(w_{3}p) > −g all zeroes of A_{1}(t) ( \overline{A}_{3}(t) ) located on the positive (negative) imaginary axis of t and the integer phases in (6.3) run over consecutive positive integers

n_{k}^{(1)} = n_{k}^{(3)} = k,  k = 1, 2, …, ∞.  (6.4)

6.2 Results

Following [3] consider the following operator valued function

Ψ_{i}(ν) = \frac{Γ(1 − g)^{−iν(1+ξ)}}{Γ(iνξ/3)Γ(−iν(ξ + 1)/3)Γ((−1 + iν)/3)Γ((1 + iν)/3)} ∫_{−i∞}^{0} \frac{dt}{t}(it)^{−iν(1+ξ)/3} A_{i}(t) ,

(6.5)

where the integration contour goes along negative imaginary axis and

ξ = \frac{g}{1 − g} .  (6.6)

This integral converges for 3(g − 1) < ℑm ν < −1, however the definition of Ψ_{j}(ν) can be extended to the whole complex plane of ν by analytic continuation. For example, using the product representation which holds for any eigenvalue A_{j}(t) of A_{j}(t)

A_{j}(t) = ∏_{n=1}^{∞} (1 − \frac{t}{t_{n}^{(j)}} ),  (6.7)

one can write the corresponding eigenvalue of Ψ_{j}(ν) as a Dirichlet series

Ψ_{j}(ν) = \frac{(Γ(1 − g))^{−iν(1+ξ)}Γ(iν(ξ + 1)/3)}{Γ(iνξ/3)Γ((−1 + iν)/3)Γ((1 + iν)/3)} ∑_{n=1}^{∞}(−it_{n}^{(j)})^{−iν(1+ξ)/3}.  (6.8)

For 0 < g < 2/3 this series converges absolutely for ℑm ν < −1 and can be analytically continued to the whole complex ν-plane by means of the standard technique [56]. We expect that the function Ψ(ν) thus defined enjoys the following analytic properties

Conjecture 1 The function Ψ(ν) is an entire function of the variable ν.

Converting the integral transform in (6.5) one obtains

log A_{i}(t) = \frac{1}{2πi} ∫_{C_{ν}} \frac{dν}{ν} Γ(iνξ/3)Γ(1 − iν(ξ + 1)/3)Γ((−1 + iν)/3) ×
Γ((1 + iν)/3)(Γ(1 − g))^{iν(1+ξ)}(it)^{iν(1+ξ)/3} Ψ(ν),  (6.9)

where the integration contour C_{ν} goes along the line ℑm ν = −1 − ε with arbitrary small positive ε.

The values of Ψ_{j}(ν) at special points on the imaginary axis of ν (where the Gammafunctions display poles) are of particular interest. For example, using (6.8), (6.7) and the definition of the nonlocal IM H_{n}^{(j)} in (4.55) it is easy to see that

Ψ_{j}(−3in(1 − g)) = \frac{(−1)^{n} n! g^{−3n}}{Γ(ng)Γ(−1/3 + n(1 − g))Γ(1/3 + n(1 − g))} H_{n}^{(j)}.  (6.10)

The other special values of Ψ_{j}(ν) of the imaginary axis are determined by the following

Conjecture 2 The function Ψ(ν) has the following special values on the positive imaginary axis

Ψ_{j}((2n − 1)i) = \frac{1}{6\sqrt{3}π} (\frac{2^{\frac{2}{3}}}{3} )^{2n−1} \frac{Γ(\frac{1}{6} − \frac{n}{3})}{Γ(\frac{1}{2} − n)} g^{n} I_{2n−1},  n = 0, 1, 2, …  (6.11)
Ψ_{j}(2ni) = \frac{1}{2\sqrt{3}π} (\frac{2^{\frac{2}{3}}}{3} )^{2n} \frac{Γ(\frac{1}{2} − \frac{n}{3})}{Γ(−\frac{1}{2} − n)} g^{n+ \frac{1}{2}} I_{2n},  n = 1, 2, …  (6.12)

Ψ_{j}(3in(g^{−1} − 1)) = \frac{(−1)^{n} n! g^{\frac{3n}{g} +1} H̃_{n}^{(j)}}{Γ(ng^{−1})Γ(−\frac{1}{3} + n(1 − g^{−1}))Γ(\frac{1}{3} + n(1 − g^{−1}))} ,  n = 1, 2, …

(6.13)

Ψ_{j}(0) = \frac{(w_{j}P)}{2π\sqrt{3}} ,  (6.14)

where j = 1, 2, 3 and I_{−1} ≡ I is the identity operator.

Below it will be more convenient to work with the variable s introduced in (4.56) instead of the spectral parameter t and exhibit all arguments of A(s)

A_{j}{s, g, ϕ(u)} ≡ exp ( − ∑_{n=1}^{∞} H_{n}^{(j)}s^{n}).  (6.15)

The above conjectures allow us to derive the asymptotic expansion of the operators A_{j}(t) for large t.

Calculating the integral (6.9) as a formal sum over residues in the upper half-plane ℑm ν ≥ −1 one has

A_{j}\{s, g, gϕ(u)\} ≃ C_{j}\{g, gϕ(u)\} s^{− \frac{w_{j} P}{g}} exp \{ ∑_{k=−1}^{∞}B_{k} s^{−k/3(1−g)} I_{k}\}A_{j}\{s^{− \frac{1}{g}} , g^{−1}, g^{−1}ϕ(u)\},

(6.16)

where

A_{j}\{s^{− \frac{1}{g}} , g^{−1}, g^{−1}ϕ(u)\} ≡ exp ( − ∑_{n=1}^{∞} H̃_{n}^{(j)}s^{− \frac{n}{g}} ),  (6.17)

and the coefficients B_{n} read

B_{2n−1} = \frac{(−1)^{n+1}Γ(\frac{2n − 1}{3(1 − g)})Γ(\frac{2n − 1}{3(1 − g^{−1})})}{3 (1 − g) n! Γ(\frac{2 − n}{3})} g^{\frac{n(1+g)−1}{g−1}} ,  n = 0, 1, 2, … (6.18)

B_{2n} = \frac{(−1)^{n+1}Γ(\frac{2n}{3(1 − g)})Γ(\frac{2n}{3(1 − g^{−1})})}{(1 − g) n! Γ( − \frac{n}{3})} g^{\frac{n(1+g)}{g−1} + \frac{1}{2}} ,  n = 1, 2, … (6.19)

An exact form for the operator C_{j}{g, gϕ(u)} is unknown. However, an asymptotic vacuum eigenvalue in a special case can be calculated (see (C.56)). Note, in particular, that the coefficient M in the leading asymptotics (6.16) is given by

M = \frac{Γ (\frac{g}{3(1−g)}) Γ (\frac{2−3g}{3(1−g)})}{Γ (\frac{2}{3})} (Γ(1 − g))^{\frac{1}{1−g}} .  (6.20)

Naturally, considerations similar to those above are valid for the operators \overline{A}_{j}(t) as well. The corresponding formulae are obtained from (6.9) simply by a replacement of A_{j}(t), Ψ_{j}(ν), H_{n}^{(j)} by their “barred” counterparts

A_{j}(t) → \overline{A}_{j}(t),  Ψ_{j}(ν) → \overline{Ψ}_{j}(ν),  H_{n}^{(j)} → \overline{H}_{n}^{(j)},  (6.21)

accompanied by sign changes

w_{j} → −w_{j},  I_{2n} → −I_{2n},  (6.22)

where j = 1, 2, 3 and n = 1, 2, …, ∞ (the sign of the “odd” local IM I_{2n−1} remains the same). Also, in the integrand of (6.9), the change t → −t needs to be made.

We can use these asymptotic results for the operators A_{i}(t), \overline{A}_{i}(t) to calculate an asymptotic expansion for 𝕋(t). Using (5.10), we can write 𝕋(t) in the following form

𝕋(t) = Λ_{i}(q^{\frac{3}{2}} t) + Λ_{i}^{−1}(q^{− \frac{1}{2}} t)\overline{Λ}_{j}(q^{\frac{1}{2}} t) + \overline{Λ}_{j}^{−1}(q^{− \frac{3}{2}} t),  i ≠ j,  (6.23)

where

Λ_{j}(t) = \frac{Q_{j}(qt)}{Q_{j}(q^{−1}t)},  \overline{Λ}_{j} = \frac{\overline{Q}_{j}(qt)}{\overline{Q}_{j}(q^{−1}t)}.  (6.24)

Then, using (6.9) we have

log Λ_{j} = 2πi(w_{j}P) + ∫_{C_{ν}} \frac{dν}{ν} \frac{Γ(1 − iν(ξ + 1)/3)Γ((−1 + iν)/3)Γ((1 + iν)/3)}{Γ(1 − iνξ/3)} ×
(Γ(1 − g))^{iν(1+ξ)}(it)^{iν(1+ξ)/3} Ψ_{j}(ν),  (6.25)

which gives via the sums of residues in the upper half-plane ℑm ν ≥ −1

log Λ_{j}(t) ≃ i m I (it)^{\frac{1}{3(1−g)}} + \begin{matrix}∞ \\ ∑ & ′ \\ k=1 \end{matrix} C_{k} (it)^{\frac{−k}{3(1−g)}} I_{k},  t → +∞  (6.26)

where

m = \frac{2πΓ(\frac{2}{3} − \frac{ξ}{3})}{Γ(1 − \frac{ξ}{3})Γ(\frac{2}{3})}(Γ(1 − g))^{\frac{1}{1−g}} ,  (6.27)

and

C_{k} = −2i g^{k(1+ξ)} sin (\frac{πξk}{3} ) (Γ(1 − g))^{− \frac{k}{1−g}} B_{k}.  (6.28)

A similar expression is obtained for log \overline{Λ}_{i}(t), using (6.21), (6.22). It is easy then to see that only the second term in (6.24) will contribute in the asymptotic limit, as the other two will be exponentially small. Thus we have

log 𝕋(t) ≃ mt^{\frac{1}{3(1−g)}} I − 2 ∑_{n=1}^{∞} C_{2n} cos (\frac{πn}{3}) (t)^{− \frac{2n}{3(1−g)}} I_{2n}
+ 2i ∑_{n=1}^{∞} C_{2n−1} sin (\frac{π(2n − 1)}{6}) (t)^{− \frac{2n−1}{3(1−g)}} I_{2n−1}.  (6.29)

The corresponding formula for \overline{𝕋}(t) is obtained from the above one by the replacement (6.22). This result can be regarded as the the quantum version of (2.29). In fact, if we take the limit ξ → 0 (equivalently q → 1), then, using the correspondence (2.20), we recover exactly the classical expression (2.29).^{6}

As in [3] the main motivation to the conjectures given above came from the study of the large p asymptotics of the vacuum eigenvalues of the operators A_{j}(t) (see Appendix C) as well as consideration of the classical limit g → 0 in the T-operator (6.23). In this way we obtain

Ψ_{j}(ν)|_{p→∞} = \frac{2^{− \frac{2iν}{3}} ( − \frac{g_{2}}{3} )^{(1−iν)/2}}{\sqrt{3π}Γ(1/2 + iν/3)}_{2}F_{1}(−\frac{1}{6} + \frac{iν}{6} , \frac{1}{6} + \frac{iν}{6} , \frac{1}{2} + \frac{iν}{3} , 1 + \frac{27g_{3}^{2}}{4g_{2}^{3}} )|_{j−th}, (6.30)

where the subscript j−th means that an appropriate branch of the hypergeometric function is chosen such that (6.14) holds, and g_{2}, g_{3} are defined by

g_{2} = x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3},  g_{3} = −x_{1}x_{2}x_{3},  (6.31)

where

x_{j} = (w_{j}P),  j = 1, 2, 3.  (6.32)

The expression (6.30) together with (6.9) allows one to derive the asymptotic expansion (6.16) in the limit p → ∞ which provided a basis for the conjectures (6.11), (6.12), (6.13), (6.14).

The formulae (6.16) allow us to obtain asymptotic expansions for the zeroes of the eigenvalues of the operators A_{j}(t) and \overline{A}_{j}(t). Let us introduce new (rescaled) variables E_{k}^{(j)}, \overline{E}_{k}^{(j)}, k = 1, 2, …, (whose meaning will become clear in Section 7.2) to denote the positions of these zeroes

E_{k}^{(j)} = −iρ t_{k}^{(j)},  \overline{E}_{k}^{(j)} = iρ \overline{t}_{k}^{(j)},  A_{j}(t_{k}^{(j)}) = \overline{A}_{j}(\overline{t}_{k}^{(j)}) = 0,  k = 1, 2, …, ∞ (6.33)

where

ρ = [(3/g)^{1−g}Γ(1 − g)]^{3}.  (6.34)

Substituting (6.16) into the Bethe Ansatz equations one obtains

N_{k}^{(j)} ≡ n_{k}^{(j)} − \frac{1}{2} + \frac{w_{j}p}{g} ≃ ∑_{n=−1}^{∞} β_{n} I_{n} (E_{k}^{(j)})^{− \frac{n}{3(1−g)}}  +∑_{n=1}^{∞} γ_{n} H̃_{n}^{(j)} (E_{k}^{(j)})^{− \frac{n}{g}} , (6.35)

\overline{N}_{k}^{(j)} ≡ \overline{n}_{k}^{(j)} − \frac{1}{2} − \frac{w_{j}p}{g} ≃ ∑_{n=−1}^{∞}(−1)^{n+1}β_{n} I_{n} (\overline{E}_{k}^{(j)})^{− \frac{n}{3(1−g)}} + ∑_{n=1}^{∞} γ_{n} ˜ \overline{H}_{n}^{(j)} (\overline{E}_{k}^{(j)})^{− \frac{n}{g}} , (6.36)

6Recall the correspondence of variables t = λ^{3}.

where

β_{n} = −π^{−1} 3^{n} g^{gn/(1−g)} sin(πn/3(1 − g)) B_{n}, γ_{k} = π^{−1} 3^{3(1−g)n/g} g^{3n} sin(πn/g),

(6.37)

the coefficients B_{n} are given in (6.18), (6.19) and I_{k}, H̃_{n}^{(j)} and ˜ \overline{H}_{n}^{(j)} are the corresponding eigenvalues of the local IM (2.8) and the dual nonlocal IM, defined in (4.70). Inverting the series (6.35), (6.36) one can express E_{k}^{(j)} and E_{k}^{(j)} as asymptotic series

E_{k}^{(j)} ≃ Υ(N_{k}^{(j)}),  \overline{E}_{k}^{(j)} ≃ \overline{Υ}(\overline{N}_{k}^{(j)}),  (6.38)

where, for example, for g < 3/5 one has

Υ(x) ≃ α_{0} x^{3−3g} (1 + α_{2}x^{−2} + α_{3}x^{−3} + O(x^{−4}) + O(x^{2−3/g})),  x → ∞ (6.39)
\overline{Υ}(x) ≃ α_{0} x^{3−3g} (1 + α_{2}x^{−2} − α_{3}x^{−3} + O(x^{−4}) + O(x^{2−3/g})),  x → ∞ (6.40)

α_{0} = (β_{−1})^{3(g−1)} = [\frac{3Γ(\frac{2}{3})Γ((ξ + 1)/3)}{gΓ(\frac{ξ}{3})}]^{3(1−g)} ,  (6.41)

α_{2} = β_{1}β_{−1} I_{1} = \frac{3(1 − g)I_{1}}{π(cot(π(ξ + 1)/3) − cot π/3)},  (6.42)

α_{3} = β_{2}β_{−1}^{2} I_{2} = \frac{3 g^{\frac{3}{2}} (1 − g) Γ( − \frac{2}{3}ξ)Γ(ξ/3)^{2}}{2Γ(\frac{2}{3})^{3} Γ( − \frac{2}{3}(ξ + 1)) Γ((ξ + 1)/3)} I_{2}.  (6.43)

The Bethe Ansatz equations (6.3) can be solved numerically. We have performed such calculations for vacuum eigenvalues of A_{1}(t) and \overline{A}_{3}(t) for a range of different values of g and p and compared the results with the asymptotic expansions (6.38). A typical example of these calculations with

g = \frac{1}{5},  p = ( _{\frac{1}{2}}, _{\frac{3}{10}} ),  (6.44)

is presented in Table 1 where the numerical values for E’s are compared with the corresponding values given by six terms of the asymptotic series (6.38) (the value of the leading term in (6.38) is also shown).

6.3 Special cases

The following special cases are related to a particular choice of the vacuum parameter p = (p, 0) such that

(w_{1}p) = p,  (w_{2}p) = 0,  (w_{3}p) = −p,  (6.45)

n  E_{n}^{(1)}(BA)  E_{n}^{(1)}(AS) E_{n}^{(1)}(LT)  \overline{E}_{n}^{(3)}(BA)  \overline{E}_{n}^{(3)}(AS) \overline{E}_{n}^{(3)}(LT)
1  426.44998  429.22624  615.63  42.72214  25.28503  147.89
2  871.85882  872.59835 1069.31  232.30688  229.38814  371.98
3 1466.71208 1466.98579 1674.60  563.02921  562.24812  723.05
4 2224.92354 2225.04588 2443.32 1039.72260 1039.43412 1216.12
5 3157.57647 3157.63845 3386.10 1673.43618 1673.30798 1864.09
6 4274.50955 4274.54393 4512.70 2474.97610 2474.91149 2678.44
7 5584.74586 5584.76628 5832.14 3454.37424 3454.33857 3669.55
8 7096.68686 7096.69967 7352.86 4620.94608 4620.92497 4846.97
9 8818.22784 8818.23622 9082.79 5983.39245 5983.37925 6219.54
10 10756.83708 10756.84278 11029.43 7549.88941 7549.88079 7795.56
11 12919.61477 12919.61876 13199.92 9328.16145 9328.15561 9582.83
12 15313.33879 15313.34165 15601.06 11325.54050 11325.53642 11588.77
13 17944.50152 17944.50363 18239.38 13549.01388 13549.01096 13820.41
14 20819.34016 20819.34174 21121.13 16005.26365 16005.26150 16284.47
15 23943.86187 23943.86307 24252.34 18700.69926 18700.69765 18987.41
16 27323.86522 27323.86615 27638.82 21641.48502 21641.48379 21935.42
17 30964.95845 30964.95918 31286.21 24833.56334 24833.56239 25134.47
18 34872.57528 34872.57586 35199.94 28282.67468 28282.67393 28590.32
19 39051.98867 39051.98914 39385.30 31994.37469 31994.37410 32308.54
20 43508.32291 43508.32328 43847.43 35974.04923 35974.04875 36294.55
21 48246.56426 48246.56456 48591.33 40226.92739 40226.92700 40553.57
22 53271.57046 53271.57072 53621.85 44758.09307 44758.09276 45090.72
23 58588.07919 58588.07940 58943.76 49572.49523 49572.49497 49910.94
24 64200.71562 64200.71580 64561.67 54674.95691 54674.95669 55019.08
25 70113.99930 70113.99945 70480.11 60070.18348 60070.18330 60419.85

Table 1: Comparison of the numerical solution of the Bethe Ansatz Equations (6.3), (6.4) for E_{n}^{(1)}, E_{n}^{(3)} with g, p given by (6.44), to the value from asymptotic series (6.38), E_{n}^{(i)}(AS), and the leading term of the series E_{n}^{(i)}(LT). and the weights (3.24) read

Δ_{2} = \frac{p^{2}}{g} + \frac{c − 2}{24},  Δ_{3} = 0.  (6.46)

The vacuum eigenvalues in this case obey the symmetry

A_{1}^{(vac)}(t) = \overline{A}_{3}^{(vac)}(−t),  A_{2}^{(vac)}(t) = \overline{A}_{2}^{(vac)}(−t),  A_{3}^{(vac)}(t) = \overline{A}_{1}^{(vac)}(−t), (6.47)
𝕋^{(vac)}(t) = \overline{𝕋}^{(vac)}(−t),  (6.48)

while the expression (4.81) reduces to

H_{1}^{(1)(vac)} = −\frac{π}{2 sin(πg)} \frac{g^{3}Γ (1 − \frac{3g}{2})}{Γ (1 − \frac{g}{2}) Γ (1 − g)} \frac{Γ(g + p)Γ(g/2 + p)}{Γ(1 − g + p)Γ(1 − g/2 + p)}.  (6.49)

Further the asymptotic expansion of the vacuum eigenvalues 𝕋^{(vac)}(t) given by (6.29) now reads

𝕋^{(vac)}(t) = m t^{\frac{1}{3(1−g)}} + 2i ∑_{n=1}^{∞} cos((n − 2)π/3) C_{2n−1} I_{2n−1}^{(vac)}t^{\frac{1−2n}{3(1−g)}} , | arg(t)| < \frac{π}{2} (1 − g).

(6.50)

6.3.1 The case g = 1/2, c = −10

Consider the case when

c = −10,  g = 1/2,  q = e^{iπg} = i.  (6.51)

Using (4.57) and (6.47) it easy to see that the Bethe Ansatz equations (6.3) in this case reduce to those for the U_{q}(ŝl_{2}) related Q-operators studied in [3]. This fact has been noticed in [20] and recently discussed in [57]. Therefore the functions A_{1}^{(vac)}(t) and A_{3}^{(vac)}(t) coincide (up to a normalisation of the spectral parameter t) with A_{+}^{(vac)}(t) and A_{−}^{(vac)}(t) from ref. [3] ^{7}. The relevant parameters used therein are related to our notation as follows

c^{sl_{2}} = −\frac{25}{2},  β^{2} = \frac{1}{4},  p^{sl_{2}} = \frac{p}{2},  Δ = \frac{Δ_{2}}{2} = p^{2} − 9/16.  (6.52)

Comparing now the asymptotic expansion (6.16) with the corresponding result of ref [3] (the eq.(4.19) therein) and using the expressions (4.73) and (6.49) one can reproduce the values of first three local IM I_{k}^{(sl_{2})(vac)}(Δ, c^{(sl_{2})} = −25/2) given in eq.(40) of ref. [2]

I_{1}^{(sl_{2})(vac)}(Δ, c^{sl_{2}}) = \frac{1}{2}I_{1}^{(vac)}(Δ_{2}, 0, c) = Δ + \frac{25}{48},  (6.53)
I_{3}^{(sl_{2})(vac)}(Δ, c^{sl_{2}}) = −\frac{3}{32}H_{1}^{(1)(vac)}(1/g, p/g) = Δ^{2} + \frac{7}{8}Δ + \frac{45}{256},  (6.54)
I_{5}^{sl(2)(vac)}(Δ, c^{sl_{2}}) = \frac{1}{8}I_{5}^{(vac)}(Δ_{2}, 0, c) = Δ^{3} + \frac{17}{16}Δ^{2} − \frac{175}{98304}Δ + \frac{15275}{774144}. (6.55)

7This correspondence extends to certain excited states as well (see also Section 8.2), but we will not elaborate this point here

6.3.2 The case g = 2/3, p = 1/6

Consider now the vacuum eigenvalue (6.50) of the operator 𝕋(t) when

c = −2,  g = 2/3,  p = 1/6.  (6.56)

The value g = 2/3 does lie in the domain (3.38) therefore the results of the previous Sections do not directly apply. In particular the definitions (4.31)-(4.34) require a renormalization since the nonlocal IM (4.33) diverge logarithmically at g = 2/3. To obtain the asymptotic of 𝕋(t) at large t in this case we set g = 2/3 − ϵ in (6.50) and let ϵ → 0 (this corresponds to an analytic regularization of divergent integrals). The first few terms of the formula (6.50) then read

log 𝕋^{(vac)}(t)|_{g=2/3,p=1/6} = −\sqrt{3} at log t + ℂ t + \frac{\sqrt{3}}{4!}(at)^{−1} − \frac{31\sqrt{3}}{8!}(at)^{−5} + … (6.57)

where ℂ is a renormalization constant and

a = Γ(1 − g)^{3} = Γ(1/3)^{3}.  (6.58)

On the other hand the eigenvalues of 𝕋(t) can be computed independently. Using the results of [58] one can show that in the case (6.56) the fusion equations (5.14) for the eigenvalues of 𝕋(t) simplify to a single equation

𝕋(tq^{\frac{1}{2}} )𝕋(tq^{− \frac{1}{2}} ) = 2 cosh(aπt) 𝕋(t),  q = e^{2πi/3}  (6.59)

which with an assumption that 𝕋(t) is an entire function of t and has the leading asymptotics at large t as in (6.57) can be completely solved

𝕋 𝕋 𝕋(t) = 𝕋^{(vac)}(t) ∏_{k=1}^{L} \frac{(1 + 2n_{k} − 2ate^{iπ/6})(1 + 2n_{k} − 2ate^{−iπ/6})}{(1 + 2n_{k} + 2ate^{iπ/6})(1 + 2n_{k} + 2ate^{−iπ/6})},  (6.60)

T^{(vac)}(t) = \frac{2πe^{ℂ′ t}}{Γ(\frac{1}{2} + ate^{iπ/6})Γ(\frac{1}{2} + ate^{−iπ/6})}.  (6.61)

where the integer L ≥ 0; {n_{k}} is an arbitrary increasing sequence of integers 0 ≤ n_{1} < n_{2} < … < n_{L} and ℂ′ is an arbitrary constant. The vacuum eigenvalue 𝕋^{(vac)}(t) is a unique solution of (6.59) with no zeroes in the strip |arg(t)| < π/3. One can easily check that its asymptotic expansion at large t perfectly agrees with (6.57).

6.3.3 The case g = 5/6

Below we will compare the asymptotic expansion (6.50) for g = 5/6 against the numerical results of [58] for several values of p

p = 1/6,  p = 1/3,  p = 2/3,  (6.62)

which corresponds to

Δ_{2} = 0,  Δ_{2} = 1/10,  Δ_{2} = 1/2,  (6.63)

respectively. Again, for g = 5/6 the formula (6.50) requires a renormalization. Using the analytic regularization as in the previous subsection one obtains

T^{(vac)}(t)|_{g=5/6} = −\frac{2}{\sqrt{3}}(at)^{2} log t + ℂ t − \frac{3\sqrt{3}}{2} (at)^{−2} I_{1}^{(vac)} + \frac{32805 \sqrt{3}}{43472}(at)^{−10} I_{5}^{(vac)} + …

(6.64)

where

a = Γ(1 − g)^{3} = Γ(1/6)^{3},  (6.65)

and I_{1}^{(vac)} and I_{5}^{(vac)} are given in (4.73).

I_{k}^{(vac)}(c, Δ_{2}, Δ_{3})  Numerical values [58]
I_{1}^{(vac)}(6/5, 0, 0)  = −\frac{1}{20}  = −0.05  −0.499999999599 10^{−1}
I_{5}^{(vac)}(6/5, 0, 0)  = −\frac{121}{10500} = −0.115238095238 … 10^{−1} −0.115238095240 10^{−1}
I_{1}^{(vac)}(6/5, 1/10, 0) = \frac{1}{20}  = 0.05  0.500000000002 10^{−1}
I_{5}^{(vac)}(6/5, 1/10, 0) = \frac{209}{42000} = 0.497619047619 … 10^{−2}  0.497619047634 10^{−2}

Table 2: Exact values of I^{(vac)}(c, Δ_{2}, Δ_{3}) for g = 5/6, and p given by (6.62), compared with the corresponding numerical results from [58].

As shown in [58] the eigenvalues of 𝕋(t) in this case satisfy the equation

𝕋(tq)𝕋(tq^{−1}) = e^{\frac{a^{2}π}{3} t^{2}}𝕋(−t) + e^{− \frac{a^{2}π}{3} t^{2}}𝕋(t).  (6.66)

In [58] this equation was transformed to TBA-like integral equations, due to Klümper and Pearce [59], and studied numerically for a few vacuum eigenvalues of 𝕋(t). In particular, the numerical values of the coefficients in (6.64) were calculated. We have verified that the values of the local IM extracted from this numerical data are in a good agreement with the corresponding exact values determined by (4.73) (see Table 2).

7 Applications

7.1 Boundary Affine Toda Theory

The Hamiltonian corresponding to the (non-equilibrium) boundary affine Toda theory (BAT) with zero bulk mass reads

H_{BAT} = H_{0} + H_{1},  (7.1)

H_{0} = \frac{1}{4πg} ∫_{−∞}^{0} dx ( Π^{2} + Φ_{x}^{2} ),  H_{1} = −\frac{κ}{2g} ∑_{i=1}^{3} u_{i} e^{i(e_{i},Φ_{B}+Vt)} ,  (7.2)

where g, κ and V = (V_{1}, V_{2}) are parameters; Φ(x) = (Φ_{1}(x), Φ_{2}(x)), Π(x) = (Π_{1}(x), Π_{2}(x)) are field operators obeying canonical commutation relations

[Π_{a}(x) , Φ_{b}(x′)] = −2πig δ(x − x′)δ_{ab},  a, b = 1, 2  (7.3)

the variables u_{1}, u_{2}, u_{3}

u_{1}u_{2} = qu_{2}u_{1},  u_{2}u_{3} = qu_{3}u_{2},  u_{3}u_{1} = qu_{1}u_{3},  u_{1}u_{2}u_{3} = q^{\frac{1}{2}} ,  (7.4)

describe boundary degree of freedom and Φ_{B} ≡ Φ(0). The way g enters the Hamiltonian (7.1) allows one to interpret it as a quantum parameter, because it always appears in the combination gℏ; in what follows we will set ℏ = 1. We will also assume that the parameter V satisfies the requirement

(e_{1}V ) > 0,  (e_{2}V ) > 0.  (7.5)

At a nonzero V and a temperature T the system (7.1) develops a stationary nonequilibrium state which can be thought of as the result of an infinite time evolution of the equilibrium state of the corresponding “free” system, with the interaction term (the last term in (7.1)) adiabatically switched on. We will denote by ⟨ A ⟩_{N} the expectation value of an observable A over this non-equilibrium stationary state.

The density matrix P(t) of the system is determined by the Schrödinger equation

∂_{t} P(t) = −i[H(t), P],  (7.6)

with the initial condition

P(t)|_{t=−∞} = P_{0} = Z_{0}^{−1} e^{−RH_{0}},  R = g/T,  (7.7)

where P_{0} is the equilibrium density matrix of the free system at the temperature T. Using the interaction representation corresponding to the Hamiltonian H_{0} in (7.1) one can write the density matrix P(t) as

P(t) = e^{−iH_{0}t} S(t, −∞) P_{0} S(−∞, t) e^{iH_{0}t} ,  (7.8)

where

S(t, t_{0}) = T exp { − i ∫_{t_{0}}^{t} dτ H_{1}^{(int)}(τ)} =
= 1 + ∑_{k=1}^{∞} (−i)^{k} ∫_{t_{0}}^{t} D_{k}({τ}) H_{1}^{(int)}(τ_{1}) H_{1}^{(int)}(τ_{2}) ⋯ H_{1}^{(int)}(τ_{n}),

(7.9) with

H_{1}^{(int)}(t) = e^{iH_{0}t} H_{1} e^{−iH_{0}t} .  (7.10)

In (7.9) and below the shorthand notation for the multiple ordered integrals

∫_{t_{0}}^{t} D_{k}({τ}) = ∫_{t_{0}}^{t} dτ_{1} ∫_{t_{0}}^{τ_{1}} dτ_{2} ⋯ ∫_{t_{0}}^{τ_{k−1}} dτ_{k}.  (7.11)

is used. The expectation value of an arbitrary operator A over the stationary nonequilibrium state (7.8) (which will be marked with the subscript N) reads

⟨A⟩_{N} = Tr_{H}[ P(t) A].  (7.12)

The trace is taken over the space H = F ⊗ U where F is the Fock space representing the Bose commutation relations (7.3) and U is the space of states of the boundary variables u_{i}, i = 1, 2, 3 (with the trace over these states defined exactly as in (4.60)). Substituting (7.8) into (7.12) one gets

⟨A⟩_{BAT} = Tr_{F}[ P_{0} S(−∞, t) A^{(int)}(t) S(t, −∞) ] = ⟨ S(−∞, t) A^{(int)}(t) S(t, −∞) ⟩_{0} ,

(7.13)

where ⟨ … ⟩_{0} denotes the expectation value over the equilibrium state (7.7) of the free system, and the superscript “(int)” means that this operator is taken in the interaction representation, i.e. A^{(int)}(t) = e^{iH_{0}t} A e^{−iH_{0}t}. Equivalently, one may write the above expectation value as

⟨ A ⟩_{N} = Tr_{F}[ P A(t) ] ,  (7.14)

where P stands for density matrix of the system at t = 0, i.e. P = P(0), and A(t) is the full Heisenberg operator

A(t) = S(0, t) A^{(int)}(t) S(t, 0) .  (7.15)

All above formulae are very well known (see e.g. [60]); the notation used here is exactly the same as in [30]. We are interested in the expectation values ⟨ 𝕍_{j} ⟩_{N} of the Heisenberg operators

𝕍_{j}(t) = S(0, t) u_{j} V_{j}^{(int)}(t) S(t, 0) .  (7.16)

where

V_{j}^{(int)}(t) = e^{i(e_{j},Φ_{B}^{(int)}(t)+Vt)},  j = 1, 2, 3  (7.17)

and Φ_{B}^{(int)}(t) is the boundary field Φ_{B} in the interaction representation. It is convenient to introduce also auxiliary operators

𝕍_{j}(t, t_{0}) = S(t_{0}, t) u_{j} V_{j}^{(int)}(t) S(t, t_{0}) ,  (7.18)

where t_{0} is a parameter. For t_{0} = 0 (7.18) coincides with the Heisenberg operators (7.16), and according to (7.13) the expectation values of (7.16) can be expressed through (7.18) as follows

⟨ 𝕍_{σ}(t) ⟩_{N} = lim_{t_{0}→−∞} ⟨ 𝕍_{σ}(t, t_{0}) ⟩_{0} ,  (7.19)

where σ = 1, 2, 3.

Using the series expansion for the evolution operators (7.9) and the commutation relations

V_{i}^{(int)}(t_{1})V_{j}^{(int)}(t_{2}) = q^{2(e_{i}e_{j})}V_{j}^{(int)}(t_{2})V_{i}^{(int)}(t_{1}), t_{1} > t_{2} (7.20)

one can represent (7.18) as a series of time-ordered integrals of products of V_{j}^{(int)}

\begin{matrix}𝕍_{σ}(t, t_{0}) = u_{σ} V_{σ}^{(int)}(t) \\ +V_{σ}^{(int)}(t) ∑_{k=1}^{∞} ∑_{σ_{1},...,σ_{k}=1,2,3} ℂ_{k}(σ, σ_{1}, …, σ_{k}) ∫_{t_{0}}^{t} D_{k}(\{t\}) V_{σ_{1}}^{(int)}(t_{1}) ⋯ V_{σ_{k}}^{(int)}(t_{k}) , \end{matrix}

(7.21)

where the sum is taken over all arrangements of the “charges” σ_{1}, …, σ_{k} = 1, 2, 3. The coefficients C_{k}(σ| σ_{1}, …, σ_{k}) can be easily calculated following the arguments of ref. [30]. From the definition (7.18) it follows that

i\frac{∂}{∂t_{0}} 𝕍_{σ}(t, t_{0}) = [H_{1}^{(int)}(t_{0}), 𝕍_{σ}(t, t_{0})].  (7.22)

Substituting the the series (7.21) into this differential equation one gets a recurrence relation for the coefficients ℂ_{k}(σ| σ_{1}, …, σ_{k}), which (with the initial condition ℂ_{0}(σ) = u_{σ}) has a unique solution

ℂ_{k}(σ, σ_{1}, …, σ_{k}) = u_{1}^{(w_{2}X_{k})} u_{2}^{−(w_{1}X_{k})} (\frac{q^{\frac{1}{2}} κ}{g})^{k} q^{F (X_{k})} ∏_{j=1}^{k} sin(πg w_{σ′_{j}}X_{j−1}) ,  (7.23)

where

σ′_{j} = σ_{j} − 1 (mod 3),  X_{j} = ∑_{s=0}^{j} e_{σ_{s}},  σ_{0} ≡ σ  (7.24)

and

F (X ) = −\frac{1}{2}(w_{1}X )^{2} − \frac{1}{2}(w_{2}X )^{2} + \frac{1}{2} .  (7.25)

Using (7.21), (7.23) one obtains an infinite series representation for the expectation values (7.19)^{8}

⟨𝕍_{σ}⟩_{N} = \frac{2πT}{κ sin πg} ∑_{n=1}^{∞}(−it)^{n} ∑_{σ_{1},...,σ_{3n−1}} ( ∏_{j=1}^{3n−1} \frac{sin(πg w_{σ′_{j}}X_{j−1})}{sin(πg)} ) J(σ, σ_{1}, …, σ_{3n−1}|p),

(7.26)

8Representations of this type for a similar problem related to the boundary sine-Gordon model were first derived in [61] by combinatorial methods.

J(σ, σ_{1}, …, σ_{3n−1}|p) = ∫_{−∞}^{0} D_{3n−1}(\{τ\})e^{−2p ∑_{j=1}^{3n−1} e_{σ_{j}} τ_{j}} ∏_{0≤j≤k≤3n−1} (2 sin (\frac{τ_{j} − τ_{k}}{2} ))^{g a_{jk}}.

(7.27)

where σ_{i} = 1, 2, 3 and the internal sum taken over all possible of values of {σ_{i}}, i = 1, …, 3n − 1 such that

e_{σ_{0}} + e_{σ_{1}} + … + e_{σ_{3n−1}} = 0 .  (7.28)

It is convenient to represent the sequence of the “charges” σ_{0}, σ_{1}, …, σ_{k} for each term in (7.21) by a path staring from the origin X = 0 of the triangular lattice

X = n_{1}e_{1} + n_{2}e_{2},  n_{1}, n_{2} ∈ Z,  (7.29)

and formed by the sequence of the “step” vector e_{σ_{0}}, e_{σ_{2}}, …, e_{σ_{k}}. The vectors X_{j} defined in (7.24) then represent the end point of the path after j + 1 steps. It is important to note, that not every such path contributes to (7.21). For example, for σ_{0} = 1 only paths which are

i) always contained in the sector (see Fig 1)

w_{1}X_{m} ≥ 0,  w_{2}X_{m} ≥ 0,  m = 0, 1, … k  (7.30)

ii) do not return to the origin (except for the final step), i.e.,

X_{m} ≠ 0,  m = 0, 1, … k − 1  (7.31)

correspond to nonzero coefficients C_{k}(σ_{0}, σ_{1}, …, σ_{k}), since otherwise one of the factors sin(πg w_{σ′_{j}}X_{j−1}) in (7.23) vanishes. Obviously, the paths (7.28) contributing to (7.26) return to the origin at the last step. As a result all integrals appearing in (7.26) converge at large τ’s for

2(e_{1}p) > −g,  2(e_{2}p) < g.  (7.32)

The expectation values (7.26) coincide with equilibrium expectation values of certain operators in the system which is similar to (7.1) but contains different boundary degrees of freedom obeying the q-oscillator algebra (4.43). Consider the Hamiltonian

\overline{H} = \frac{1}{4πg} ∫_{−∞}^{0} dx ( Π^{2} + Φ_{x}^{2} ) + \frac{1}{2}hV − \frac{κ}{2g} ∑_{i=1}^{3} y_{i} e^{ie_{i}Φ_{B}} ,  (7.33)

where Φ(x), Π(x) are again the Bose field operators obeying the same commutation relations (7.3)

h = \frac{2}{3}(h_{1}e_{1} + h_{2}e_{2} + h_{3}e_{3}),  (7.34)

Figure 1: The shaded area is the region defined by (7.30). We also show an admissible path X_{5} = e_{1} + e_{1} + e_{3} + e_{3} + e_{2} + e_{2}.

and

\begin{aligned}y_{1} & = & (q − q^{−1})a_{2}^{−}, & y_{2} & = & (q − q^{−1})a_{1}^{+}q^{H_{2}}, & y_{3} & = & q^{\frac{1}{2}} (q − q^{−1})^{2}q^{−H_{2}}a_{1}^{−}a_{2}^{+}, \\ h_{1} & = & −H_{1} − 2H_{2}, & h_{2} & = & 2H_{1} + H_{2}, & h_{1} & = & −H_{1} + H_{2}, \end{aligned}

(7.35)

where the operators a_{i}^{±} and H_{i}, i = 1, 2, commute with Φ(x), Π(x) and satisfy the commutation relations (4.43). Let ρ be any representation of the algebra (4.43) such that the spectra of H_{1} and H_{2} are both real and bounded from above. The Hamiltonian (7.1) acts in the space

\overline{H} = F ⊗ ρ,  (7.36)

where F is the Fock space representing the Bose commutation relations (7.3) and for the values of V satisfying (7.5) this Hamiltonian is bounded from below. Then the system (7.1) has a thermal equilibrium state described by the standard density matrix

\overline{P} = \overline{Z}(κ, V )^{−1}e^{−R\overline{H}},  R = g/T  (7.37)

where

\overline{Z}(κ, V ) = Tr\overline{H}e^{−R\overline{H}},  (7.38)

is the partition function. Denote by ⟨A⟩_{E} the expectation value of an observable A over this thermal equilibrium state

⟨A⟩_{E} = Tr\overline{H} (A\overline{P}).  (7.39)

We will be particularly interested in the expectation value ⟨\overline{𝕍}_{1}⟩_{E} of the operator

\overline{𝕍}_{1} = y_{1} e^{ie_{1}Φ_{B}},  (7.40)

and will show below that

⟨\overline{𝕍}_{1}⟩_{E} = ⟨𝕍_{1}⟩_{N},  (7.41)

where ⟨𝕍_{1}⟩_{N} is given by (7.26) above.

The proof is closely parallel to that used in the proof of the analogous result for the Boundary Sine-Gordon model, found in [30]. We will use the interaction representation with the Hamiltonian

\overline{H}_{0} = \frac{1}{4πg} ∫_{−∞}^{0} dx ( Π^{2} + Φ_{x}^{2} ) + \frac{1}{2}hV ,  (7.42)

corresponding to κ = 0 in (7.33). Consider an auxiliary operator

\overline{𝕍}_{1}(t, t_{0}) = \overline{S}(t_{0}, t) e^{i\overline{H}_{0}t} \overline{𝕍}_{1} e^{−i\overline{H}_{0}t}\overline{S}(t, t_{0}),  (7.43)

where \overline{S}(t, t_{0}) is defined as in (7.9) but with the interaction Hamiltonian replaced by

\overline{H}_{1}^{(int)}(t) = e^{i\overline{H}_{0}t} \overline{H}_{1} e^{−i\overline{H}_{0}t},  \overline{H}_{1} = −\frac{κ}{2g} ∑_{i=1}^{3} y_{i} e^{ie_{i}Φ_{B}}.  (7.44)

The operator (7.43) can be represented in a form similar to (7.21)

\begin{matrix}\overline{𝕍}_{1}(t, t_{0}) = a_{2}^{−} V_{1}^{(int)}(t) \\ +V_{σ}^{(int)}(t) ∑_{k=1}^{∞} ∑_{σ_{1},...,σ_{k}=1,2,3} \overline{ℂ}_{k}(σ, σ_{1}, …, σ_{k}) ∫_{t_{0}}^{t} D_{k}(\{t\}) V_{σ_{1}}^{(int)}(t_{1}) ⋯ V_{σ_{k}}^{(int)}(t_{k}) , \end{matrix}

(7.45)

where

\overline{ℂ}_{k}(σ, σ_{1}, …, σ_{k}) = (a_{2}^{−})^{(w_{2}X_{k})}(a_{1}^{−}q^{−H_{2}})^{−(w_{1}X_{k})} (\frac{q^{\frac{1}{2}} κ}{g})^{k} q^{F (X_{k})} ∏_{j=1}^{k} sin(πg w_{σ′_{j}}X_{j−1}) ,

(7.46) and the rest of notation is the same as in (7.21). In particular, the same remarks about the paths contributing to (7.21) apply to (7.45). We can now use the representation (7.45) in (7.39). Repeating the arguments which used in derivation of eq.(3.15) in ref. [30] one obtains

⟨\overline{𝕍}_{1}⟩_{E} = lim_{t_{0}→−∞} \frac{Tr\overline{H}[e^{−R\overline{H}_{0}} \overline{𝕍}_{1}(0, t_{0})]}{Tr\overline{H}[e^{−R\overline{H}_{0}}]}  (7.47)

Using now (7.45) and the simple property of the trace

Tr\overline{H}[e^{−R\overline{H}_{0}}(a_{1}^{−})^{n}(a_{2}^{−})^{m}] = δ_{n,0}δ_{m,0}Tr\overline{H}[e^{−R\overline{H}_{0}}]  (7.48)

one arrives exactly to the series (7.26) for the expectation value (7.47). This proves the relation (7.41).

Further, comparing the Matsubara representation for the partition function with the vacuum eigenvalues of the operators A_{i}(t) defined by (4.49) one can conclude that

\frac{\overline{Z}(κ, V )}{\overline{Z}(0, V )} = A_{3}^{(vac)}(t, p),  (7.49)

where the parameters t and p related to κ, V and the temperature T as

t = i[ \frac{κ sin πg}{2πT} (\frac{g}{2πT} )^{−g} ]^{3},  p = −i\frac{g}{4πT} V .  (7.50)

Expressing now the equilibrium expectation value ⟨\overline{𝕍}_{1}⟩_{E} through the partition function (7.38)

< \overline{𝕍_{1}} >_{E}= \frac{2}{3}T ∂_{κ} log \overline{Z}(κ, V ),  (7.51)

and using (7.41) we obtain

⟨𝕍_{1}⟩_{N} = \frac{2}{3}T ∂_{κ} log A_{3}^{(vac)}(t, p).  (7.52)

In a similar way one can show that

⟨𝕍_{2}⟩_{N} = \frac{2}{3}T ∂_{κ} log A_{1}^{(vac)}(t, p),  ⟨𝕍_{3}⟩_{N} = \frac{2}{3}T ∂_{κ} log A_{2}^{(vac)}(t, p).  (7.53)

Finally note the g → 1/g duality properties of the expectation values (7.52), (7.53). Introduce the following combinations expectation values

J_{α} = iπκ ⟨ 𝕍_{σ−1}⟩_{N} − iπκ ⟨𝕍_{σ}⟩_{N},  α = 1, 2, 3  (7.54)

then

J_{α} = 2πiT t∂_{t} log \frac{A_{β}^{(vac)}(t, p)}{A_{γ}^{(vac)}(t, p)},  (7.55)

where (α, β, γ) is a cyclic permutation of (1, 2, 3).

According to (6.16) the quantities (7.54) satisfy the following “strong weak” duality relations

J_{α}(g, V , −κ) = −e_{α}V − g^{−1}J_{α}(g^{−1}, gV , −C(g)(−κ)^{− \frac{1}{g}}),  (7.56)
C(g) = \frac{Γ(g^{−1})}{π}(\frac{Γ(g)}{π})^{\frac{1}{g}} .  (7.57)

7.2 Relation to third order differential equations

Consider the differential equation

\{∂_{x}^{3} + \frac{ℓ_{1}ℓ_{2} + ℓ_{1}ℓ_{3} + ℓ_{2}ℓ_{3} − 2}{x^{2}}∂_{x} − \frac{ℓ_{1}ℓ_{2}ℓ_{3}}{x^{3}} + x^{3α}\}Ψ(x) = EΨ(x),  (7.58)

where

ℓ_{1} + ℓ_{2} + ℓ_{3} = 3.  (7.59)

For

ℜe(ℓ_{1} + 3) > ℜe(ℓ_{2}),  ℜe(ℓ_{1} + 3) > ℜe(ℓ_{3}),  (7.60)

the equation (7.58) has a unique solution satisfying the condition

ψ(x, E, ℓ_{1}, ℓ_{2}, ℓ_{3}) = x^{ℓ_{1}} + O(x^{ℓ_{1}+3}),  (7.61)

This solution can be analytically continued outside the domain (7.60). Obviously the functions ψ(x, E, ℓ_{2}, ℓ_{1}, ℓ_{3}) and ψ(x, E, ℓ_{3}, ℓ_{1}, ℓ_{2}) defined in this way satisfy the same equation (7.58) and for generic ℓ_{1}, ℓ_{2}, ℓ_{3} the solutions

ψ_{1}(x) = ψ(x, E, ℓ_{1}, ℓ_{2}, ℓ_{3}),  ψ_{2}(x) = ψ(x, E, ℓ_{2}, ℓ_{1}, ℓ_{3}),  ψ_{3}(x) = ψ(x, E, ℓ_{3}, ℓ_{1}, ℓ_{2}),

(7.62)

are linearly independent since

Wr [ψ_{1}(x), ψ_{2}(x), ψ_{3}(x)] = (ℓ_{1} − ℓ_{2})(ℓ_{2} − ℓ_{3})(ℓ_{3} − ℓ_{1}),  (7.63)

where Wr denotes the usual Wronskian.

Further, for all E the equation (7.58) has a unique solution χ(x, E) which decays at x → +∞. We normalise this solution as

χ(x) = x^{−α} exp \{ − \frac{x^{α+1}}{α + 1} + O(x^{α−2})\}.  (7.64)

It can be expanded in the basis (7.62)

χ(x, E) = R_{1}(E)ψ_{1} + R_{2}(E)ψ_{2} + R_{3}(E)ψ_{3}.  (7.65)

It was conjectured in [20] that the connection coefficients R_{i}(E) are simply related to the vacuum eigenvalues of the Q-operators (4.49). The exact correspondence is as follows:

\frac{R_{i}(E)}{R_{i}(0)} = A_{i}^{(vac)}(t, p),  i = 1, 2, 3  (7.66)

where

α = \frac{1}{g} − 1,  E = i ρ t,  ℓ_{i} = −\frac{3(pw_{i})}{g} + 1,  (7.67)

with the parameter ρ and the vectors w_{i} defined in (2.34) and (6.34) respectively. The conjecture (7.66) was supported in [20] by extensive numerical work, but no analytical proof is currently known (see Section 8.3 below for a discussion).

\begin{matrix}n & E_{n}^{(1)} [20] & E_{n}^{(1)} (AS) & E_{n}^{(3)} [20] & E_{n}^{(3)} (AS) \\ 1 & 2.88682816 & 2.8787210 & 0.890944821 & 0.66892875 \\ 2 & 5.15225193 & 5.1513633 & 3.643901890 & 3.64527385 \\ 3 & 7.13037330 & 7.1300266 & 5.817136367 & 5.81718647 \\ 4 & 8.94036211 & 8.9401849 & 7.737709470 & 7.73772076 \\ 5 & 10.6356087 & 10.6355029 & 9.508485338 & 9.50848916 \\ 6 & 12.2451641 & 12.2450946 & 11.17453999 & 11.17454166 \\ 7 & 13.7870634 & 13.7870146 & 12.76111150 & 12.76111235 \\ 8 & 15.2734900 & 15.2734540 & 14.28420813 & 14.28420862 \\ 9 & 16.7131734 & 16.7131460 & 15.75481968 & 15.75481998 \end{matrix}

Table 3: Numerical values for E_{n}^{(1)} and E_{n}^{(3)} from Table 2 in [20], truncated to 8 decimal places, compared with E_{n}^{(1)}(AS) and E_{n}^{(3)}(AS), obtained from our asymptotic formula, (7.71).

Note that, due to the correspondence (7.66), the formulae (6.16) provide asymptotic expansions for the zeroes of the coefficients R_{i}(E). As an example consider the following particular case of (7.58),

∂_{x}^{3}Ψ(x) + x Ψ(x) = EΨ(x),  (7.68)

which corresponds to

α = 1/3,  ℓ_{1} = 0,  ℓ_{2} = 1,  ℓ_{3} = 2,  (7.69)

and

g = 3/4, (pw_{1}) = 1/4, (pw_{2}) = 0, (pw_{3}) = −1/4. (7.70)

The zeroes R_{1}(E) and R_{3}(E) in this case have been extensively studied numerically both from the solutions of the differential equation as well as from the non-linear integral equation [18–20] which is equivalent to the Bethe Ansatz equations (6.3), (6.4) for the vacuum eigenvalues A_{i}^{(vac)}(t). We have compared this numerical data against the asymptotic expansion (6.38). For g = 3/4 the expressions (6.35), (6.36) allow us to derive only one non-trivial term in (6.39), (6.40) (note that the first dual nonlocal IM also contribute). For the values of p given in (7.70) the corresponding formulae read

\begin{aligned}E_{n}^{(1)} & = & (\frac{8π}{3\sqrt{3}})^{\frac{3}{4}} (n − \frac{1}{6})^{3/4} [1 + \frac{21}{256π^{2}} (n − \frac{1}{6})^{−2} + O(n^{−4})], \\ E_{n}^{(3)} & = & (\frac{8π}{3\sqrt{3}})^{\frac{3}{4}} (n − \frac{5}{6})^{3/4} [1 − \frac{15}{256π^{2}} (n − \frac{5}{6})^{−2} + O(n^{−4})]. \end{aligned}  (7.71)

The numerical values are presented in Table 3.

As shown in [62] the correspondence (7.66) allows us to derive a certain duality relation for the vacuum eigenvalues A_{i}^{(vac)}(t, p). It is convenient to make a change of variables Ψ(x) = e^{y}Ψ̃(y), x = e^{y} which brings the equation (7.58) to the form

{∂_{y}^{3} − (ν_{1}ν_{2} + ν_{1}ν_{3} + ν_{2}ν_{3})∂_{y} + iν_{1}ν_{2}ν_{3} + e^{3(α+1)y} − Ee^{3y}}Ψ̃(y) = 0,  (7.72)

where

ν_{j} = −i(ℓ_{j} − 1) = \frac{3i(pw_{j})}{g} .  (7.73)

It is easy to see that the change of variables

y = gy′ + y_{0},  e^{−3y_{0}} = −g^{3}E,  (7.74)

transform (7.72) to an equation of the same form but with the parameters g, {ν_{i}} and replaced by

g → g^{−1},  ν_{i} = gν_{i},  E → −g^{3}g^{− \frac{3}{g}} (−E)^{− \frac{1}{g}} .  (7.75)

Using this fact in (7.65) one obtains the following duality relation for the connection coefficients R_{i}

\frac{e^{iy_{0}ν_{α}} R_{α}(g, ν_{1}, ν_{2}, ν_{3}, E)}{e^{iy_{0}ν_{β}} R_{β}(g, ν_{1}, ν_{2}, ν_{3}, E)} = \frac{R_{α}(g^{−1}, gν_{1}, gν_{2}, gν_{3}, −c(g) (−E)^{− \frac{1}{g}})}{R_{β}(g^{−1}, gν_{1}, gν_{2}, gν_{3}, −c(g) (−E)^{− \frac{1}{g}})}, c(g) = g^{3(1−g^{−1})},

(7.76)

where α, β = 1, 2, 3. Note that due to (7.66), this relation is in agreement with (6.16) and (7.56).

8 Discussion

In conclusion we make a few remarks on some open problems relevant to the topics of this paper.

8.1 Truncation of the functional relations

It is well known that the fusion equation for the transfer matrices in solvable lattice models truncate when q^{2} is a root unity. A similar truncation, of course takes place for the CFT T-operators. For the theories related to the quantum affine U_{q}(ŝl(2)) this truncation is well understood. For example, consider the CFT T-operators T_{j}(λ), j = 0, \frac{1}{2}, 1, \frac{3}{2} …, of ref. [2], associated with 2j + 1-dimensional representations of U_{q}(ŝl(2)). If q = e^{iπβ^{2}} is a primitive root of 1 or −1, such that q^{N} = ±1 with some integer N ≥ 2 then one can easily show (see, e.g., [4]) that

T_{\frac{N}{2}} (λ) = 2 cos(2πNp) + T_{\frac{N}{2} −1}(λ) ,  (8.77)

where p and β are related to the Virasoro highest weight Δ and the central charge ĉ as

Δ = \frac{p^{2}}{β^{2}} + \frac{ĉ − 1}{24},  ĉ = 1 − 6(β − β^{−1})^{2},  q = e^{iπβ^{2}}  (8.78)

As is shown in [3] this allows one to bring the fusion relations for T_{j}(λ) to the form identical to the functional TBA equations (the Y -system) of D_{N} type [29,63].

Further reduction occurs at particular values of the vacuum parameter p

T_{\frac{N}{2} − \frac{1}{2}} (λ) = 0,  when z^{N} = ±1,  z = e^{2πip}  (8.79)

which corresponds to the RSOS reduction in the six-vertex lattice model [64]. This latter RSOS-type reduction is also well known for the case of U_{q}(ŝl(3)) [16]. For example, it readily follows from the determinant formulae (5.1), (5.2): when the vacuum parameters z_{i} defined in (3.31) become roots of unity z_{i}^{N} = 1, i = 1, 2, 3 along with q^{N} = 1 and either μ_{1} + 2 = N or μ_{2} + 1 = N the determinants (5.1), (5.2) vanish. However, no simple analogues of the reduction (8.77) valid for arbitrary values of z_{i} is known (except for the particular case of Sect 6.3 which accidentally reduces to U_{q}(ŝl(2))). Similar remarks apply to related problems such as the string structure of the Bethe Ansatz equations and the TBA equations. In particular, no U_{q}(ŝl(3)) analogues of the D_{N} type TBA equations are known.

8.2 Connection with U_{q}(A_{2}^{(2)}) related IQFT

The set of classical IM (1.2) admits consistent reductions V (u) = 0 or V (u) = U′(u)/2 causing all even IM I_{k}^{(cl)}, k = 2, 4 (mod 6) to vanish. Remarkably, the non-vanishing IM I_{k}^{(cl)}, k = 1, 5 (mod 6) remain in involution [10] with respect to Virasoro Poisson bracket of the field U(u). The quantization of these conserved quantities leads to new sets (with respect to the quantum KdV case [2, 52, 53, 65]) of local IM in c < 1 CFT.

Actually, there are two new non-equivalent sets of IM related to the different reductions mentioned above. This structure of the local IM naturally arises in the conformal limit of the massive IQFT associated with the twisted quantum affine U_{q}(A_{2}^{(2)}) [65, 66]. The universal CFT L-operator for this case reads

L = e^{iπP h}P exp { ∫_{0}^{2π} ( : e^{−2ϕ(u)} : y_{0}+ : e^{ϕ(u)} : y_{1})du},  (8.80)

where ϕ(u) is the chiral Bose field normalised exactly as in [2] and h, y_{0} and y_{1} are the generators of the Borel subalgebra of U_{q}(A_{2}^{(2)}) where q is related to the Virasoro central charge ĉ as in (8.78). In ref. [67] Fioravanti, Ravanini and Stanishkov constructed an infinite set of finite-dimensional representations π_{s}(t), s = 0, 1, 2, …, of this subalgebra (with the dimensions dim[π_{s}(t)] = (s + 1)(s + 2)/2) and defined a set of T-operators

T_{s}(t) = Tr_{π_{s}(t)}[e^{iπP h}L],  s = 0, 1, 2, …  (8.81)

Here we use the spectral parameter t which is related to λ of [67] as t = λ^{3}. These operators act in the Virasoro highest weight module V_{Δ} with Δ defined in (8.78); they are entire functions of t and their leading asymptotics at large t can be easily estimated

log T_{s}(t) ∼ m_{s} t^{2/(6−3β^{2})},  |t| → ∞  (8.82)

where m_{s} are some numerical constants. Further, at small t one has

T_{1}(t) = 1 + 2 cos(2πp) + G_{1} t + O(t^{2}),  (8.83)

where the vacuum eigenvalue G_{1} in V_{Δ} can be explicitly calculated

G_{1}^{(vac)}|_{A_{2}^{(2)}} = 2π e^{−2πip} ∫_{0}^{2π} du_{1} ∫_{0}^{u_{1}} du_{2} \frac{e^{ip(u_{1}+u_{2})}(2 sin (\frac{u_{1} − u_{2}}{2} ))^{\frac{β^{2}}{2}}}{(2 sin \frac{u_{1}}{2} )^{β^{2}}(2 sin \frac{u_{2}}{2} )^{β^{2}}}

= \frac{4π^{3}Γ(1 + β^{2}/2)Γ(1 − 3β^{2}/4)Γ(1 − β^{2})}{Γ(1 + β^{2}/4)Γ(1 + p − β^{2}/2)Γ(1 − p − β^{2}/2)Γ(1 + p − β^{2}/4)Γ(1 − p − β^{2}/4)}. (8.84)

In [67] it was shown^{9} that these operators satisfy the functional relation [68]

T_{s}(tq^{1/2})T_{s}(tq^{−1/2}) = T_{s}(−t) + T_{s−1}(t)T_{s+1}(t),  q = e^{iπβ^{2}}  (8.85)

Now turn back to the W_{3} theory. Consider the W_{3} module V_{Δ_{2},0}, with the weight Δ_{3} = 0. Let V_{sym} be a subspace V_{sym} ∈ V_{Δ_{2},0} of this module invariant with respect to the action of the automorphism σ_{13} defined in Sect. 4.1 Then from the definitions (4.21), (4.36) and the relations (4.18), (4.11) one concludes that the operators 𝕋_{m}(t) and \overline{𝕋}_{m}(t) from (4.38), (4.39) obey the relation

9Although the calculations in [67] have been performed in the lowest non-trivial order in t the relation (8.85) is expected to hold in all orders in t.

𝕋_{m}(t) = \overline{𝕋}_{m}(−t)  (8.86)

when acting in the subspace V_{sym}. It follows then from (6.29) and the discussion there that all even local IM I_{k}, k = 2, 4 (mod 6), vanish in this subspace. Denote by 𝕋_{m}(t|g, p)) the eigenvalues 𝕋_{m}(t) in V_{sym} ∈ V_{Δ_{2},0} where g is related to the central charge c of the W_{3} algebra (3.22) and p parameterises the weight Δ_{2} = p^{2}/g + (c − 2)/24. Taking into account (8.86) the functional equations (5.14) become equivalent to (8.85) provided one makes a replacement q → q^{\frac{1}{2}} . Thus we have good reasons to expect that with a suitable redefinition of parameters the above eigenvalues 𝕋_{m}(t|g, p)) of 𝕋_{m}(t) are identical to the eigenvalues T_{s}(t|β^{2}, p) of the operators (8.81) acting in the Virasoro module V_{Δ}(ĉ) with p and ĉ already defined in (8.78). The exact correspondence is

T_{s}(t|β^{2}, p) = 𝕋_{s}(ρ_{1}t|β^{2}/2, p),  s = 0, 1, 2, …  (8.87)

where the factor

ρ_{1} = \frac{Γ(1 − \frac{β^{2}}{4} ) Γ(1 + \frac{β^{2}}{2} ) Γ(1 − β^{2})}{Γ(1 + \frac{β^{2}}{4} ) Γ^{2}(1 − \frac{β^{2}}{2} )},  (8.88)

is calculated from the comparison of (8.84) with (4.84). Further, the subspace V_{sym} should support the action of the Virasoro algebra with the central charge ĉ. It would be interesting to obtain an explicit realization of this Virasoro algebra in terms of the generators of the W_{3} algebra. It would also be interesting to obtain an explicit construction of the Q operators for the U_{q}(A_{2}^{(2)})-related T operators (8.81).

8.3 Concluding remarks

The study of the integrable structure of CFT in [2–4] and in this paper once again demonstrates the fundamental role of Baxter’s Q-operators in the theory of integrable quantum systems. Some further applications of Q-operators in CFT can be found in [69].

Here we adopted the approach of [3,4] to construct the Q-operators as traces of certain quantum monodromy matrices associated with infinite-dimensional representations of the q-oscillator algebra. As remarked in [4] this construction is rather general and can be applied to the lattice models as well. In particular, as the structure of the functional relations considered in Section 5 is completely determined by the decomposition properties of the products of representation of the quantum affine algebra U_{q}(ŝl(3)), all these functional relations remain valid for the associated lattice models, with minor modifications related to the normalisation conventions of the lattice transfer matrices. Note that for certain lattice models there exists an alternative approach to Q-operators [70] (based on Baxter’s original work [71]) which allows the explicit calculation of their matrix elements in a simple product form (see also [72–78] for further examples). It would be interesting to understand a relationship between the above two approaches to the Q-operators.

The conjecture (7.66) of [20] is a generalisation of an earlier conjecture of the same authors [79] which established an exact relation between the spectral determinants of the (second order) Schrödinger operator with homogeneous potentials [80,81] and the vacuum eigenvalues of the Q-operators in the c < 1 CFT [3]. This earlier conjecture of [79] has been extended for the more general Schrödinger operator (which includes the “angular momentum” term)

∂_{x}^{2}Ψ(x) + \{ E − x^{2α} − \frac{l(l + 1)}{x^{2}} \}Ψ(x) = 0  (8.89)

and proved in [62]. Here the parameters l and α are related to the Virasoro vacuum parameter p and central charge ĉ defined in (8.78) as l = 2p/β^{2} − 1/2 and α = 1/β^{2} − 1. The most essential ingredients of this proof can be summarized as follows:

(i) The analytic properties (both in the spectral parameter E and the angular momentum l). For the spectral determinants of (8.89) these properties are deduced from the standard theory of the Schrödinger equations (see, e.g., [82]). On the other hand, in [4] and [30] the same analytic properties for the vacuum eigenvalues of the Qoperators were establish in c < 1 CFT. Note that the analysis of these analytic properties in the variable p in [30] (corresponding to l in (8.89)) was based on the “real time perturbation theory” results (analogous to the representations (7.26) of this paper).

(ii) The quasiclassical approximation. The same real time perturbation theory results were used in [30] to obtain the leading large p asymptotics of the coefficient of the expansion of the eigenvalues of the Q-operators in the spectral parameter. These asymptotics exactly matched [62] the corresponding asymptotics for the spectral determinants of (8.89) calculated from the standard quasiclassical approximation to (8.89) for large values of l.

(iii) Quantum Wronskian relation and the Riemann-Hilbert problem. The final step of the proof of [62] was based on the key observation of [79] that the functional relation for spectral determinants of the Schrödinger equation obtained in [80,81] (see also [62] for the case of general l) exactly coincide with the quantum Wronskian relation for the eigenvalues of the Q-operators in c < 1 CFT obtained in [3]. In fact, it was shown [30] that the above properties (i) and (ii) transform this functional relation to a simple Riemann-Hilbert problem which has a unique solution.

It should be noted that no additional assumptions (like the phase assignment in the Bethe Ansatz equations for vacuum eigenvalues, similar to (6.4)) were used in [62].

It would be interesting to see whether is it possible to modify the above arguments to prove the conjecture (7.66) related to the third order differential equation (7.58). It seems that there no difficulties in generalising the above properties (i) and (ii). In particular, we have verified that the asymptotics of the connection coefficients R_{i}(E) in (7.65) obtained from the quasiclassical approximation to the differential equation (7.58) for large values of ℓ_{i} exactly match the p → ∞ asymptotics of the eigenvalues of the Q-operators obtained from (6.9) and (6.30). It is also easy to prove that the connection coefficients R_{i}(E)/R_{i}(0) satisfy the same quantum Wronskian relation (1.13). However, it is currently not clear how transform this relation to a tractable Riemann-Hilbert problem which now should involve two complex “angular momentum” variables.

Acknowledgements

VB thanks S.L.Lukyanov and A.B.Zamolodchikov for useful discussions. The work of S.Khoroshkin was supported by RFBR grant 01-01-00539, grant for the support of scientific schools 00-15-96557, and INTAS 99-1705.

Appendix A. Universal R-matrix

Here we give a short proof of Proposition 1, which allows us to derive the basic properties of the universal L-operator (3.34).

Due to (3.6) and (3.14) the basic property (3.9) of the universal R-matrix implies the following relations:

(1 ⊗ y_{i} + y_{i} ⊗ q^{−h_{i}})\overline{R} = \overline{R}(1 ⊗ y_{i} + y_{i} ⊗ q^{h_{i}}),  (A.1)

that is

[1 ⊗ y_{i}, \overline{R}] = \overline{R}(y_{i} ⊗ q^{h_{i}}) − (y_{i} ⊗ q^{−h_{i}})\overline{R}.  (A.2)

Sum up (A.2) over i and put \overline{I} = ∑_{i} y_{i}. We have

[\overline{I}, \overline{R}] = \overline{R}(∑_{i=1}^{3} y_{i} ⊗ q^{h_{i}}) − (∑_{i=1}^{3} y_{i} ⊗ q^{−h_{i}})\overline{R}.  (A.3)

Let us imagine now that we check equality (A.3) directly. For this we compute the commutator on the left hand side using the relations

[y_{i}, x_{j}] = δ_{i,j} \frac{q^{h_{i}} − q^{−h_{i}}}{q − q^{−1}} .  (A.4)

After application of the Leibnitz rule we have certain series over y_{i} ⊗ 1 and 1 ⊗ x_{i} with q^{h_{i}} and q^{−h_{i}} inserted somewhere in the right tensor space.

We move all q^{−h_{i}} to the left and all q^{h_{i}} to the right using the relations

q^{h_{i}}x_{j} = q^{−a_{ij}}x_{j}q^{h_{i}},  q^{−h_{i}}x_{j} = q^{a_{ij}}x_{j}q^{−h_{i}}.  (A.5)

The result should coincide with right hand side of (A.3).

Suppose now that we have some I, A_{i} and B_{i}, satisfying the conditions (3.16), (3.17) of Proposition 1, and we try to compute [I ⊗ 1, \overline{R}]. Again, we first use (3.16), to get monomials over y_{i} ⊗1 and 1⊗x_{i} with A_{i} and B_{i} inserted in the second tensor component, then move A_{i} to the left and B_{i} to the right by means of (3.17). Clearly, we should get the same result as in the first calculation with q^{h_{i}} ⊗ 1 replaced by −B_{i} ⊗ 1 and q^{−h_{i}} ⊗ 1 replaced by −A_{i} ⊗ 1, since we used identical commutation relations. This proves (3.18).

Appendix B. Functional relations

In this appendix we derive the determinant formulas (5.1), (5.2) for the operators T_{μ}(t), and \overline{T}_{μ}(t) and the functional relations (5.4), (5.5). The main ingredient of the proof is similar to that of [4] and consists of decomposition of the tensor product of certain degenerated Verma modules into a sum of shifted evaluation Verma modules.

Let π_{μ}^{+}(t) be the evaluation Verma module of the algebra B_{+} with highest weight μ = (μ_{1}, μ_{2}, μ_{3}) with respect to evaluation map (4.15). We fix the basis v\overline{n}, where \overline{n} ∈ ℤ_{≥0}^{3}, that is \overline{n} = (n_{1}, n_{2}, n_{3}) and n_{i} ≥ 0, n_{i} ∈ ℤ, which is given by the action of the generators F_{α},F_{β} and F_{α+β} of U_{q}(gl(3)):

v\overline{n} = t^{n_{2}+ \frac{n_{1}+n_{3}}{2}} q^{− \frac{n_{2}}{2}} F_{α}^{n_{1}}F_{α+β}^{n_{2}}F_{β}^{n_{3}}v_{0},  (B.1)

where v_{0} is the vacuum vector. Then the action of the generators of the algebra B_{+} is given by the relations:

\begin{aligned}y_{1}v\overline{n} & = & t^{\frac{1}{2}} [n_{3}]_{q}[μ_{β} − n_{3} + 1]_{q}v_{\overline{n}−\overline{ε}_{3}} + t^{\frac{1}{2}} q^{μ_{β}−2n_{3}}[n_{2}]_{q}v_{\overline{n}+\overline{ε}_{1}−\overline{ε}_{2}}, \\ y_{2}v\overline{n} & = & q^{n_{3}+μ_{1}+μ_{3}+ \frac{1}{2}} v_{\overline{n}+\overline{ε}_{2}}, \\ y_{3}v\overline{n} & = & t^{\frac{1}{2}} [n_{1}]_{q}[μ_{α} − n_{1} − n_{2} + n_{3} + 1]_{q}v_{\overline{n}−\overline{ε}_{1}} − t^{\frac{1}{2}} q^{n_{2}−n_{3}−2−μ_{α}}[n_{2}]_{q}v_{\overline{n}−\overline{ε}_{2}+\overline{ε}_{3}}, \\ h_{1}v\overline{n} & = & (μ_{β} + n_{1} − n_{2} − 2n_{3})v\overline{n}, \\ h_{2}v\overline{n} & = & (−μ_{α+β} + n_{1} + 2n_{2} + n_{3})v\overline{n}, \\ h_{3}v\overline{n} & = & (μ_{α} − 2n_{1} − n_{2} + n_{3})v\overline{n}, \end{aligned} (B.2)

where

\begin{aligned}μ_{α} & = & μ_{1} − μ_{2}, & μ_{β} & = & μ_{2} − μ_{3}, & μ_{α+β} & = & μ_{1} − μ_{3}, \\ \overline{ε}_{1} & = & (1, 0, 0), & \overline{ε}_{2} & = & (0, 1, 0), & \overline{ε}_{3} & = & (0, 0, 1). \end{aligned}  (B.3)

Any weight μ defines a shift automorphism p_{μ} of the algebra B_{+}:

\begin{aligned}p_{μ}(y_{i}) & = & y_{i}, & p_{μ}(h_{1}) & = & h_{1} − μ_{2} + μ_{3}, \\ p_{μ}(h_{2}) & = & h_{2} + μ_{1} − μ_{3}, & p_{μ}(h_{3}) & = & h_{2} − μ_{1} + μ_{2}. \end{aligned}  (B.4)

For a representation π of B_{+} denote by π[μ] the shifted representation π·p_{μ}. In particular, denote by π̃_{μ}^{+}(t) the shifted evaluation Verma module π_{μ}^{+}(t)[μ]. In this module the action of generators y_{i} is given by (B.2) while the vacuum vector is of zero weight and

\begin{matrix}h_{1}v\overline{n} = (n_{1} − n_{2} − 2n_{3})v\overline{n},  h_{2}v\overline{n} = (n_{1} + 2n_{2} + n_{3})v\overline{n}, \\ h_{3}v\overline{n} = (n_{3} − 2n_{1} − n_{2})v\overline{n}, \end{matrix}  (B.5)

so the traces are related by

Tr_{π_{μ}^{+}(t)} e^{iπP h}L = z_{1}^{μ_{1}}z_{2}^{μ_{2}}z_{3}^{μ_{3}}Tr_{π̃_{μ}^{+}(t)} e^{iπP h}L.  (B.6)

Let us define three degenerations π̃^{i}(t), i = 1, 2, 3 of shifted evaluation Verma module π̃_{μ}^{+}(t). They correspond to the limit q → +∞ in three different asymptotic zones

μ_{α} ≫ 0, μ_{β} ≫ 0;  μ_{α} ≪ 0, μ_{β} ≫ 0  and μ_{α} ≪ 0, μ_{β} ≪ 0  (B.7)

under assumptions that all the numbers are real.

All the degenerated Verma modules π̃^{i}(t) are equipped with bases v\overline{n}, \overline{n}_{i} ∈ ℤ_{≥0}^{3}, on which the Cartan generators act as in (B.5) and

\begin{aligned}y_{1}v\overline{n} & = & (q − q^{−1})^{−1}q^{x−n_{3}+1}[n_{3}]_{q}v_{\overline{n}−\overline{ε}_{3}} + q^{x−2n_{3}}[n_{2}]_{q}v_{\overline{n}+\overline{ε}_{1}−\overline{ε}_{2}}, \\ y_{2}v\overline{n} & = & q^{n_{3}+ \frac{1}{2}} v_{\overline{n}+\overline{ε}_{2}}, \\ y_{3}v\overline{n} & = & (q − q^{−1})^{−1}q^{x−n_{1}−n_{2}+n_{3}+1}[n_{1}]_{q}v_{\overline{n}−\overline{ε}_{1}}, \end{aligned}  (B.8)

for the module π̃^{1}(t), where we use the notation t = q^{2x};

\begin{aligned}y_{1}v\overline{n} & = & (q − q^{−1})^{−1}q^{x−n_{3}+1}[n_{3}]_{q}v_{\overline{n}−\overline{ε}_{3}} + q^{x−2n_{3}}[n_{2}]_{q}v_{\overline{n}+\overline{ε}_{1}−\overline{ε}_{2}}, \\ y_{2}v\overline{n} & = & q^{n_{3}+ \frac{1}{2}} v_{\overline{n}+\overline{ε}_{2}}, \\ y_{3}v\overline{n} & = & −(q − q^{−1})^{−1}q^{x+n_{1}+n_{2}−n_{3}−1}[n_{1}]_{q}v_{\overline{n}−\overline{ε}_{1}} − q^{x+n_{2}−n_{3}−2}[n_{2}]_{q}v_{\overline{n}−\overline{ε}_{2}+\overline{ε}_{3}}, \end{aligned}  (B.9)

for the module π̃^{2}(t) and

\begin{aligned}y_{1}v\overline{n} & = & −(q − q^{−1})^{−1}q^{x+n_{3}−1}[n_{3}]_{q}v_{\overline{n}−\overline{ε}_{3}}, \\ y_{2}v\overline{n} & = & q^{n_{3}+ \frac{1}{2}} v_{\overline{n}+\overline{ε}_{2}}, \\ y_{3}v\overline{n} & = & −(q − q^{−1})^{−1}q^{x+n_{1}+n_{2}−n_{3}−1}[n_{1}]_{q}v_{\overline{n}−\overline{ε}_{1}} − q^{x+n_{2}−n_{3}−2}[n_{2}]_{q}v_{\overline{n}−\overline{ε}_{2}+\overline{ε}_{3}}, \end{aligned}  (B.10)

for the module π̃^{3}(t).

The determinant formula (5.1) is based on the decomposition of the tensor product

π̃^{1}(tq^{2μ_{1}}) ⊗ π̃^{2}(tq^{2μ_{2}}) ⊗ π̃^{2}(tq^{2μ_{3}}).  (B.11)

The claim is that this tensor product admits a filtration, labelled by six arbitrary nonnegative integers k_{1}, k_{2}, l_{1}, l_{2}, m_{1}, m_{2} with factormodules, isomorphic to

π̃_{μ}^{+}(t)[(k_{1} + k_{2})α + (l_{1} + l_{2})β + (m_{1} + m_{2})(α + β)],

where α = (1, −1, 0), β = (0, 1, −1) are simple roots for the Lie algebra sl(3). On the level of characters this means an equality of the form

\begin{matrix}π̃^{1}(tq^{2μ_{1}}) ⊗ π̃^{2}(tq^{2μ_{2}}) ⊗ π̃^{2}(tq^{2μ_{3}}) = \\ ∑_{k_{i},l_{i},m_{i}≥0} π̃_{μ}^{+}(t)[(k_{1} + k_{2} + m_{1} + m_{2})α + (l_{1} + l_{2} + m_{1} + m_{2})β], \end{matrix}  (B.12)

or, if we put Ã_{i}(t) = Tr_{π̃^{i}(t)} e^{iπP h}L, which converges in a region (4.48),

Ã_{1}(tq^{2μ_{1}})Ã_{2}(tq^{2μ_{2}})Ã_{3}(tq^{2μ_{3}}) = \overline{Δ}^{−2}Tr_{π̃_{μ}^{+}(t)} e^{iπP h}L,  (B.13)

where

\overline{Δ} = (1 − z_{2}/z_{1})(1 − z_{3}/z_{1})(1 − z_{3}/z_{2}).  (B.14)

We construct the decomposition of tensor product (B.11) in two steps. First, consider the second order tensor product π̃^{1}(t_{1}) ⊗ π̃^{2}(t_{2}), where we choose the basis v\overline{m} ; \overline{n}, \overline{n}, \overline{m} ∈ ℤ_{≥0}^{3}

v\overline{m} ; \overline{n} = ∑_{k_{1}=0}^{n_{1}} ∑_{k_{2}=0}^{n_{2}} \binom{n_{1}}{k_{1}}_{q}\binom{n_{2}}{k_{2}}_{q}q^{Ω(\overline{n},\overline{m},\overline{k})} (t_{1}/t_{2})^{k_{1}/2} v_{\overline{m}+\overline{k}} ⊗ v_{\overline{n}−\overline{k}},  (B.15)

where \overline{k} = (k_{1}, k_{2}, 0),

(n)_{q} = \frac{q^{2n} − 1}{q^{2} − 1},  \binom{n}{k}_{q} = \frac{(n)_{q}!}{(k)_{q}!(n − q)_{q}!},  (B.16)

and the phase Ω(\overline{n}, \overline{m}, \overline{k}) is

Ω(\overline{n}, \overline{m}, \overline{k}) = (n_{1} − k_{1})k_{2} + (k_{2} − k_{1})m_{3} + k_{2}m_{2}.  (B.17)
Define the following partial order in ℤ_{≥0}^{3}: \overline{m}′ ≤ \overline{m} if
m′_{3} ≤ m_{3}, m′_{2} + m′_{3} ≤ m_{2} + m_{3} and m′_{1} + m′_{2} + m′_{3} ≤ m_{1} + m_{2} + m_{3}. (B.18)

Let V \overline{m} be B_{+} submodules of π̃^{1}(t_{1}) ⊗ π̃^{2}(t_{2}), generated by the free action of the y_{i}, i = 1, 2, 3,

V \overline{m} = ∑_{\overline{m}′≤\overline{m}} B_{+}(v_{\overline{m}}′ ⊗ v_{0}).  (B.19)

Then the vectors v\overline{m} ; \overline{n}, \overline{n} ∈ ℤ_{≥0}^{3} form a basis of the factormodule

Ṽ \overline{m} = V \overline{m}/ ∑_{\overline{m}′≤\overline{m}}^{\overline{m}′≠\overline{m}} V^{\overline{m}′}  (B.20)

and the action of the generators of B_{+} in Ṽ \overline{m} is given by

\begin{aligned}y_{1}v\overline{m} ; \overline{n} & = & (q − q^{−1})^{−1}q^{x_{2}−n_{3}+1}[n_{3}]_{q}v\overline{m} ; \overline{n}−\overline{ε}_{3} + q^{x_{2}−2n_{3}}[n_{2}]_{q}v\overline{m} ; \overline{n}+\overline{ε}_{1}−\overline{ε}_{2}, \\ y_{2}v\overline{m} ; \overline{n} & = & q^{n_{3}+ \frac{1}{2}} v\overline{m} ; \overline{n}+\overline{ε}_{2}, \\ y_{3}v\overline{m} ; \overline{n} & = & q^{x_{1}}[x_{1} − x_{2} − n_{1} − n_{2} + n_{3} + 1]_{q}[n_{1}]_{q}v\overline{m} ; \overline{n}−\overline{ε}_{1}− \\ & & −q^{x_{2}+n_{2}−n_{3}−2}[n_{2}]_{q}v\overline{m} ; \overline{n}−\overline{ε}_{2}+\overline{ε}_{3}, \\ h_{1}v\overline{m} ; \overline{n} & = & (n_{1} − n_{2} − 2n_{3} + m_{1} − m_{2} − 2m_{3})v\overline{m} ; \overline{n}, \\ h_{2}v\overline{m} ; \overline{n} & = & (n_{1} + 2n_{2} + n_{3} + m_{1} + 2m_{2} + m_{3})v\overline{m} ; \overline{n}, \\ h_{3}v\overline{m} ; \overline{n} & = & (−2n_{1} − n_{2} + n_{3} − 2m_{1} − m_{2} + m_{3})v\overline{m} ; \overline{n}. \end{aligned}  (B.21)

where t_{1} = q^{2x_{1}}, t_{2} = q^{2x_{2}}.

Let π̃^{12}(t_{1}, t_{2}) be the following representation of B_{+} with a basis v\overline{n}, \overline{n} ∈ ℤ_{≥0}^{3}:

\begin{aligned}y_{1}v\overline{n} & = & (q − q^{−1})^{−1}q^{x_{2}−n_{3}+1}[n_{3}]_{q}v_{\overline{n}−\overline{ε}_{3}} + q^{x_{2}−2n_{3}}[n_{2}]_{q}v_{\overline{n}+\overline{ε}_{1}−\overline{ε}_{2}}, \\ y_{2}v\overline{n} & = & q^{n_{3}+ \frac{1}{2}} v_{\overline{n}+\overline{ε}_{2}}, \\ y_{3}v\overline{n} & = & q^{x_{1}}[x_{1} − x_{2} + 1 − n_{1} − n_{2} + n_{3}]_{q}[n_{1}]_{q}v_{\overline{n}−\overline{ε}_{1}} − q^{x_{2}+n_{2}−n_{3}−2}[n_{2}]_{q}v_{\overline{n}−\overline{ε}_{2}+\overline{ε}_{3}}, \end{aligned}

(B.22)

(the Cartan generators act as in (B.5)). Then, the relations (B.21) mean that the tensor product π̃^{1}(t_{1}) ⊗ π̃^{2}(t_{2}) admits a filtration with factormodules isomorphic to

π̃^{12}(t_{1}, t_{2})[(m_{1} + m_{2})α + (m_{2} + m_{3})β], m_{i} ≥ 0

and

Ã_{1}(t_{1})Ã_{2}(t_{2}) = \overline{Δ}^{−1}Tr_{π̃^{12}(t_{1},t_{2})} e^{iπP h}L.  (B.23)

Next, we decompose the tensor product π̃^{1,2}(t_{1}, t_{2}) ⊗ π̃^{3}(t_{3}). Here we use the basis v\overline{m} ; \overline{n}, \overline{n}, \overline{m} ∈ ℤ_{≥0}^{3},

\begin{aligned}v\overline{m} ; \overline{n} & = & ∑_{j=0}^{min(n_{1},n_{3})} ∑_{k_{2}=0}^{n_{2}} ∑_{k_{3}=0}^{n_{3}−j}(−1)^{j}(j)_{q}!\binom{n_{2}}{k_{2}}_{q}\binom{n_{3} − j}{k_{3}}_{q}\binom{n_{3}}{j}_{q}\binom{n_{1}}{j}_{q}× \\ & & ×q^{Ω_{j}(\overline{n},\overline{m},\overline{k})} (t_{3}/t_{2})^{k_{3}/2} v_{\overline{n}−\overline{k}−j(\overline{ε}_{1}+\overline{ε}_{3})} ⊗ v_{\overline{m}+\overline{k}+j\overline{ε}_{2}}, \end{aligned}  (B.24)

where \overline{k} = (0, k_{2}, k_{3}) and the phase Ω_{j}(\overline{n}, \overline{m}, \overline{k}) is

\begin{aligned}Ω_{j}(\overline{n}, \overline{m}, \overline{k}) & = & j(n_{2} − k_{2} − n_{3} + k_{3} − m_{1} − 2m_{2} − m_{3}) + k_{2}(k_{3} − n_{3} − m_{1} − 2m_{2} + m_{3}) \\ & & +n_{1}(2m_{1} + m_{2} − m_{3}) + n_{2}(m_{1} + 2m_{2}) + n_{3}(m_{2} − m_{1} + 2m_{3}) + \frac{3j(j+1)}{2}. \end{aligned}

In this case we define the another partial order in ℤ_{≥0}^{3}: \overline{m}′ ≤ \overline{m} if

m′_{1} ≤ m_{1}, m′_{1} + m′_{2} ≤ m_{1} + m_{2} and m′_{1} + m′_{2} + m′_{3} ≤ m_{1} + m_{2} + m_{3}.

Let V \overline{m} be the following B_{+} submodules of π̃^{12}(t_{1}, t_{2}) ⊗ π̃^{3}(t_{3}):

V \overline{m} = ∑_{\overline{m}′≤\overline{m}} B_{+}(v_{0} ⊗ v_{\overline{m}}′).  (B.25)

The vectors v\overline{m} ; \overline{n}, \overline{n} ∈ ℤ_{≥0}^{3} form a basis of the factormodule

Ṽ \overline{m} = V \overline{m}/ ∑_{\overline{m}′≤\overline{m}}^{\overline{m}′≠\overline{m}} V^{\overline{m}′}  (B.26)

in which

\begin{aligned}y_{1}v\overline{m} ; \overline{n} & = & q^{x_{3}}[x_{2} − x_{3} − n_{3} + 1]_{q}[n_{3}]_{q}v\overline{m} ; \overline{n}−\overline{ε}_{3} + q^{x_{2}−2n_{3}}[n_{2}]_{q}v\overline{m} ; \overline{n}+\overline{ε}_{1}−\overline{ε}_{2}, \\ y_{2}v\overline{m} ; \overline{n} & = & q^{n_{3}+ \frac{1}{2}} v\overline{m} ; \overline{n}+\overline{ε}_{2}, \\ y_{3}v\overline{m} ; \overline{n} & = & q^{x_{1}}[x_{1} − x_{2} − n_{1} − n_{2} + n_{3} + 1]_{q}[n_{1}]_{q}v\overline{m} ; \overline{n}−\overline{ε}_{1}− \\ & & −q^{x_{2}+n_{2}−n_{3}−2}[n_{2}]_{q}v\overline{m} ; \overline{n}−\overline{ε}_{2}+\overline{ε}_{3}, \end{aligned}  (B.27)

with t_{i} = q^{2x_{i}} and the Cartan generators acting as in (B.21). We see that the factormodules are shifts of the same B_{+}-module which is isomorphic to π̃_{μ}^{+}(t), if we put t_{i} = tq^{2μ_{i}}. This in combination with (B.23), proves the main statements (B.12) and (B.13).

In order to deduce (5.1) from (B.13) we notice that the degenerated evaluation Verma modules π̃^{i}(t_{i}) are not irreducible; each of them admits a filtration indexed by a single nonnegative integer such that all the factors are the shifts of oscillator representations: the representation π̃^{1}(t) admits an increasing filtration by submodules V^{n}, where V^{n} is generated over B_{+} by the vectors v_{(0,0,k)}, k ≤ n with factormodules isomorphic to representation ρ_{1}(q^{2}t)[nβ] ,n ≥ 0; the representation π̃^{3}(t) admits an increasing filtration by submodules V^{n}, which are generated over B_{+} by the vectors v_{(k,0,0)}, k ≤ n with factormodules isomorphic to representation ρ_{3}(q^{−2}t)[nα] ,n ≥ 0 and the representation π̃^{2}(t) admits a decreasing filtration by submodules V^{n}, which are generated over B_{+} by the vector

∑_{k=0}^{n} \frac{[n]_{q}!}{[k]_{q}![n − k]_{q}!}q^{\frac{−k(k−1)}{2}} (q^{−1} − q)^{k}v_{(k,n−k,k)}.  (B.28)

The factormodules are isomorphic to ρ_{3}(t)[n(α + β)], n ≥ 0.

This means that

Ã_{1}(t) = (1 − z_{3}/z_{2})^{−1}Tr_{ρ_{1}(q^{2}t)} e^{iπP h}L = \overline{Δ}^{−1}A_{1}(q^{2}t),
Ã_{2}(t) = (1 − z_{3}/z_{1})^{−1}Tr_{ρ_{2}(t)} e^{iπP h}L = \overline{Δ}^{−1}A_{2}(t),
Ã_{3}(t) = (1 − z_{2}/z_{1})^{−1}Tr_{ρ_{3}(q^{−2}t)} e^{iπP h}L = \overline{Δ}^{−1}A_{3}(q^{−2}t).

(B.29)

The relation (B.13) combined with (B.29) gives

Tr_{π̃_{μ}^{+}(t)} e^{iπP h}L = \overline{Δ}^{−1}A_{1}(tq^{2μ_{1}+2})A_{2}(tq^{2μ_{2}})A_{3}(tq^{2μ_{3}−2}).  (B.30)

The determinant formula (5.1) is a direct corollary of (B.30) and of the Bernstein-Gel’fand-Gel’fand resolution of finite-dimensional module π_{μ}, which implies the relation

T_{μ}(t) = ∑_{σ∈S_{3}}(−1)^{l(σ)}T_{σ(μ+ρ)−ρ}^{+}(t),  (B.31)

where ρ = (1, 0, −1).

The determinant formula (5.2) can be deduced in an analogous manner with the use of the second evaluation map or can be deduced from (5.1) in the following way. Due to the definition (4.18), we conjugate the action of the generators of B_{+} in the shifted Verma module π̃_{μ}^{+}(t) by σ_{13}. Taking into account the fact that the U_{q}(gl(3))-Verma module π_{μ_{1},μ_{2},μ_{3}} · σ_{13} is isomorphic to the Verma module π_{−μ_{3},−μ_{2},−μ_{1}}, we see that

\overline{T}_{μ}(t) = Tr_{π_{−σ_{13}(μ)}(−t)·(1⊗σ_{13})} e^{iπP h}L and T_{μ}^{+}(t) = Tr_{π_{−σ_{13}(μ)}^{+}(−t)·(1⊗σ_{13})} e^{iπP h}L, which means that we can use the statement of (5.1) for the highest weight −σ_{13}(μ) combined with automorphism σ_{13} of the algebra B_{+}. Due to the definition (4.46) we get the statement of (5.2).

Let us turn to the relations (5.4), (5.5). Their proof consists of an interpretation of both sides of the equalities as of B_{+}-modules. Consider, for instance, the tensor product

π̃^{2}(t_{2}) ⊗ π̃^{3}(t_{3}),

of degenerated evaluation Verma modules. Analogously to (B.22), (B.23) it is isomorphic, on the level of characters, to a sum

∑_{m_{i}≥0} π̃^{23}(t_{2}, t_{3})[(m_{1} + m_{2})α + (m_{2} + m_{3})β)],

where the module π̃^{23}(t_{2}, t_{3}) is the following representation of B_{+} with a basis v\overline{n}, \overline{n} ∈ ℤ_{≥0}^{3}:

\begin{aligned}y_{1}v\overline{n} & = & q^{x_{3}}[x_{2} − x_{3} − n_{3} + 1]_{q}[n_{3}]_{q}v_{\overline{n}−\overline{ε}_{3}} + q^{x_{2}−2n_{3}}[n_{2}]_{q}v_{\overline{n}+\overline{ε}_{1}−\overline{ε}_{2}}, \\ y_{2}v\overline{n} & = & q^{n_{3}+ \frac{1}{2}} v_{\overline{n}+\overline{ε}_{2}}, \\ y_{3}v\overline{n} & = & −(q − q^{−1})^{−1}q^{x_{2}+n_{1}+n_{2}−n_{3}−1}[n_{1}]_{q}v_{\overline{n}−\overline{ε}_{1}} − q^{x_{2}+n_{2}−n_{3}−2}[n_{2}]_{q}v_{\overline{n}−\overline{ε}_{2}+\overline{ε}_{3}}, \end{aligned} (B.32)

where h_{i} act as in (B.5), t_{i} = q^{2x_{i}}. When t_{2} = t_{3} = t, π̃^{23}(t_{2}, t_{3}) has a submodule generated by the vector v_{\overline{ε}_{3}} with a basis ṽ\overline{n} = v_{\overline{n}+\overline{ε}_{3}}, \overline{n} ∈ ℤ_{≥0}^{3}. One can see from (B.33), (B.5) that this submodule is isomorphic to π̃^{23}(q^{−2}t, q^{2}t)[β]. The factormodule

π̃^{23}(t, t)/π̃^{23}(q^{−2}t, q^{2}t)[β]

has a basis v_{n,m}, n, m ≥ 0 with the action of B_{+} given by

\begin{aligned}y_{1}v_{n,m} & = & q^{x−2p}[m]_{q}v_{n+1,m−1}, \\ y_{2}v_{n,m} & = & q^{\frac{1}{2}} v_{n,m+1}, \\ y_{3}v_{n,m} & = & −(q − q^{−1})^{−1}q^{x+n+m−1}[n]_{q}v_{n−1,m}, \end{aligned}  (B.33)

h_{1}v_{n,m} = (n − m)v_{n,m},  h_{2}v_{n,m} = (n + 2m)v_{n,m},  h_{3}v_{n,m} = (−2n − m)v_{n,m}.

From (B.33) we can check directly that this factormodule is isomorphic to the representation \overline{ρ}_{1}^{+}(q^{−1}t), so

\overline{Δ} (Ã_{2}(t)Ã_{3}(t) − z_{3}/z_{2}Ã_{2}(q^{−2}t)Ã_{3}(q^{2}t)) = (1 − z_{3}/z_{1})^{−1}(1 − z_{2}/z_{1})^{−1}\overline{A}_{1}(q^{−1}t),

or

A_{2}(qt)A_{3}(q^{−1}t) − z_{3}/z_{2}A_{2}(q^{−1}t)A_{3}(qt) = (1 − z_{3}/z_{2})\overline{A}_{1}(t),  (B.34)

which is equivalent to one of the relations (5.4). The others follow from (B.34) after application of the symmetries (4.9) and (4.10).

Appendix C. The p → ∞ asymptotics of Ψ(ν).

In this Appendix we derive the leading asymptotics of the function Ψ(ν) at large values of p = (p_{1}, p_{2}). We assume p_{1} and p_{2} tend to infinity while their ratio p_{2}/p_{1} is kept fixed

p_{1}, p_{2} → ∞,  p_{2}/p_{1} = finite.  (C.1)

First consider the classical case. Writing the formula (2.28) in full one gets

T^{(cl)}(λ) = Λ_{1}^{(cl)}(λ) + Λ_{2}^{(cl)}(λ) + Λ_{3}^{(cl)}(λ),  Λ_{1}^{(cl)}(λ)Λ_{2}^{(cl)}(λ)Λ_{3}^{(cl)}(λ) = 1,  (C.2)

where Λ_{j}^{(cl)}(λ), j = 1, 2, 3 are the eigenvalues of the classical monodromy matrix M(λ). The leading p → ∞ contribution to these eigenvalues can be easily calculated directly from the definition (2.27). According to (2.44) the potentials U(u) and V (u) in (2.26) in this limit become independent of u

U(u)|_{p→∞} ∼ x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3},  V (u)|_{p→∞} ∼ ix_{1}x_{2}x_{3},  (C.3)

where

x_{j} = (w_{j}p),  j = 1, 2, 3  (C.4)

with w_{j} defined in (2.34). Therefore the eigenvalues of M(λ)

Λ_{j}^{(cl)}(λ) = e^{2πix_{j}(λ)},  (C.5)

are expressed though the roots x_{j}(λ), j = 1, 2, 3, of the cubic equation

x(λ)^{3} + g_{2} x(λ) + g_{3} − iλ^{3} = 0,  (C.6)

where we have introduced the notation

g_{2} = x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3},  g_{3} = −x_{1}x_{2}x_{3}.  (C.7)

The roots of (C.6) can be written as

x_{j}(λ) = 2[( − \frac{g_{2}}{3} )^{\frac{1}{2}} _{2}F_{1}(−\frac{1}{6}, \frac{1}{6}, \frac{1}{2}; 1 + \frac{27 (g_{3} − iλ^{3})^{2}}{4 g_{2}^{2}} )]_{j−th},  (C.8)

where _{2}F_{1} stands for the Gauss hypergeometric function. The expression in the square brackets above is a multivalued function and the symbol [...]_{j−th} denotes an appropriate branch of this function fixed by the requirement x_{j}(0) = x_{j} where x_{j}, j = 1, 2, 3 defined in (C.4). Expanding this expression in a series in λ and using the Sommerfeld-Watson transformation one can rewrite (C.8) as

log Λ_{j}^{(cl)}(λ) = 2πir_{j} + ∫_{C_{ν}} \frac{dν}{ν} Γ(1 − \frac{iν}{3} )Γ(−\frac{1}{3} + \frac{iν}{3} )Γ(\frac{1}{3} + \frac{iν}{3} )(−iλ)^{\frac{iν}{3}} Ψ_{j}^{(cl)}(ν), (C.9)

where

Ψ_{j}^{(cl)}(ν)|_{p→∞} = \frac{2^{− \frac{2iν}{3}}}{\sqrt{3π}Γ(1/2 + iν/3)} ×

[( − \frac{g_{2}}{3} )^{(1−iν)/2} _{2}F_{1}(−\frac{1}{6} + \frac{iν}{6} , \frac{1}{6} + \frac{iν}{6} , \frac{1}{2} + \frac{iν}{3} , 1 + \frac{27g_{3}^{2}}{4g_{2}^{3}} )]_{j−th}, (C.10)

where the integration contour goes along the line ℑm ν = −1 − ϵ and the branches of the expression in the square brackets are chosen in accordance with (C.8)

Ψ_{j}^{(cl)}(0) = \frac{(w_{j}p)}{2π\sqrt{3}} .  (C.11)

Now turn to the quantum case. As before we restrict our considerations to the case 0 < g < \frac{2}{3}. Examining expressions (6.9) it is easy to conclude that the leading p → ∞ asymptotics of Ψ(ν) is the same for all its eigenvalues, i.e. Ψ(ν)|_{p→∞} is proportional to the unit operator I. Further we make an assumption (which again can be justified by studying the p → ∞ of the Coulomb integrals (4.75)) that the p and g dependence of Ψ(ν)|_{p→∞} factorizes in the form

Ψ(λ)|_{p→∞} ∼ ψ_{1}(ν, g)ψ_{2}(ν, p) I.  (C.12)

Comparing (C.9) with the g → 0 limit of (6.9) one gets

Ψ_{j}(ν)|_{p→∞} = \frac{ψ_{1}(ν, g)}{ψ_{1}(ν, 0)}Ψ_{j}^{(cl)}(ν)|_{p→∞}.  (C.13)

To calculate the g-dependent multiplier in the last formula we will use the DDV equation. Since the asymptotics (6.16) are the same for all eigenvalues of Ψ(ν) it is enough to consider the vacuum eigenvalue. Moreover, since the multiplier we want to calculate does not depend on p we will restrict ourselves to the case p_{2} = 0 and denote

p_{1} = p,  x_{1} = p,  x_{2} = 0,  x_{3} = −p.  (C.14)

In this case the vacuum eigenvalues A_{1}^{(vac)}(t) and \overline{A}_{3}^{(vac)}(t) of the operator A_{1}(t) and \overline{A}_{3}(t) satisfy the relation

\overline{A}_{3}^{(vac)}(t) = A_{1}^{(vac)}(−t).  (C.15)

Define the function

a(t) = e^{2πip} \frac{A_{1}^{(vac)}(itq^{2})}{A_{1}^{(vac)}(itq^{−2})} \frac{A_{1}^{(vac)}(−itq^{−1})}{A_{1}^{(vac)}(−itq)}.  (C.16)

Recalling the assumption about the structure of the zeros of A_{1}(t) in the beginning of Section 6 (the zeroes of A_{1}^{(vac)}(t) lie on the positive imaginary axis of t) and using (6.7) we can show for real p

a(t)^{∗} = a(t^{∗})^{−1}.  (C.17)

Due to (6.1) a(t) has the asymptotics

log a(t) ∼ −2iM sin (\frac{π(1 + ξ)}{3}) t^{\frac{1+ξ}{3}} ,  t → ∞,  | arg(t)| < min[2πg, π(1 − g)]

(C.18)

while for small t, we obviously have

a(t) = 2πip + O(t),  t → 0.  (C.19)

Again using (6.7) we have

log a(t) = 2πip + ∑_{k} F (itt_{k}^{−1}),  (C.20)

where

F (t) = log (\frac{1 − tq^{2}}{1 − tq^{−2}} \frac{1 + tq^{−1}}{1 + tq} ) .  (C.21)

Using the Bethe Ansatz equations (6.3), (6.4) we can write the sum in (C.20) as a contour integral

log a(t) = 2πip + ∫_{C} \frac{dv}{2πi}F (tv^{−1})∂_{v} log(1 + a(v)),  (C.22)

where the contour C goes from +∞ to zero above the real axis, winds about zero, and returns to +∞ below the real axis. Integrating by parts (boundary terms vanish) one obtains

log a(t) − 2πip = \frac{1}{2πi} ∫_{0}^{∞} \frac{dv}{v} t∂_{t}F (tv^{−1}) \{log(1 + a(v − i0)) − log(1 + a(v + i0))\} ,

= ∫_{0}^{∞} \frac{dv}{πv} t∂_{t}F (tv^{−1})ℑm log(1 + a(v − i0)) − ∫_{0}^{∞} \frac{dv}{2πiv} t∂_{t}F (tv^{−1}) log a(v),

where we have used (C.17) to obtain the second line.

We now introduce new variables

t = e^{\frac{3θ}{1+ξ}} , v = e^{\frac{3θ′}{1+ξ}} ,  (C.23)

and recalling that q = exp(iπg), we obtain

log a(θ) = 2πip − 2i ∫_{−∞}^{∞} R(θ − θ′)ℑm log(1 + a(θ′ − i0)) dθ′
+ ∫_{−∞}^{∞} R(θ − θ′) log a(θ′) dθ′,  (C.24)

where

R(θ) = \frac{3i}{2π(1 + ξ)}t∂_{t}F (t).  (C.25)

With the standard notation for the convolution integral the Eq. (C.24) becomes

K ∗ log a(θ) = 2πip − 2iR ∗ ℑm log(1 + a(θ − i0)),  (C.26)

with K(θ) = δ(θ) − R(θ). We now apply the inverse of the integral operator K(θ) to both sides of (C.26). Since K(θ) is a singular operator one has to add an appropriate zero mode of K to the RHS of the resulting equation. The required zero mode is completely fixed by the asymptotic conditions (C.18) and (C.19). In this way one obtains

i log a(θ) = −\frac{2πp}{g} − 2G ∗ ℑm log(1 + a(θ − i0)) + 2M sin (\frac{π(1 + ξ)}{3}) e^{θ},  (C.27)

where G(θ) = −(K^{−1} ∗ R)(θ).

Now define

B(p) = \frac{(1 + ξ)}{3} log(−it_{1}),  (C.28)

where t_{1} is the smallest (in absolute value) zero of A_{1}^{vac}(t). The convolution term in (C.27) can be written as

G ∗ ℑm log(1 + a^{vac}(θ − i0)) = ∫_{−∞}^{B(p)} G(θ − θ′)ℑm log(1 + a^{vac}(θ′ − i0)) dθ′
+ ∫_{B(p)}^{∞} G(θ − θ′)ℑm log(1 + a^{vac}(θ′ − i0)) dθ′.(C.29)

Let us make the technical assumption that

B(p) ∼ const log p, p → ∞.  (C.30)

Then one can show the second term in (C.29) vanishes when p → +∞ and therefore can be dropped in the leading large p approximation. The Eq. (C.27) then becomes

−\frac{πp}{g} + M sin (\frac{π(1 + ξ)}{3}) e^{θ} = ∫_{−∞}^{B(p)} \frac{dθ′}{2πi}∂_{θ} log S(θ − θ′)ℑm log (1 + a(θ′)) , (C.31)

where we have used the notation

G(θ) = δ(θ) + \frac{1}{2πi}∂_{θ} log S(θ),  (C.32)

log S(θ) = −i ∫_{−∞}^{∞} \frac{dν}{ν} sin(νθ)\begin{matrix}sinh & (\frac{πν(1+ξ)}{3}) & cosh & (\frac{πν}{6} ) \\ sinh & (\frac{πνξ}{3}) & cosh & (\frac{πν}{2} ) \end{matrix},  (C.33)

and the relation

log a^{vac}(θ) = 2i log ℑm log(1 + a^{vac}(θ)), θ < B(p)  (C.34)

which holds for θ < B(p). The function S(θ) is Smirnov’s kink-kink S-matrix for the algebra A_{2}^{(2)}, [66].

The Fourier transform of the integral kernel in (C.31) factorizes as

−\frac{1}{2πi} ∫_{−∞}^{∞} e^{−iνθ}∂_{θ} log S(θ)dθ = \frac{1}{K_{+}(ν)K_{−}(ν)},  (C.35)

where

K_{+}(ν) = \frac{1}{\sqrt{g}} \frac{Γ (−\frac{iν(1+ξ)}{3}) Γ (\frac{1}{2} − \frac{iν}{6} ) e^{iνδ}}{Γ (−\frac{iνξ}{3}) Γ (\frac{1}{2} − \frac{iν}{2} )} ,  K_{−}(ν) = K_{+}(−ν),  (C.36)
3δ = (1 + ξ) log(1 + ξ) − ξ log ξ + log \frac{2}{3\sqrt{3}}.  (C.37)

The function K_{+}(ν) is analytic and nonzero in the upper half-plane ℑm ν > −1 where it has the asymptotics

K_{+}(ν) ∼ 1 + O (\frac{1}{ν}) ,  ν → ∞.  (C.38)

The Eq.(C.31) is an equation of Wiener-Hopf type, which can be solved by the standard technique. Defining the Fourier transform

f_{+}(ν) = ∫_{−∞}^{B(p)} ℑm log(1 + a^{vac}(θ))e^{−iθν} dθ,  (C.39)

one obtains

\frac{f_{+}(ν)}{K_{+}(ν)}e^{iνB(p)} = \frac{1}{2πi} ∫_{−∞+iϵ}^{∞+iϵ} \frac{1}{ν′ − ν} K_{−}(ν′)G_{+}(ν′) dν′,  (C.40)

where

G_{+}(ν) = ∫_{−∞}^{0} (\frac{πp}{g} − M sin (\frac{π(1 + ξ)}{3}) e^{θ+B(p)}) e^{−iθν} dν,
= \frac{iπp}{gν} − \frac{iM}{ν + i} sin (\frac{π(1 + ξ)}{3}) e^{B(p)},  (C.41)

and ϵ is a small positive constant. Completing the contour in the lower half-plane, we have

\frac{f_{+}(ν)}{K_{+}(ν)}e^{iνB(p)} = \frac{iπp}{\sqrt{g}ν} − \frac{iMπe^{B(p)−δ}}{\sqrt{g}(ν + i)} \frac{Γ (\frac{2}{3})}{Γ (\frac{ξ}{3}) Γ (\frac{2−ξ}{3})}.  (C.42)

As follows from the definition (C.39) the leading 1/ν term in the ν → ∞ asymptotic expansion of the RHS of (C.42) is independent of p. This gives

(−it_{1})^{\frac{1+ξ}{3}} = p\frac{Γ (\frac{ξ}{3}) Γ (\frac{2−ξ}{3})}{MΓ (\frac{2}{3})}e^{δ},  (C.43)

and thus we have

f_{+}(ν) = \frac{iπp}{2ν} \frac{Γ (1 − \frac{iν(1+ξ)}{3}) Γ (\frac{1}{2} − \frac{iν}{6} )}{Γ (1 − \frac{iξν}{3} ) Γ (\frac{3}{2} − \frac{iν}{2} )}e^{iνδ}(−it_{1})^{\frac{−iν(1+ξ)}{3}} .  (C.44)

Given the function f_{+}(ν), (C.44), we can recover log A_{1}(t) from

log A_{1}(t) = −\frac{i}{2π} ∫_{C_{ν}} cosh (\frac{πν}{6} ) ; cosh (\frac{πν}{2} ) sinh (\frac{πξν}{3} )f_{+}(ν)(it)^{\frac{iν(1+ξ)}{3}} dν,  (C.45)

where the contour C_{ν} is along the line ℑm ν = −1 − ϵ, with ϵ is a small positive constant. Thus we have

log A_{1}^{vac}(t) = −\frac{3p}{4πξ} ∫_{C_{ν}} \frac{e^{iνδ}}{ν^{2}} \frac{Γ (1 − \frac{iν(1+ξ)}{3}) Γ (−\frac{1}{2} + \frac{iν}{2} ) Γ (1 + \frac{iξν}{3} )}{Γ (\frac{1}{2} + \frac{iν}{6} )} (−\frac{t}{t_{1}} )^{\frac{iν(1+ξ)}{3}} dν.

(C.46)

This expression can be written in the form of (6.9)

log A_{i}(t) = \frac{1}{2πi} ∫_{C_{ν}} \frac{dν}{ν} Γ(iνξ/3)Γ(1 − iν(ξ + 1)/3)Γ((−1 + iν)/3) ×
Γ((1 + iν)/3)(Γ(1 − g))^{iν(1+ξ)}(it)^{iν(1+ξ)/3} Ψ(ν)|_{p→∞}, (C.47)

where

Ψ(ν)|_{p→∞} = \frac{2^{− \frac{2iν}{3}}}{\sqrt{3}} (\frac{p^{2}}{3} )^{(1−iν)/2} \frac{1}{Γ(1/3 + iν/6)Γ(2/3 + iν/6)}.  (C.48)

Comparing this with the specialization of (C.10) for the case (C.14), one concludes that the factor in (C.13)

\frac{ψ_{1}(ν, g)}{ψ_{1}(ν, 0)} ≡ 1.  (C.49)

Taking into account (C.13), we obtain the expression (6.30) given in the main text.

To get a small t expansion of A_{1}^{vac}(t), we complete the contour in the lower-half plane, and evaluate the residues at the points ν = 3i(g − 1)n, n = 1, 2, 3 …, and we obtain the large p asymptotic equivalent of (4.55). Thus, we have from (C.46)

H_{1}^{(1)(vac)}|_{p→∞} = −\frac{p}{2} \frac{e^{3(1−g)δ}}{Γ^{3}(1 − g)} \frac{Γ (1 − \frac{3}{2}g) Γ(g)}{Γ (1 − \frac{1}{2}g)}g^{3}(−it_{1})^{−1}.  (C.50)

However, we know H_{1}^{(1)(vac)} exactly (4.85). Taking the large p asymptotic of (4.85), comparing with (C.50), and using (C.43) we see that we must take

M = \frac{Γ (\frac{ξ}{3}) Γ (\frac{2−ξ}{3})}{Γ (\frac{2}{3})} (Γ(1 − g))^{\frac{1}{1−g}} ,  (C.51)

and then

H_{n}^{(1)(vac)}|_{p→∞} = \frac{(−1)^{n}}{2n!} \frac{Γ (\frac{3(1−g)n−1}{2}) Γ(gn)}{Γ (\frac{(1−g)n+1}{2})}g^{3n}p^{1−3(1−g)n}.  (C.52)

We also generate an asymptotic expansion for large t by completing the contour in the upper half-plane. Here we have poles at the following points:

ν = i(2n − 1), n = 0, 1, 2, …, ν = 3i(g^{−1} − 1)n, n = 0, 1, 2, 3 …,  (C.53)

and we obtain

A_{1}^{vac}(t) = C^{vac}(g, p)(s)^{− \frac{p}{g}} exp \{∑_{n=0}^{∞} B_{2n−1}I_{2n−1}^{vac}|_{p→∞}(s)^{− \frac{2n−1}{3(1−g)}} \}
× exp \{− ∑_{n=1}^{∞} H̃_{n}^{(1)(vac)}|_{p→∞}(s)^{− \frac{n}{g}} \} ,  (C.54)

where

B_{2n−1} = \frac{(−1)^{n+1}}{3(1 − g)(n)!} \frac{Γ (\frac{2n−1}{3(1−g)}) Γ (\frac{2n−1}{3(1−g^{−1})})}{Γ (\frac{2−n}{3})}g^{\frac{n(1+g)−1}{g−1}} ,  (C.55)

C^{(vac)}(g, p) = (\frac{2^{\frac{2}{3}} p}{e})^{\frac{3p}{g} −3p} (g)^{− \frac{3p}{g}} ,  H̃_{n}^{(1)(vac)}(g, p) = H_{n}^{(1)(vac)}(g^{−1}, g^{−1}p). (C.56)

and I_{2n−1}^{vac}|_{p→∞}, n = 0, 1, 2, … are the remaining Integrals of Motion after the A_{2}^{(2)} specialization has been taken into account (W (u) ≡ 0),

I_{−1}^{vac} ≡ 1,  I_{2n−1}^{vac}|_{p→∞} = \frac{p^{2n}}{g^{n}} , n = 1, 2, 3, …  (C.57)

Note also that C^{(vac)}(g, p) is the vacuum asymptotic value of the operator C_{j}{g, gϕ(u)} defined in (6.16).

Appendix D. Representation Matrices for the Fundamental Representations

For the first fundamental representation π_{ω_{1}} we have

π_{ω_{1}}(H_{α}) = ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 0 \end{matrix} ) π_{ω_{1}}(E_{α}) = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) π_{ω_{1}}(F_{α}) = ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )

(D.1)

π_{ω_{1}}(H_{β}) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) π_{ω_{1}}(E_{β}) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) π_{ω_{1}}(F_{β}) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) (D.2)

while for π_{ω_{2}} we have

π_{ω_{2}}(H_{α}) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ) π_{ω_{2}}(E_{α}) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) π_{ω_{2}}(F_{α}) = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ) (D.3)

π_{ω_{2}}(H_{β}) = ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 0 \end{matrix} ) π_{ω_{2}}(E_{β}) = ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) π_{ω_{2}}(F_{β}) = ( \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) (D.4)

References

[1] Zamolodchikov, A. B. and Zamolodchikov, A. B. Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Physics 120 (1979) 253–291.

[2] Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe Ansatz. Comm. Math. Phys. 177 (1996) 381–398. [hep-th/9412229].

[3] Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. Integrable structure of conformal field theory. II. Q-operator and DDV equation. Comm. Math. Phys. 190 (1997) 247–278. [hep-th/9604044].

[4] Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. Integrable structure of conformal field theory. III. The Yang-Baxter Relation. Comm. Math. Phys. 200 (1999) 297–324. [hep-th/9805008].

[5] Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. Quantum field theories in finite volume: excited state energies. Nuclear Phys. B 489 (1997) 487–531.

[hep-th/9607099].

[6] Sklyanin, E. K., Tahtadžjan, L. A., and Faddeev, L. D. Quantum inverse problem method. I. Teoret. Mat. Fiz. 40 (1979) 194–220.

[7] Baxter, R. J. Exactly Solved Models in Statistical Mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1982.

[8] Zamolodchikov, A. B. Infinite additional symmetries in two-dimensional conformal quantum field theory. Teoret. Mat. Fiz. 65 (1985) 347–359. [English translation: Theor. Math. Phys. 65 (1985) 1205-1213].

[9] Lukyanov, S. L. and Fateev, V. V. Additional symmetries and exactly-soluble models in two-dimensional conformal field theory. Sov. Sci. Rev. A. Phys. 15 (1990) 1–117.

[10] Kupershmidt, B. A. and Mathieu, P. Quantum Korteweg-de Vries like equations and perturbed conformal field theories. Phys. Lett. B 227 (1989) 245–250.

[11] Baxter, R. J. Partition function of the eight-vertex lattice model. Ann. Physics 70 (1972) 193–228.

[12] Kulish, P. P. and Reshetikhin, N. Y. On GL_{3}-invariant solutions of the Yang-Baxter equation and associated quantum systems. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 120 (1982) 92–121. Questions in Quantum Field Theory and Statistical Physics, 3.

[13] Krichever, I., Lipan, O., Wiegmann, P., and Zabrodin, A. Quantum integrable models and discrete classical Hirota equations. Comm. Math. Phys. 188 (1997)

267–304. [hep-th/9604080].

[14] Pronko, G. P. and Stroganov, Y. G. Families of solutions of the nested Bethe Ansatz for the A_{2} spin chain. J. Phys. A 33 (2000) 8267–8273. [hep-th/9902085].

[15] Sutherland, B. Model for a multicompontent quantum system. Phys. Rev. B 12 (1975) 3795–3805.

[16] Bazhanov, V. V. and Reshetikhin, N. Restricted solid-on-solid models connected with simply laced algebras and conformal field theory. J. Phys. A 23 (1990) 1477–1492.

[17] Kuniba, A., Nakanishi, T., and Suzuki, J. Functional relations in solvable lattice models. I. Functional relations and representation theory. Internat. J. Modern Phys. A 9 (1994) 5215–5266. [hep-th/9309137].

[18] Destri, C. and de Vega, H. J. Unified approach to thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories. Nuclear Phys. B 438 (1995) 413–454. [hep-th/9407117].

[19] Klümper, A., Batchelor, M. T., and Pearce, P. A. Central charges of the 6- and 19vertex models with twisted boundary conditions. J. Phys. A 24 (1991) 3111–3133.

[20] Dorey, P. and Tateo, R. Differential equations and integrable models: the SU(3) case. Nuclear Phys. B 571 (2000) 583–606. [hep-th/9910102 ].

[21] Ghoshal, S. and Zamolodchikov, A. Boundary S matrix and boundary state in twodimensional integrable quantum field theory. Internat. J. Modern Phys. A 9 (1994)

3841–3885. [hep-th/9306002].

[22] Caldeira, A. O. and Leggett, A. J. Influence of dissipation on quantum tunneling in macroscopic systems. Phys. Rev. Lett 46 (1981) 211–214.

[23] Schmid, A. Diffusion and localization in a dissipative quantum system. Phys. Rev. Lett. 51 (1983) 1506–1509.

[24] Callan, C. G. and Thorlacius, L. Open string theory as dissipative quantum mechanics. Nuclear Phys. B. 329 (1990) 117–138.

[25] Fisher, M. P. A. and Zwerger, W. Quantum Brownian motion in a periodic potential. Phys. Rev. B 32 (1990) 6190–6206.

[26] Kane, C. L. and Fisher, M. P. A. Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas. Phys. Rev. B 46 (1992) 15233– 15262.

[27] Moon, K., Yi, H., Kane, C. L., Girvin, S. M., and Fisher, M. P. A. Resonant tunneling between quantum Hall edge states. Phys. Rev. Lett. 27 (1993) 4381–4384.

[28] Fendley, P., Ludwig, A. W. W., and H., S. Exact conductance through point contacts in the ν = 1/3 fractional quantum Hall effects. Phys. Rev. Lett. 74 (1995) 3005–3008.

[cond-mat/9408068].

[29] Fendley, P., Ludwig, A. W. W., and Saleur, H. Exact non-equilibrium transport through point contacts in quantum wires and fractional quantum Hall devices. Phys. Rev. B 52 (1995) 8934–8950. [cond-mat/9503172].

[30] Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. On non-equilibrium states in QFT model with boundary interaction. Nuclear Phys. B 549 (1999) 529–

545. [hep-th/9812091].

[31] Corrigan, E., Dorey, P. E., Rietdijk, R. H., and Sasaki, R. Affine Toda field theory on a half-line. Phys. Lett. B 333 (1994) 83–91. [hep-th/9404108].

[32] Penati, S. and Zanon, D. Quantum integrability in two-dimensional systems with boundary. Phys. Lett. B 358 (1995) 63–72. [hep-th/9501105].

[33] Fateev, V. A. and Lykyanov, S. L. The models of two-dimensional conformal quantum field theory with Z_{n} symmetry. Internat. J. Modern Phys. A 3 (1988) 507–520.

[34] Boussinesq, J. Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizonatal. J. Math. Pures. Appl. 7 (1872) 55–108.

[35] Zakharov, V. On stochastization of one-dimensional chains of nonlinear oscillators. Zh. Eksp. Teor. Fiz 65 (1973) 219–225.

[36] Gel′fand, I. M. and Dikiĭ, L. A. A family of Hamiltonian structures connected with integrable nonlinear differential equations. Akad. Nauk SSSR Inst. Prikl. Mat. Preprint (1978) 41.

[37] Mathieu, P. Extended classical conformal algebras and the second Hamiltonian structure of Lax equations. Phys. Lett. B 208 (1988) 101–106.

[38] McKean, H. P. Boussinesq’s equation as a Hamiltonian system. In Topics in Functional Analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), pages 217–226. Academic Press, New York, 1978.

[39] Drinfel’d, V. G. and Sokolov, V. V. Lie algebras and equations of Korteweg de Vries type. In Current Problems in Mathematics, Vol. 24, pages 81–180. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.

[40] Sklyanin, E. On the complete integrability of the Landau-Lifshitz equation. Preprint LOMI, E-3-79, Leningrad (1979).

[41] Faddeev, L. D. and Takhtajan, L. A. Hamiltonian Methods in the Theory of Solitons. Springer-Verlag, Berlin, 1987.

[42] Drinfel’d, V. G. Quantum groups. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), pages 798–820, Providence, RI, 1987. Amer. Math. Soc.

[43] Khoroshkin, S. M. and Tolstoy, V. N. The uniqueness theorem for the universal R-matrix. Lett. Math. Phys. 24 (1992) 231–244.

[44] Tolstoĭ, V. N. and Khoroshkin, S. M. Universal R-matrix for quantized nontwisted affine Lie algebras. Funktsional. Anal. i Prilozhen. 26 (1992) 85–88.

[45] Zhang, Y. Z. and Gould, M. D. Quantum affine algebras and universal R-matrix with spectral parameter. Lett. Math. Phys. 31 (1994) 101–110. [hep-th/9307007].

[46] Fateev, V. A. and Zamolodchikov, A. B. Conformal quantum field theory models in two dimensions having Z_{3} symmetry. Nuclear Phys. B 280 (1987) 644–660.

[47] Fateev, V. A. and Lukyanov, S. L. Poisson-Lie groups and classical W -algebras. Internat. J. Modern Phys. A 7 (1992) 853–876.

[48] Fateev, V. A. and Lukyanov, S. L. Vertex operators and representations of quantum universal enveloping algebras. Internat. J. Modern Phys. A 7 (1992) 1325–1359.

[49] Bouwknegt, P., McCarthy, J., and Pilch, K. The W_{3} Algebra. Springer-Verlag, Berlin, 1996. Lecture Notes in Physics. New Series m: Monographs, 42 [hep-th/9509119].

[50] Frenkel, E. and Smirnov, F. A. Private communication, 1998.

[51] Faddeev, L. D. Discrete Heisenberg-Weyl group and modular group. Lett. Math. Phys. 34 (1995) 249–254. [hep-th/9504111].

[52] Sasaki, R. and Yamanaka, I. Virasoro algebra, vertex operators, quantum sine-Gordon and solvable quantum field theories. In Conformal Field Theory and Solvable Lattice Models (Kyoto, 1986) Advanced Studies in Pure Mathematics (Tokyo, Japan) Vol 16, pages 271–296. Academic Press, Boston, MA, 1988.

[53] Eguchi, T. and Yang, S. Deformations of conformal field theories and soliton equations. Phys. Lett. B 224 (1989) 373–378.

[54] Feigin, B. and Frenkel, E. Integrals of motion and quantum groups. In Integrable Systems and Quantum Groups (Montecatini Terme, 1993), pages 349–418. Springer, Berlin, 1996. [hep-th/9310022].

[55] Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. Higher Transcendental Functions. Vols. I, II. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman.

[56] Mumford, D. Tata lectures on Theta. I, II. Birkhäuser Boston Inc., Boston, Mass., 1983, 1984.

[57] Saleur, H. and Wehefritz-Kaufmann, B. Thermodynamics of the complex su(3) Toda theory. Phys. Lett. B 481 (2000) 419–426.

[58] Ellem, R. M. and Bazhanov, V. V. Excited state TBA for the ϕ_{2,1} perturbed M_{3,5} model. In preparation.

[59] Klümper, A. and Pearce, P. A. Analytic calculation of scaling dimensions: tricritical hard squares and critical hard hexagons. J. Statist. Phys. 64 (1991) 13–76.

[60] Keldysh, L. V. Diagram technique for nonequilibrium processes. Soviet Physics JETP 20 (1965) 1018–1026.

[61] Weiss, U., Egger, R., and Sassetti, M. Low- temperature nonequilibrium transport in a Luttinger liquid. Phys. Rev. B 52 (1995) 16707–16719. [cond-mat/9509150].

[62] Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. Spectral determinants for Schrödinger equation and Q-operators of conformal field theory. J.Statist.Phys. 102 (2001) 567–576. [hep-th/9812247].

[63] Zamolodchikov, A. B. On the thermodynamic Bethe Ansatz equations for reflectionless ADE scattering theories. Phys. Lett. B 253 (1991) 391–394.

[64] Bazhanov, V. V. and Reshetikhin, N. Y. Critical RSOS models and conformal field theory. Internat. J. Modern Phys. A 4 (1989) 115–142.

[65] Zamolodchikov, A. B. Integrable field theory from conformal field theory. In Integrable Systems in Quantum Field Theory and Statistical Mechanics, Advanced Studies in Pure Mathematics (Tokyo, Japan) Vol 19, pages 641–674. Academic Press, Boston, MA, 1989.

[66] Smirnov, F. A. Exact S-matrices for ϕ_{1,2}-perturbated minimal models of conformal field theory. Internat. J. Modern Phys. A 6 (1991) 1407–1428.

[67] Fioravanti, D., Ravanini, F., and Stanishkov, M. Generalized KdV and quantum inverse scattering description of conformal minimal models. Phys. Lett. B 367 (1996)

113–120. [hep-th/9510047].

[68] Kuniba, A. and Suzuki, J. Functional relations and analytic Bethe Ansatz for twisted quantum affine algebras. J. Phys. A 28 (1995) 711–722. [hep-th/9408135].

[69] Smirnov, F. A. Quasi-classical study of form factors in finite volume. In L. D. Faddeev’s Seminar on Mathematical Physics, pages 283–307. Amer. Math. Soc., Providence, RI, 2000. [hep-th/9802132].

[70] Bazhanov, V. V. and Stroganov, Y. G. Chiral Potts model as a descendant of the six-vertex model. J. Statist. Phys. 59 (1990) 799–817.

[71] Baxter, R. J. Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I. Some fundamental eigenvectors.II. Equivalence to a generalized ice-type lattice model. III. Eigenvectors of the transfer matrix and Hamiltonian. Ann. Phys. 76 (1973) 1–24, 25–47, 48–71.

[72] Baxter, R. J., Bazhanov, V. V., and Perk, J. H. H. Functional relations for transfer matrices of the chiral Potts model. Internat. J. Modern Phys. B 4 (1990) 803–870.

[73] Bazhanov, V. V., Kashaev, R. M., Mangazeev, V. V., and Stroganov, Y. G. (Z_{N}×)^{n−1} generalization of the chiral Potts model. Comm. Math. Phys. 138 (1991) 393–408.

[74] Faddeev, L. D. and Kashaev, R. M. Generalized Bethe Ansatz equations for Hofstadter problem. Comm. Math. Phys. 169 (1995) 181–191. [hep-th/9312133].

[75] Pron’ko, G. P. On Baxter Q-operators for Toda Chain. [nlin.SI/0003002].

[76] Smirnov, F. A. Dual Baxter equations and quantization of affine Jacobian.

[math-ph/0001032].

[77] Kashaev, R. M. The non-compact quantum dilogarithm and the Baxter equations. J.Statist.Phys. 102 (2001) 923–936.

[78] Hikami, K. Baxter equation for quantum discrete Boussinesq equation.

[nlin.SI/0102021].

[79] Dorey, P. and Tateo, R. Anharmonic oscillators, the thermodynamic Bethe ansatz and nonlinear integral equations. J. Phys. A 32 (1999) L419–L425.

[hep-th/9812211].

[80] Voros, A. Spectral zeta functions. In Zeta functions in geometry (Tokyo, 1990), pages 327–358. Kinokuniya, Tokyo, 1992.

[81] Voros, A. Exact quantization condition for anharmonic oscillators (in one dimension). J. Phys. A 27 (1994) 4653–4661.

[82] Newton, R. G. The Complex j-Plane. Complex Angular Momentum in Nonrelativistic Quantum Scattering Theory. W. A. Benjamin, Inc., New York-Amsterdam, 1964. The Mathematical Physics Monograh Series.