FUNCTIONAL EQUATIONS AND QUANTUM SEPARATION OF VARIABLES FOR 3D SPIN MODELS

S. SERGEEV

Abstract. The recently proposed invariant formulation of the auxiliary linear problem for 3d integrable models provides several new ideas in the solution of the spectral problem of 3d spin models, e.g. the Zamolodchikov-Bazhanov-Baxter model in its vertex formulation. This paper is an announce of the results following from the invariant formulation. Here we formulate the class of 3d spin models which is in fact the appropriately parameterized inhomogeneous Zamolodchikov-Bazhanov-Baxter model, give an expression for the generating function of the complete set of matrices commuting with the transfer matrix of ZBB model (integrals of motion), give the functional equations defining the eigenvalues of the integrals of motion and the transfer matrices, describe explicitly the groupoid of isospectral transformations of the initial system of integrals of motion and finally give an explicit parameterization of a projector to the separated states in the sense of the quantum separation of variables (functional Bethe ansatz).

1. Introduction

Local invariant formulation of auxiliary linear problem for quantum evolution models in wholly discrete 2+1 dimensional space-time, suggested in [1], provides a powerful method for solution of the corresponding spectral problems. The algebra of observables in the 2+1 quantum evolution models is isomorphic to the local Weyl algebra, which admits the finite dimensional unitary representations when Weyl’s parameter q is the primitive root of unity,

q = e^{2πi/N} . (1)

Date: Mai 21, 2002. 1991 Mathematics Subject Classification. 82B23. Key words and phrases. 3d integrable models, Zamolodchikov-Bazhanov-Baxter model, Chiral Potts

model, Baxter equation, Quantum separation of variables.

This work was supported by the grants INTAS OPEN 00-00055, RFBR 01-01-00201, CGP CRDF RM1-2334-MO-02.

2 S. SERGEEV

In this situation the basic intertwiner, solving the local zero curvature condition, is equivalent to the R-matrix of the Zamolodchikov-Bazhanov-Baxter model in its vertex formulation, see [2, 3] for the details. Global operators, acting in the whole Hilbert space, such as an evolution operator or layer-to-layer transfer matrix, are the solutions of some dynamics of the algebra of observables and may be constructed with the help of local R-matrices.

Local auxiliary linear problem provides in this case the proof of the exact integrability of ZBB model (in particular, the tetrahedron equation for R-matrices is the trivial consequence of the zero curvature condition) and the method of derivation of a set of functional equations for the eigenvalues of ZBB transfer matrices.

N-th powers of the Weyl elements at the root of unity are the centers, which are just the complex parameters for the finite dimensional representations considered. N-th powers of Weyl elements coincide with the modularly dual Weyl pair [4, 5] and therefore have the similar dynamics as the dynamics of primary Weyl generators. ZBB model corresponds to the case of a constant solution of the classical dynamics. But for a finite 3d lattice a complete solution of the classical equations of motion may be derived. In general it gives a parameterization of N-th powers in terms of theta-functions on the Jacobian of a classical spectral curve, depending on a size, form and boundary conditions of 3d lattice. Corresponding 3d spin models belong to a new class of integrable models, and we will discuss them separately.

Simple models like ZBB correspond to the rational limit of the classical spectral curve, and classical equations of motion may still have nontrivial solutions. Corresponding spin counterpart contains modified layer-to-layer transfer matrices and their ordered products, helping one to construct a complete groupoid of isospectral deformations of initial ZBB model. Surprisingly, this groupoid contains special degenerative elements corresponding to projectors to separated states in the sense of the method of quantum separation of variables [6, 7]. Such phenomenon was recently observed for the quantum Toda chain at the root of unity in [8].

This paper is to be considered as the brief announce of all these results. We decided to skip all the proofs for the sake of shortness and so hide completely the method we used. The fragments of the method the reader may find in [1, 8].

Here we first recall the vertex formulation of ZBB model [2, 3], and then step by step we will define the objects and parameterizations necessary for our considerations and simply declare their properties without any kind of proof. Therefore we will formulate a lot of conjectures, but the reader should take into account that all these conjectures are proven, their proof will appear in the forthcoming papers.

2. Formulation of the 3d spin model

2.1. Cubic lattice. Let the cubic lattice is formed by the vectors

n = n_{1}e_{1} + n_{2}e_{2} + n_{3}e_{3} (2)

with integer n_{1}, n_{2}, n_{3} and some non-complanar e_{1}, e_{2}, e_{3}. Partition function is by the definition the following sum:

(3) Z = ∑_{{σ}} ∏_{n} ⟨σ_{1,n}, σ_{2,n}, σ_{3,n}|R_{n}|σ_{1,n+e_{1}}, σ_{2,n+e_{2}}, σ_{3,n+e_{3}}⟩ ,

where each spin variable σ_{j,n} belongs to a finite set (namely, σ_{j,n} ∈ ℤ_{N}), and the local “Boltzmann weights” are written as matrix elements ⟨σ_{1}, σ_{2}, σ_{3}|R_{n}|σ′_{1}, σ′_{2}, σ′_{3}⟩ of some N^{3} × N^{3} matrix R_{n}, whose matrix elements depend in general on n.

If the periodical b.c. are implied in the definition (3), then for the lattice of the size N_{1} × N_{2} × N_{3} one puts n_{1} ∈ ℤ_{N_{1}}, n_{2} ∈ ℤ_{N_{2}}, n_{3} ∈ ℤ_{N_{3}}.

As a particular case one may consider the homogeneous lattice, when the matrix elements of the Boltzmann weight matrix do not depend on the vertex n: R_{n} = R.

The “monodromy” and “transfer” matrices may be defined for the cubic lattice. We choose the first direction as the “time”, i.e. consider the layer of n = n_{1}e_{1} +n_{2}e_{2} +n_{3}e_{3} with n_{1} frozen, so that σ_{1,n} = σ_{1:n_{2},n_{3}} and σ_{1,n+e_{1}} = σ′_{1:n_{2},n_{3}} Then by definition, the “monodromy” layer-to-layer matrix is defined by

\begin{matrix}⟨σ_{1}, σ_{2}, σ_{3}|Q̂_{n_{1}}|σ′_{1}, σ′_{2}, σ′_{3}⟩ = \\ = ∑_{σ_{2},σ_{3}} ∏_{n_{2},n_{3}}⟨σ_{1:n_{2},n_{3}}, σ_{2:n_{2},n_{3}}, σ_{3:n_{2},n_{3}}|R_{n}|σ′_{1:n_{2},n_{3}}, σ_{2:n_{2}+1,n_{3}}, σ_{3:n_{2},n_{3}+1}⟩ , \end{matrix} (4)

where the matrix elements of Q̂_{n_{1}} are taken between the sets

σ_{1} = {σ_{1:n_{2},n_{3}}}, σ_{2} = {σ_{2:0,n_{3}}}, σ_{3} = {σ_{3:n_{2},0}}, n_{2} = 0, ..., N_{1} − 1, n_{3} = 0, ..., N_{3} − 1 (5) and

σ′_{1} = {σ′_{1:n_{2},n_{3}}}, σ′_{2} = {σ_{2:N_{2},n_{3}}}, σ′_{3} = {σ_{3:n_{2},N_{3}}}, n_{2} = 0, ..., N_{2} − 1, n_{3} = 0, ..., N_{3} − 1 , (6)

and the summation is taken with respect to all “internal” σ_{2:n_{2},n_{3}}, σ_{3:n_{2},n_{3}}, 0 < n_{2} < N_{2}, 0 < n_{3} < N_{3}. The “layer-to-layer” transfer matrix is the result of the tracing of the monodromy matrix:

⟨σ_{1}|Q_{n_{1}}|σ′_{1}⟩ = ∑_{σ_{2},σ_{3}} ⟨σ_{1}, σ_{2}, σ_{3}|Q̂_{n_{1}}|σ′_{1}, σ_{2}, σ_{3}⟩ , (7)

where the summation is taken over all σ_{2} = {σ_{2:0,n_{3}} = σ_{2:N_{2},n_{3}}} and σ_{3} = {σ_{3:n_{2},0} = σ_{3:n_{2},N_{3}}}. Due to this periodicity we will always imply below

n_{2} ∈ ℤ_{N_{2}} , n_{3} ∈ ℤ_{N_{3}} (8)

The partition function is

Z = Tr Z_{1} , (9)

where Z_{1} is the partition function for the cubic lattice, open in the “first” direction:

Z_{1} = Q_{0} · Q_{1} · . . . Q_{N_{1}−1} . (10)

Integrability of simple 3d spin model is based on the existence of the commutative set of the transfer matrices:

[Q, Q′] = 0 , (11)

where Q is formed by the weights R_{n_{2},n_{3}} and Q′ is formed by analogous but different weights R′_{n_{2},n_{3}}. The commutativity is provided by the Tetrahedron equation for R-s. The model formulated is integrable if all Q_{n_{1}} in (10) belong to the same commutative family.

Complicated 3d spin models are governed by the so called “modified tetrahedron equation”, leading at first to the modified exchange relations for the layer-to-layer transfer matrices, but finally the integrability means the existence of the commutative family of Z_{1}-matrices (10).

The main example of the 3d spin model is the Zamolodchikov-Bazhanov-Baxter model. Below the explicit form of its vertex weight is given.

2.2. Fermat curve and the function W. Let p be a point on the Fermat curve F

p =^{def} (x, y) ∈ F ⇔ x^{N} + y^{N} = 1 . (12)

W_{p}(n), p ∈ F, n ∈ ℤ_{N}, we define as follows:

\frac{W_{p}(n)}{W_{p}(n − 1)} = \frac{y}{1 − x q^{n}} , W_{p}(0) = 1 . (13)

The root of unity q is given by (1).

Function W_{p}(n), being “q - gamma function”, “q - exponent” and “q - dilogarithm” at the root of unity, has a lot of remarkable properties, see e.g. the Appendix of [3] for an introduction into “q - root of unity - hyper-geometry”. 2.3. R-matrix. Define N^{3} × N^{3} matrix R as the matrix function of four Fermat points p_{1}, p_{2}, p_{3}, p_{4} as follows [3]:

\begin{matrix}⟨σ_{1}, σ_{2}, σ_{3}|R|σ′_{1}, σ′_{2}, σ′_{3}⟩ =^{def} R_{σ_{1},σ_{2},σ_{3}}^{σ′_{1},σ′_{2},σ′_{3}} = \\ = δ_{σ_{2}+σ_{3},σ′_{2}+σ′_{3}} q^{(σ′_{1}−σ_{1}) σ′_{3}} \frac{W_{p_{1}}(σ_{2} − σ_{1}) W_{p_{2}}(σ′_{2} − σ′_{1})}{W_{p_{3}}(σ′_{2} − σ_{1}) W_{p_{4}}(σ_{2} − σ′_{1})} , \end{matrix} (14)

where x-coordinates of four Fermat curve points are identically related by x_{1} x_{2} = q x_{3} x_{4} . (15)

R-matrix (14) is considered as the universal building block for 3d spin integrable models. In the expressions for the “monodromy” or “transfer” matrices (4,7), as well as in the expression for the partition function (3), the dependence on n means simply R_{n} = R(p_{1,n}, p_{2,n}, p_{3,n}, p_{4,n}) . (16) It is known for the homogeneous case, p_{k,n} = p_{k}, two transfer matrices Q and Q′, defined via p_{1}, p_{2}, p_{3}, p_{4} ↦ eq. (14) : R ↦ eqs. (4,7) : Q (17) and p′_{1}, p′_{2}, p′_{3}, p′_{4} ↦ R′ ↦ Q′, commute if x_{3}/x_{1} = x′_{3}/x′_{1} [3].

6 S. SERGEEV

2.4. Inhomogeneous ZBB model. Define the symbol e(X, Y), X, Y ∈ ℂ as

e(X, Y)^{N} = X − Y . (18)

Conjecture 1. The inhomogeneous ZBB model corresponds to the following parameterization of R_{n} in (3) in the sense of (16):

x_{1,n} = q^{−1} \frac{e(X_{n_{1}}, Z′_{n_{3}})e(Y_{n_{2}}, Z_{n_{3}})}{e(X_{n_{1}}, Z_{n_{3}})e(Y_{n_{2}}, Z′_{n_{3}})} , y_{1,n} = \frac{e(X_{n_{1}}, Y_{n_{2}})e(Z_{n_{3}}, Z′_{n_{3}})}{e(X_{n_{1}}, Z_{n_{3}})e(Y_{n_{2}}, Z′_{n_{3}})} ,
x_{2,n} = \frac{e(X′_{n_{1}}, Z′_{n_{3}})e(Y′_{n_{2}}, Z_{n_{3}})}{e(X′_{n_{1}}, Z_{n_{3}})e(Y′_{n_{2}}, Z′_{n_{3}})} ,  y_{2,n} = \frac{e(X′_{n_{1}}, Y′_{n_{2}})e(Z_{n_{3}}, Z′_{n_{3}})}{e(X′_{n_{1}}, Z_{n_{3}})e(Y′_{n_{2}}, Z′_{n_{3}})} ,
x_{3,n} = q^{−1} \frac{e(X_{n_{1}}, Z′_{n_{3}})e(Y′_{n_{2}}, Z_{n_{3}})}{e(X_{n_{1}}, Z_{n_{3}})e(Y′_{n_{2}}, Z′_{n_{3}})} , y_{3,n} = q \frac{e(X_{n_{1}}, Y′_{n_{2}})e(Z_{n_{3}}, Z′_{n_{3}})}{e(X_{n_{1}}, Z_{n_{3}})e(Y′_{n_{2}}, Z′_{n_{3}})} ,
x_{4,n} = q^{−1} \frac{e(X′_{n_{1}}, Z′_{n_{3}})e(Y_{n_{2}}, Z_{n_{3}})}{e(X′_{n_{1}}, Z_{n_{3}})e(Y_{n_{2}}, Z′_{n_{3}})} , y_{4,n} = \frac{e(X′_{n_{1}}, Y_{n_{2}})e(Z_{n_{3}}, Z′_{n_{3}})}{e(X′_{n_{1}}, Z_{n_{3}})e(Y_{n_{2}}, Z′_{n_{3}})} .

(19)

Here the pair of complex parameters X_{n_{1}}, X′_{n_{1}} corresponds to n_{1}-th layer in the “first” direction, Y_{n_{2}}, Y′_{n_{2}} corresponds to n_{2}-th layer in the “second” direction and so on. So this pairs may be considered as the free spectral parameters of the model, they are related to but do not coincide with Zamolodchikov’s dihedral angles.

2.5. Eigenvalue and eigenvector problem.

Conjecture 2. Any two transfer matrices Q_{n_{1}} and Q_{m_{1}}, defined by (4,7) with the parameterization (19), commute if they have the same set of Y_{n_{2}}, Y′_{n_{2}}, Z_{n_{3}}, Z′_{n_{3}} (i.e. the same e(Y_{n_{2}}^{#}, Z_{n_{3}}^{#})) and arbitrary X_{n_{1}}, X′_{n_{1}} and X_{m_{1}}, X′_{m_{1}}.

It means that they have the same set of eigenvectors |Ψ_{t}⟩,

Q_{n_{1}}|Ψ_{t}⟩ = |Ψ_{t}⟩Q_{n_{1},t} , (20)

where the symbol t stands as a formal labeling of eigenvectors (i.e. a multi-index), and Q_{n_{1},t} is the eigenvalue of Q_{n_{1}} for this eigenvector. |Ψ_{t}⟩ depends on the set of Y_{n_{2}}, Y′_{n_{2}}, Z_{n_{3}}, Z′_{n_{3}} and do not depend on X_{n_{1}}, X′_{n_{1}}. Together with the right eigenvectors one may use the left ones, defining the spectral decomposition of Q_{n_{1}}:

Q_{n_{1}} = ∑_{t} |Ψ_{t}⟩ Q_{n_{1},t} ⟨Ψ_{t}| . (21)

The general problem is to find out the eigenvalues and eigenvectors of Q_{n_{1}}. To solve this problem explicitly is quite hopeless.

In this paper we will give the solution for more simple problems. At the first, we will define the structure of t, i.e. we will derive the complete set of independent matrices, commuting with Q_{n_{1}} (i.e. the “integrals of motion”).

Next we will give a set of equations for the spectrum of the integrals of motion as well the equations connecting the spectrum of Q_{n_{1}} and the spectrum of the integrals of motion (such type equations are called usually the functional equations for the spectrum).

Finally we will construct (explicitly) a projector to a “separated state” |Φ_{α}⟩ such that

⟨Ψ_{t}|Φ_{α}⟩ = ∏_{n_{1}} Q_{n_{1},t}(α) (22)

for some special values of X_{n_{1}} and some special value of N_{1}. α here is a multi-index, defined by some particular choices of N-th roots of e-symbol in the definition of Q_{n_{1},t}. Equation (22) is called usually the quantum separation of variables.

3. Exact integrability

3.1. A sample Q-matrix. We will investigate one separate Q-matrix from the set of Q_{n_{1}}- matrixes. For this we will omit for a moment the index n_{1}. Q̂ as well as Q are given by (4,7) with R_{n} = R_{n_{2},n_{3}} parameterized by (19) with X_{n_{1}} = X, X′_{n_{1}} = X′.

Besides the usual transfer matrix Q we need a set of degenerative transfer matrices Q(α, β) defined by

⟨σ_{1}|Q(α, β)|σ′_{1}⟩ = ∑_{σ_{2} : ∑_{n_{3}} σ_{2:n_{3}} =−β ; σ_{3} : ∑_{n_{2}} σ_{3:n_{2}} =α}⟨σ_{1}, σ_{2}, σ_{3}|Q̂|σ′_{1}, σ_{2}, σ_{3}⟩ , (23)

so that

Q = ∑_{α,β} Q(α, β) . (24)

see (4,7).

Conjecture 3. Matrices Q(α, β) also form the commutative family with respect to X, X′ and α, β.

3.2. Local Weyl algebra. The structure of Q, defined by its matrix elements between the vectors

⟨σ_{1}| = ⨂_{n_{2},n_{3}}⟨σ_{1:n_{2},n_{3}}| and |σ′_{1}⟩ = ⨂_{n_{2},n_{3}} |σ′_{1:n_{2},n_{3}}⟩ (25)

belonging to the vector spaces V^{∗} and V correspondingly, such that each σ-variable is ℤ_{N}- invariant, implies that we may define the algebra if observables as the direct product of simple Weyl algebra at the root of unity.

Namely, let the simple Weyl algebra is formed by the N × N matrices x and z, defined in the basis |σ⟩ = |σ mod N⟩, ⟨σ|σ′⟩ = δ_{σ,σ′},

x|σ⟩ = |σ⟩q^{σ} , z|σ⟩ = |σ + 1⟩ , x z = q zx . (26)

The corresponding components in the tensor product of Δ = N_{2}N_{3} copies of the simple Weyl algebra we will denote as

x_{1:n_{2},n_{3}} = 1 ⊗ 1 ⊗ ... ⊗ \underbrace{x}_{{(n_{2},n_{3})−th \atop place}} ⊗ ... , z_{1:n_{2},n_{3}} = 1 ⊗ 1 ⊗ ... ⊗ \underbrace{z}_{{(n_{2},n_{3})−th \atop place}} ⊗ ... , (27)

so that the structure of the indices in (25) matches exactly the structure of observables in (27).

3.3. Matrix ℓ and the determinant. Define now the operator-valued Δ × Δ matrix ℓ (the auxiliary linear problem coefficients matrix) as follows:

\begin{matrix}ℓ_{n_{2}+1,n_{3}|n_{2},n_{3}} = a_{n_{2}} , ℓ_{n_{2},n_{3}+1|n_{2},n_{3}} = b_{n_{3}} v_{1:n_{2},n_{3}} , \\ ℓ_{n_{2},n_{3}|n_{2},n_{3}} = − w_{1:n_{2},n_{3}} , ℓ_{n_{2}+1,n_{3}+1|n_{2},n_{3}} = − a_{n_{2}}b_{n_{3}} u_{1:n_{2},n_{3}} , \end{matrix} (28)

where

\begin{matrix}v_{1:n_{2},n_{3}} = \frac{e(Y_{n_{2}}, Z′_{n_{3}})}{e(Y′_{n_{2}}, Z_{n_{3}})} x_{1:n_{2},n_{3}}z_{1:n_{2},n_{3}} , \\ w_{1:n_{2},n_{3}} = \frac{e(Y_{n_{2}}, Z_{n_{3}})}{e(Y′_{n_{2}}, Z_{n_{3}})} z_{1:n_{2},n_{3}} , u_{1:n_{2},n_{3}} = \frac{e(Y′_{n_{2}}, Z′_{n_{3}})}{e(Y′_{n_{2}}, Z_{n_{3}})} x_{1:n_{2},n_{3}} . \end{matrix} (29)

Here as usual n_{2} ∈ ℤ_{N_{2}}, n_{3} ∈ ℤ_{N_{3}}. a_{n_{2}} and b_{n_{3}} are extra parameters. Since the elements of different columns of the matrix ℓ commute, the formal determinant of ℓ is well defined.

Parameters a_{n_{2}} and b_{n_{3}} enters this determinant only via

A = ∏_{n_{2}} a_{n_{2}} , B = ∏_{n_{3}} b_{n_{3}} . (30)

The determinant is a polynomial with respect to A, B. Further we will need two normalizations of it:

J(A, B) = det ||ℓ|| ∏_{n_{2},n_{3}} (−a_{n_{2}}b_{n_{3}}u_{1:n_{2},n_{3}})^{−1} (31)

and

J̃(A, B) = det ||ℓ|| ∏_{n_{2},n_{3}} (−w_{1:n_{2},n_{3}})^{−1} (32)

Matrices J and J̃ in the special points play the important role:

a_{n_{2}} = \frac{e(X, Y_{n_{2}})}{e(X, Y′_{n_{2}})} , b_{n_{3}} = \frac{e(X, Z_{n_{3}})}{e(X, Z′_{n_{3}})} ⇒ J(A, B) =^{def} J(X) , (33)

and

a_{n_{2}} = \frac{e(X′, Y_{n_{2}})}{e(X′, Y′_{n_{2}})} , b_{n_{3}} = \frac{e(X′, Z_{n_{3}})}{e(X′, Z′_{n_{3}})} ⇒ J̃(A, B) =^{def} J̃(X′) , (34)

3.4. Integrals of motion. J defined by (31) is the polynomial with respect to A^{−1} and B^{−1}:

J(A, B) = ∑_{m_{2}=0}^{N_{2}} ∑_{m_{3}=0}^{N_{3}} A^{−m_{3}}B^{−m_{2}} J_{m_{2},m_{3}} , J_{0,0} = 1 . (35)

Conjecture 4. If Q is the transfer matrix prepared with the help of the same e(Y_{n_{2}}^{#}, Z_{n_{3}}^{#}) as J, then J · Q = Q · J (36) for any X, X′ entering Q and any A, B entering J. Conjecture 5. Coefficients J_{m_{2},m_{3}} form the following algebra: J_{m_{2},m_{3}}J_{m′_{2},m′_{3}} = q^{m′_{2}m_{3}−m′_{3}m_{2}} J_{m′_{2},m′_{3}}J_{m_{2},m_{3}} . (37)

10 S. SERGEEV

Note, J̃ is equivalent to J due to J̃ ∼ JJ_{N_{2},N_{3}}^{−1}.

N_{2} + N_{3} remarkable simple matrices belong to the system of J_{m_{2},m_{3}}:

U_{n_{2}} = ∏_{n_{3}} x_{1:n_{2},n_{3}} , W_{n_{3}} = ∏_{n_{2}} z_{1:n_{2},n_{3}} . (38)

Since U_{n_{2}}W_{n_{3}} = qW_{n_{3}}U_{n_{2}}, they contain one “mass center”. Let J_{m_{2},m_{3}} = W_{0}^{m_{3}}U_{0}^{−m_{2}} t_{m_{2},m_{3}} . (39) Conjecture 6. The set t_{m_{2},m_{3}} is commutative, and each t_{m_{2},m_{3}} commute with U_{0} and W_{0}. Conjecture 7. Matrices t_{m_{2},m_{3}} for the perimeter of Newton’s polygon 0 ≤ m_{2} ≤ N_{2}, 0 ≤ m_{3} ≤ N_{3} are polynomials with respect to U_{0}^{−1}U_{n_{2}} and W_{0}^{−1}W_{n_{3}}. Conjecture 8. g = (N_{2} −1)(N_{3} −1) nontrivial matrices t_{m_{2},m_{3}} inside the Newton’s polygon are algebraically independent. Conjecture 9. g matrices from inside the Newton’s polygon, N_{2} − 1 matrices U_{0}^{−1}U_{n_{2}}, N_{3} − 1 matrices W_{0}^{−1}W_{n_{3}} and any non-degenerative combination of U_{0}, W_{0} is the complete set of commutative matrices. Additionally one has Conjecture 10. t_{m_{2},m_{3}} Q(α, β) = Q(α, β) t_{m_{2},m_{3}} , (40) and moreover, t_{m_{2},m_{3}} and Q(α, β) may be diagonalized simultaneously. The “mass center” simply governs the phases of A, B:

(41) U_{0} J(A, B) = J(q^{−1}A, B) U_{0} , W_{0} J(A, B) = J(A, q^{−1}B) W_{0} , and analogously

(42) U_{0} Q(α, β) = Q(α − 1, β) U_{0} , W_{0} Q(α, β) = Q(α, β − 1) W_{0} . Conjecture 11. Q(α, β) and t_{m_{2},m_{3}} satisfy the following system: J(X) Q(0, 0) = Q(0, 0) J̃(X′) = 0 , (43) where J(X) and J̃(X′) are defined by (33,34).

Conjecture 12. In the basis of common eigenvectors of Q(α, β) and t_{m_{2},m_{3}} (43) is 2N equations in the matrix block U_{0}, W_{0}, and this system of functional equations allows one (in principle) to find the spectra of t and Q.

Each of equations (43) is exactly Baxter’s t − Q equation for N_{2} = 2 or N_{3} = 2 and generic integer N. For N_{2} > 2 and N_{3} > 2 (43) are equivalent to all the chain of nested Bethe ansatz system.

4. Isospectrality and separation

4.1. Modified Q-operators.

Conjecture 13. The system of two algebraic equations for two variables P, P′ ∈ C

∏_{n_{2}=0}^{N_{2}−1} \frac{P′ − Y′_{n_{2}}}{P − Y′_{n_{2}}} = ∏_{n_{2}=0}^{N_{2}−1} \frac{P′ − Y_{n_{2}}}{P − Y_{n_{2}}} , ∏_{n_{3}=0}^{N_{3}−1} \frac{P′ − Z′_{n_{3}}}{P − Z′_{n_{3}}} = ∏_{n_{3}=0}^{N_{3}−1} \frac{P′ − Z_{n_{3}}}{P − Z_{n_{3}}} (44)

has g = (N_{2} − 1)(N_{3} − 1) inequivalent solutions.

Let us enumerate these solutions as P_{k}, P′_{k}, k = 0, ..., g − 1.

For some N_{1} define the matrix Z_{1} by (10) with the following local parameterization of R_{n}:

x_{1,n} = q^{−1} \frac{e(X_{n_{1}},Z′_{n_{3}})e(Y_{n_{2}},Z_{n_{3}})}{e(X_{n_{1}},Z_{n_{3}})e(Y_{n_{2}},Z′_{n_{3}})} \frac{τ_{1,n}τ_{2,n+e_{3}}}{τ_{1,n+e_{3}}τ_{2,n}} , y_{1,n} = \frac{e(X_{n_{1}},Y_{n_{2}})e(Z_{n_{3}},Z′_{n_{3}})}{e(X_{n_{1}},Z_{n_{3}})e(Y_{n_{2}},Z′_{n_{3}})} \frac{λ_{n}τ_{3,n}}{τ_{1,n+e_{3}}τ_{2,n}} ,
x_{2,n} = \frac{e(X′_{n_{1}},Z′_{n_{3}})e(Y′_{n_{2}},Z_{n_{3}})}{e(X′_{n_{1}},Z_{n_{3}})e(Y′_{n_{2}},Z′_{n_{3}})} \frac{τ_{1,n+e_{2}}τ_{2,n+e_{1}+e_{3}}}{τ_{1,n+e_{2}+e_{3}}τ_{2,n+e_{1}}} , y_{2,n} = \frac{e(X′_{n_{1}},Y′_{n_{2}})e(Z_{n_{3}},Z′_{n_{3}})}{e(X′_{n_{1}},Z_{n_{3}})e(Y′_{n_{2}},Z′_{n_{3}})} \frac{λ_{n}τ_{3,n+e_{1}+e_{2}}}{τ_{1,n+e_{2}+e_{3}}τ_{2,n+e_{1}}} ,
x_{3,n} = q^{−1} \frac{e(X_{n_{1}},Z′_{n_{3}})e(Y′_{n_{2}},Z_{n_{3}})}{e(X_{n_{1}},Z_{n_{3}})e(Y′_{n_{2}},Z′_{n_{3}})} \frac{τ_{1,n+e_{2}}τ_{2,n+e_{3}}}{τ_{1,n+e_{2}+e_{3}}τ_{2,n}} , y_{3,n} = q \frac{e(X_{n_{1}},Y′_{n_{2}})e(Z_{n_{3}},Z′_{n_{3}})}{e(X_{n_{1}},Z_{n_{3}})e(Y′_{n_{2}},Z′_{n_{3}})} \frac{λ_{n}τ_{3,n+e_{2}}}{τ_{1,n+e_{2}+e_{3}}τ_{2,n}} ,
x_{4,n} = q^{−1} \frac{e(X′_{n_{1}},Z′_{n_{3}})e(Y_{n_{2}},Z_{n_{3}})}{e(X′_{n_{1}},Z_{n_{3}})e(Y_{n_{2}},Z′_{n_{3}})} \frac{τ_{1,n}τ_{2,n+e_{1}+e_{3}}}{τ_{1,n+e_{3}}τ_{2,n+e_{1}}} , y_{4,n} = \frac{e(X′_{n_{1}},Y_{n_{2}})e(Z_{n_{3}},Z′_{n_{3}})}{e(X′_{n_{1}},Z_{n_{3}})e(Y_{n_{2}},Z′_{n_{3}})} \frac{λ_{n}τ_{3,n+e_{1}}}{τ_{1,n+e_{3}}τ_{2,n+e_{1}}} ,

(45)

where

τ_{1:n}^{N} = H^{(g)} (\{\frac{I_{k}(n)}{σ_{k}(X_{n_{1}})}\}_{k=0}^{g−1}) , τ_{2:n}^{N} = H^{(g)} (\{\frac{I_{k}(n)}{σ_{k}(Y_{n_{2}})}\}_{k=0}^{g−1}) ,
τ_{3:n}^{N} = H^{(g)} (\{\frac{I_{k}(n)}{σ_{k}(Z_{n_{3}})}\}_{k=0}^{g−1}) , λ_{n} = H^{(g)} (\{\frac{I_{k}(n)σ_{k}(Z′_{n_{3}})}{σ_{k}(X_{n_{1}})σ_{k}(Y_{n_{2}})}\}_{k=0}^{g−1}) ,

(46)

12 S. SERGEEV

with

H^{(g)}(\{f_{k}\}_{k=0}^{g−1}) = \frac{det |P_{j}^{i} − f_{j} P_{j}^{′i}|_{i,j=0}^{g−1}}{∏_{i>j} (P_{i} − P_{j})} , and σ_{k}(X) = \frac{P′_{k} − X}{P_{k} − X} , (47)

and I_{k}(n) defined as follows:

I_{k}(n) = f_{k} ( ∏_{j_{1}=0}^{n_{1}−1} \frac{σ_{k}(X′_{j_{1}})}{σ_{k}(X_{j_{1}})}) ( ∏_{j_{2}=0}^{n_{2}−1} \frac{σ_{k}(Y′_{j_{2}})}{σ_{k}(Y_{j_{2}})}) ( ∏_{j_{3}=0}^{n_{3}−1} \frac{σ_{k}(Z′_{j_{3}})}{σ_{k}(Z_{j_{3}})}) . (48)

g coefficients f_{k} are generic complex numbers. Obviously this newly defined Z_{1} coincides with the previous definition if all f_{k} = 0. But now one has no commutativity of Q_{n_{1}}. Instead there exists an exchange relation which is equivalent to the following proposition:

Conjecture 14. The newly defined Z_{1} is the symmetrical matrix function with respect to any permutation of the pairs (X_{n_{1}}, X′_{n_{1}}), n_{1} = 0, ..., N_{1} − 1.

In the newly defined Z_{1} any such permutation changes τ_{j,n} and so it changes significantly the set of R_{n} and so the set of Q_{n_{1}}.

Corresponding determinants should be defined by the same formulae (28,31) with

v_{1:n_{2},n_{3}}(n_{1}) = \frac{e(Y_{n_{2}}, Z′_{n_{3}})}{e(Y′_{n_{2}}, Z_{n_{3}})} \frac{τ_{2,n}τ_{3,n+e_{2}}}{τ_{2,n+e_{3}}τ_{3,n}} x_{1:n_{2},n_{3}}z_{1:n_{2},n_{3}} ,

w_{1:n_{2},n_{3}}(n_{1}) = \frac{e(Y_{n_{2}}, Z_{n_{3}})}{e(Y′_{n_{2}}, Z_{n_{3}})} \frac{τ_{3,n+e_{2}}}{τ_{3,n}} z_{1:n_{2},n_{3}} , u_{1:n_{2},n_{3}}(n_{1}) = \frac{e(Y′_{n_{2}}, Z′_{n_{3}})}{e(Y′_{n_{2}}, Z_{n_{3}})} \frac{τ_{2,n}}{τ_{2,n+e_{3}}} x_{1:n_{2},n_{3}} .

(49)

Now the determinant depends on n_{1} via the dependence of τ_{j,n}, and we may denote it as J_{n_{1}}(A, B).

Conjecture 15. For given ordering of (X_{n_{1}}, X′_{n_{1}}) we have

J_{n_{1}}(A, B) Q_{n_{1}} = Q_{n_{1}} J_{n_{1}+1}(A, B) . (50)

4.2. Isospectrality. Let now

N_{1} = g = (N_{2} − 1)(N_{3} − 1) . (51)

Change the normalization of the coefficients f_{k},

f_{k} = f′_{k} σ_{k}(X_{k}) (52) and consider the limit when

(53) X_{k} ↦ P′_{k} , f′_{k} being generic complex numdes. Then in the set of arguments of τ_{2,n} and τ_{3,n} the number of nonzero amplitudes I_{k}(n) is exactly n_{1}. In particular J_{0}(A, B) ≡ J(A, B) , (54) where J is our initial τ-independent determinant (31). Applying (50) g times, one obtains J(A, B) Z_{1} = Z_{1} J_{g}(A, B) , (55) giving the following Conjecture 16. ℂ^{g}-parametric family of operators J_{g}(A, B) is isospectral to J(A, B), and newly constructed Z_{1} is the similarity operator, non-degenerative for generic f⃗′ ∈ ℂ^{g}. Let |Ψ_{t}⟩ be the set of eigenvectors for initial t_{m_{2},m_{3}}, consisting J, and |Ψ_{t}^{(g)}⟩ be the set of eigenvectors for the final t_{m_{2},m_{3}}^{(g)}, consisting J_{g}. Then one has Conjecture 17.

Z_{1} = ∑_{t} |Ψ_{t}⟩Q_{0,t}Q_{1,t}...Q_{g−1,t}⟨Ψ_{t}^{(g)}| . (56)

4.3. Separation. After (52) and (53) consider further the limit when f′_{k} ↦ 1 . (57) In this limit we may construct all the modified Q_{n_{1}}-operators, but now we are interesting in the modified operators

Q_{n_{1}}(α) = ∑_{β} Q_{n_{1}}(α, β) . (58)

Due to the summation over β, W_{0}Q_{n_{1}}(α) = Q_{n_{1}}(α)W_{0}. Let in this limit

(59) Z_{1}(α⃗) = Q_{0}(α_{0})Q_{1}(α_{1}) … Q_{g−1}(α_{g−1}) with X_{k} ↦ P′_{k}, f′_{k} ↦ 1 . Quantum separation of variables is based on

14 S. SERGEEV

Conjecture 18. In the limit (57) the set of final eigenvectors degenerates, and matrix Z_{1}(α) has the following decomposition:

Z_{1}(α⃗) = ∑_{t} |Ψ_{t}⟩Q_{0,t}(α_{0}) … Q_{g−1,t}(α_{g−1})⟨χ_{t′}| , (60)

where t′ is the subset of t corresponding to N_{2} + N_{3} − 1 commutative operators U_{0}^{−1}U_{n_{2}} and W_{n_{3}}.

Note, ⟨χ| depends on X′_{n_{1}} and therefore has a significant freedom in the definition. In (60) one may make the summation with respect t \ t′:

|Φ_{α⃗,t′}⟩ = ∑_{t\t′} |Ψ_{t}⟩ Q_{0,t}(α_{0}) … Q_{g−1,t}(α_{g−1}) . (61)

The basis of vectors |Φ_{α⃗,t′}⟩ is called the separated basis. Now we will try to explain its origin.

Turn back to the matrix ℓ_{s|v} (28,29). Let A_{v|s} be its matrix of algebraic supplements, defined by

∑_{v∈ℤ_{N_{2}}×ℤ_{N_{3}}} ℓ_{s|v} A_{v|s′} = δ_{s|s′} J(A, B) , (62)

where J is given by (31).

“Usual” way to derive the quantum separation of variables [6, 7] is to construct a rather complicated procedure that may be described symbolically as

(63) “quantum resultant”_{with respect to B}(J(A, B), A_{s|v}(A, B)) = b̃_{v}(A) b_{s}(A) ,

where b_{s}(A) is a g-th power polynomial with respect to A. (61) are the eigenvectors of b_{s}(A) for s = (0, 0). Fortunately one may get rid of the consideration of the “quantum resultant”, choosing the following direct way.

Let V_{s}(A, B) be the subspace of the complete vector space V defined by

(64) |Φ⟩ ∈ V_{s}(A, B) ⇔ J(A, B) |Φ⟩ = 0 and A_{v|s} |Φ⟩ = 0 ∀v .

Conjecture 19. dim V_{s}(A, B) = N^{Δ−2} (Δ = N_{2}N_{3}) if

A = A_{k} =^{def} q^{α_{k}} ∏_{n_{2}} \frac{e(P′_{k}, Y_{n_{2}})}{e(P′_{k}, Y′_{n_{2}})} , B = B_{k} =^{del} q^{β_{k}} ∏_{n_{3}} \frac{e(P′_{k}, Z_{n_{3}})}{e(P′_{k}, Z′_{n_{3}})} , (65)

for any k = 0, ..., g − 1.

Let further

V_{s}(k, α_{k}) = ⋃_{n∈ℤ_{N}} V_{s}(A_{k}, q^{n}B_{k}) , V_{s}(α⃗) = ⋂_{k=0}^{g−1} V_{s}(k, α_{k}) . (66)

Obviously, due to the construction, dim V_{s}(α⃗) = N^{Δ−g}, and b_{s}(A_{k}) · V_{s}(α⃗) = 0.

Our approach to the quantum separation of variables is based technically on the following main

Conjecture 20. The first modified operator Q_{0}(α_{0}, β_{0}) in the sequence (59) obeys in the limit (53)

J(A_{0}, B_{0}) · Q_{0}(α_{0}, β_{0}) = A_{v|s}(A_{0}, B_{0}) · Q_{0}(α_{0}, β_{0}) = 0 ∀v (67)

for s = (n_{2}, n_{3}) if τ_{1,0e_{1}+n_{2}e_{2}+n_{3}e_{3}} = 0.

Therefore Q_{0}(α_{0}) is the projector to V_{s}(0, α_{0}) and due to the conjecture 14 on the page 12, eq. (59) gives the projector to V_{s}(α⃗). The limit (57) corresponds to the choice s = (0, 0).

Acknowledgments I should like to thank cordially professors J. Satsuma and T. Tokihiro for their kind hospitality during my stay in the Graduated School of Mathematical Sciences in Tokyo University, where the main part of this work was done.

Also I would like to thank P. Kulish, M. Semeniv-Tyan-Shanski and my co-authors G. von Gehlen and S. Pakuliak for many stimulating discussions.

I am grateful to A. Belavin, D. Lebedev, V. Matveev, S. Kharchev, A. Gerasimov, A. Volkov, R. Kashaev, G. Pronko, Yu. Stroganov and H. Boos for their interest to this subject and valuable advises.

Special thanks are devoted to professors G. Harder, G. Faltings, Yu. Manin and D. Zagier for the possibility to live in the atmosphere of the scientific activity in the Max Plank Institute for Mathematics.

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Sergei Sergeev, Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Moscow reg., Russia

E-mail address: sergeev@thsun1.jinr.ru

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

E-mail address: sergeev@ms.u-tokyo.ac.jp

Max Plank Institute fur Mathematic, Vivatsgasse 7, D-53111, Bonn, Germany

E-mail address: sergeev@mpim-bonn.mpg.de