Quantum curve in q-oscillator model

A lattice model of interacting q-oscillators, proposed in [V. Bazhanov, S. Sergeev, arXiv:hep-th/0509181], is the quantum mechanical integrable model in 2+1 dimensional space-time. Its layer-to-layer transfer-matrix is a polynomial of two spectral parameters, it may be regarded in the terms of quantum groups both as a sum of sl(N) transfer matrices of a chain of length M and as a sum of sl(M) transfer matrices of a chain of length N for reducible representations. The aim of this paper is to derive the Bethe Ansatz equations for the q-oscillator model entirely in the framework of 2+1 integrability providing the evident rank-size duality.


Introduction
The q-oscillator lattice model was formulated recently in [1,2]. It describes a system of interacting q-oscillators situated in the vertices of two dimensional lattice, and therefore it is the quantum mechanical system in 2 + 1 dimensional space-time in the same way, as a chain of interacting particles (or spins) is regarded as a model in 1 + 1 dimensional space-time. Formulation of the q-oscillator model provides a definition of a layer-to-layer transfer matrix as a polynomial of two spectral parameters. This transfer matrix may be interpreted in the terms of quantum inverse scattering method and quantum groups, so that both sizes of the two dimensional lattice may be interpreted as either a length of an effective chain or as symmetry group's rank. This was called in [1] the "rank-size" duality.
The appearance of a complete set of fundamental transfer matrices for U q ( sl) series is a signal that the layer-to-layer transfer matrix of q-oscillator model is closely related to Bethe Ansatz in the form of generalized Baxter's "T-Q" equations. The subject of this paper is the derivation of such equations in the framework of 2+1 dimensional integrability.
Below in this Introduction we formulate the answer, i.e. we give an explicit form of "T-Q" equations in the terms of a given layer-to-layer transfer matrix. To do this, we need to repeat the structure of the layer-to-layer transfer matrix for q-oscillator model in more details.
The q-oscillator model describes a system of interacting q-oscillators H v , The two dimensional lattice may be identified with a layer (or a section) of three dimensional cubic lattice, further we call it either the layer or the auxiliary lattice.
Auxiliary matrices L α,β [H v ], acting in C 2 ⊗ C 2 ⊗ F v , were introduced in [1]. The layer transfer matrix T(u, v) may be constructed as a trace of an 2d ordered product of auxiliary matrices L[H v ]. The transfer matrix is a polynomial of two spectral parameters, (2) T(u, v) = And let C-numerical parameters of q-oscillator lattice are inhomogeneous enough.
Then "T-Q" equation for sl M is The statement is that if t n,m take their eigenvalues, then there exist M special values v 1 , . . . , v M of v, such that corresponding Q 1 (u), . . . , Q M (u) in (7) are polynomials 1 . Degrees of the polynomials are uniquely defined by certain occupation numbers of oscillators.
In its turn, equivalent "T-Q" equation for sl N is If t n,m take their eigenvalues, then there exist N special values u 1 , . . . , u N of u, such that corresponding Q 1 (v), . . . , Q N (v) in (8) are polynomials. All the other forms of nested Bethe Ansatz equations follow from (7) or (8).
Polynomials Q(u) and Q(v) may be denoted in the quantum mechanical way as "wave functions" of states Q| and |Q : where |u and v| serve the simple Weyl algebra W: Then (7) and (8) are correspondingly The formulation of the q-oscillator model and definition of T(u, v) are locally 3d invariant, the quantum group interpretation (3) is the secondary one. In this paper we will derive (11) without any quantum group technique.
To explain our method, we need to comment a little the classical limit.
In the classical limit q → 1, the local q-oscillator generators become the classical dynamical variables, the q-oscillator model becomes a model of classical mechanics, quantum evolution operators become Baecklund transformations for the dynamical variables. In particular, T (u, v) may be understood as a partition function of a completely inhomogeneous free fermion six-vertex model on the square lattice (but it should not be regarded as a model of statistical mechanics). In its turn, J(u, v) becomes a free fermion determinant (the sign (−) nm+n+m counts the number of fermionic loops). There exists a well known formula in the theory of two dimensional free fermion models, relating T and J: In the classical limit, equation J(u, v) = 0 defines the spectral curve. Dynamical variables may be expressed in the terms of θ-functions on the Jacobian of the spectral curve. The sequence of Baecklund transformations, which is the "discrete time" in the classical model, is a sequence of linear shifts of a point on the Jacobian. The classical model was formulated and solved by I. G. Korepanov [3].
Classical integrability is based on an auxiliary linear problem. Equation J(u, v) = 0 is the condition of the existence of a solution of the linear problem. Our point is that in quantum q = 1 case, the linear problem is still the basic concept of the solvability. Quantum J(u, v) is a well defined determinant of an operator-valued matrix, and J(u, v)|Ψ = 0 is again the condition of the existence of a solution of a quantum linear problem. The polynomial structure of e.g. v|Ψ follows from a more detailed consideration of the quantum linear problem in a special basis of diagonal "quantum Baker-Akhiezer function" (related to a quantum separation of variables).
The structure of the paper is the following. In the first section we recall briefly some basic notions of the classical model [3]: the linear problem, spectral curve and details of the combinatorial representation of the spectral curve. In the second section we repeat the definition of the quantum model and its integrability [1,2]. In particular, our definition of the spectral parameters differs from that of [1]. Quantum linear problem, derivation of (11) and properties of various forms of (12) are given in the third section. The fourth section includes an example.

The Korepanov model
We start with a short review of the integrable model of classical mechanics in discrete 2 + 1 dimensional space-time [3]. The main purpose of this section is to recall the relation between Korepanov's linear problem, spectral determinant and partition function for free fermion model. Another aim is to fix several useful definition and notations.
1.1. Linear problem. Consider a two dimensional lattice formed by the intersection of straight lines enumerated by the Greek letters. Let the vertices of the lattice are enumerated in some way.
Consider a particular vertex with a number v formed by the intersection of lines α and β, as it is shown in Fig. 1. It was mentioned in the Introduction, an auxiliary lattice is a section of three dimensional lattice, the vertices on the auxiliary lattice are equivalent to the edges of the three-dimensional one. In Fig. 1, the dashed lines are the lines of auxiliary lattice, while the solid sprout from the vertex v is the edge of the three dimensional lattice.
Let four free C-valued variables (14) A are associated with vertex v. In addition, let C-valued variables ψ α and ψ β are associated with the ingoing edges, and C-valued variables ψ ′ α and ψ ′ β are associated with the outgoing edges, as it is shown in Fig. 1 (a certain orientation of auxiliary lines is implied). The local linear problem is a pair of linear relations binding the edge variables. Its standard form, the right hand side of Fig. 1 in matrix notations, is A v belong to the edges of the 3d lattice. Therefore A v are distinguished from A ′ v , but linear variables on outer edges ψ α , . . . , ψ ′ γ , in both top and bottom auxiliary planes are identified. Consider the bottom plane first. The linear problem rule (15) may be applied three times for excluding internal edges, as the result one obtains an expression of the "primed" linear variables in the terms of "unprimed": where (cf. the ordering of α, β, γ in column vectors) The top plane of Fig. 2 may be considered in the same way, where the matrices X #,# are given by (17) with . The associativity condition of linear problems (16) and (18) is the Korepanov equation relating the set of 12 variables A v with the set of 12 variables A ′ v , v = 1, 2, 3. Equation In what follows, such "direct sum" imbedding of 2 × 2 matrices X into higher dimensional unity matrices will always be implied.   quasi-periodical boundary conditions for them: where u and v are C-valued spectral parameters.
Linear equations (21) may be iterated for the whole lattice as follows. Let Then the repeated use of (21) gives where, in the terms of matrix imbedding discussed right after (19), the (N +M)× (N +M) monodromy matrix X α,β may be written as The boundary conditions (22) give ψ The whole linear problem has a solution if and only if where the "action" is , and the measure is Spectral parameters appear in (29) via In the terms of Grassmanian variables, the exponent of a quadratic form is a normal symbol of some operator L, The fermionic coherent states are defined by and the extra summands in (30) correspond to the unity operators It is important to note that the indices of the operator L α,β [A v ] (33) label copies of twodimensional vector spaces (34) C 2 ∋ x|0 + y|1 . Thus, L α,β acts in the tensor product of two-dimensional vector spaces, while X α,β acts in the tensor sum of one-dimensional vector spaces. In the basis of the fermionic states Besides, the Korepanov equation (19) is the equality of the exponents of the normal symbol form of the local Yang-Baxter equation Turn now to the expression of the determinant (28) in the terms of operators L. Let 2 N +M × 2 N +M matrix L α,β be the ordered product of local L-s: This is related to the monodromy matrix (25) by means of (cf. (33)) Define now the boundary matrices for L α,β , and let By the construction, T (u, v) is the partition function for a free-fermion lattice model with the inhomogeneous Boltzmann weights -matrix elements of L αn,βm [A n,m ] -and u, v-boundary conditions. It is the polynomial of u and v: Sometimes a pure combinatorial representation of the partition function is very useful.
Any monomial in T (u, v) (43) corresponds to a non-self-intersecting path on the toroidal lattice. A path may go through a vertex in one of five different ways as it is shown in Fig.   5 (or do not go through at all). A factor f v is associated with each variant, these factors Any non-self-intersecting path on the toroidal lattice has a homotopy class where A is the toroidal cycle along the α-lines, and B is the toroidal cycle along the β-lines of Fig. 3. Then the element t n,m of (44) is where the sign (−) nm+n+m counts the number of fermionic loops on the toroidal square lattice. The determinant J may be expressed in the terms of T and vice versa: The last equality is very well known in the two dimensional free-fermion model as the formula relating the lattice partition function and fermionic determinant.

Quantum model
In the previous section, we did not pay any attention to the structure of vertex variables A v (14), they were defined simply as the list of elements of X (15). The key point for the quantization of the model is that it is possible do define a local Poisson structure on A v [1] such that the transformation defined by Korepanov equation (19), is a symplectic map. Symplectic structure admits an immediate quantization. We will skip here all the details and proceed directly to the The aim of this section is just to give precise definition of T(u, v) (2).

Quantum Korepanov and Tetrahedron equations.
The local q-oscillator algebra H is defined by (1). The Fock space F representation for q-oscillators corresponds to Quantized dynamical variables (14) are the q-oscillator generators and a pair of C-valued Quantized X (15) and L (37) are given by One may verify directly, the quantum Korepanov equation is equivalent to the auxiliary tetrahedron equation (the quantum local Yang-Baxter equation) Here the intertwining operator R = R 123 acts in the tensor product of representation (55) and (56) are equivalent to the following set of six equations: Classical equations (19) and (38) follow from (55) and (56) in the q → 1 limit of the For irreducible representations of H v , R is defined uniquely, its matrix elements for the Fock space representation are given in [1].
Remarkably, Fig. 2  Since the arguments of L-s are local q-oscillator generators, T(u, v) ∈ H ⊗N M is by definition a layer-to-layer transfer matrix, its graphical representation is again the classical Fig. 3. The model is integrable since i.e. the coefficients t n,m in (60) form the set of the integrals of motion.
Now we give the technical proof of the commutativity (61). The commutativity of layer-to-layer transfer matrices follow from a proper tetrahedron equation [4]. In addition where A 0 ∼ (H 0 ; λ 0 , µ 0 ). The constant tetrahedron equation for L and L, may be verified directly, it is just 16×16 matrix equation with the operator-valued entries.
Another technical relation is where D is given by (42). Combining (63) for the whole lattice, we come to and taking into account (64), we get . Now taking the trace over the representation space F 0 of H 0 and denoting we come to the final similarity relation and therefore two transfer matrices T(u, v) = Trace In the limit q → 1 coefficients of (60) become the involutive moduli of the spectral curve α,α ′ , and therefore In the same way, the other direction of the lattice may be chosen, and the dual Lax may be considered.
Matrix R β,β ′ (77) has the block structure as well, where λ 0 , µ 0 are extra parameters of A 0 (77), and R ω m ,ω m ′ is the U q ( sl M ) R-matrix for the representations π ω m ⊗ π ω m ′ . In particular, in the sector π ω 1 ⊗ π ω 1 with the basis ϕ j (79), one can obtain the fundamental R-matrix (we used the Fock space (51) for H 0 for the calculation of the trace in (77)): Lax operator L where J is the eigenvalue of J , and π Jω 1 is the rank-J symmetrical tensor representation of U q ( sl M ) (dominant weight Jω 1 ).
We would like to conclude this subsection by the example of M = 2 containing the six-vertex model. The block-diagonal structure of L (74) is where index n of (74) is omitted, and 2 × 2 central block is Fixed integer J = h 1 + h 2 in the quantum space corresponds to spin J /2 representation of sl 2 . For spin 1/2 representation (J = 1), matrix elements of (88) may be presented by One may easily see, in the limit q → 1 (92) becomes (48). Operator J(u, v) belongs to where J(u, v) ∈ W, (11). Define a linear space Ψ by The linear space may be decomposed with respect to the basis | t , The point is that Q t | and |Q t have a simple and predictable structure, in particular Q t |u and v|Q t may be defined as polynomials.
Functions Q|u and v|Q (index t is usually omitted) obey the linear difference equations (95), which are exactly Eqs. (7) and (8).
In this section, we will derive (92) and (95) may be rewritten for the whole lattice in a matrix form, The fragments in (99), corresponding to (98), are In the quantum world, matrix elements ℓ k,j are non-commutative operators ℓ k,j . Therefore (99) may have two slightly different forms, Let the operators ℓ k,j obey the following exchange relations: The aim of this subsection is to establish fundamental properties of such ℓ and the form of solution of (101) A) and B).
where σ are the permutations of the indices 1, 2, 3, . . . . According to the first relation of (102), the definition (103) is invariant with respect to the ordering of j, for instance . . , and in general where τ is any permutation of 1, 2, 3, . . . . Define next the algebraic supplements A j,k of ℓ (adjoint matrix) as (103)-determinants of the minors of ℓ, The first equality here follows from (105), the last one in the definition of A j,k . The second line of (102) provides Therefore (det ℓ) −1 A j,k and A j,k (det ℓ) −1 are two variants of inverse matrices, or, since the inverse matrix must be both left-and right-inverse, The main property of the elements of inverse matrices is the commutativity of their matrix elements with the same k 0 , e.g. for the variant A), It is a particular case of the following statement: To prove (111), consider Due to the first relation of (102) and definition (111), j,j ′ |k,k ′ is the matrix of the second algebraical supplements of ℓ: , what proves the commutativity of m j . The commutativity (110) corresponds to ε k = δ k,k 0 .

Extended algebra of observables.
To establish the relation between the algebra of matrix elements (102) and our case of X (53), we have to introduce the extended where q hv , x v , y v are generators of q-oscillator H v (1) and λ v , µ v , ν v are the generators of Weyl algebra W v : The q-oscillators and the Weyl algebra elements for different vertices always commute.
where u and v are C-valued spectral parameters introduced according to (22). Let locally (124) In the combinatorial representation of the determinant (Fig. 5, eqs. (45,47,48)), consider a path C n,m of the homotopy class nA + mB. A monomial summand, corresponding to this path, may be factorized as J Cn,m = t Cn,m e φ(Cn,m) .
Monomial t Cn,m gathers all the q-oscillators and C-valued parameters λ v , µ v and −q −1 λ v µ v , and therefore it is exactly the C n,m -monomial of T(u, v) (up to unessential renormalization of x v and y v ). Operator φ is a sum of local Q v and P v . One may easily see, Let C 1,0 and C 0,1 be two particular fixed paths. Due to (126), all commute. Therefore one may diagonalize them simultaneously and without lost of gen- On this condition, (129) φ(C n,m ) → nφ(C 1,0 ) + mφ(C 0,1 ) ≡ nQ 0 + mP 0 , and exponent of φ(C n,m ), together with the combinatorial factor (−) nm+n+m u n v m , be- According to (110,114,115), each of |ψ j and ψ j | belong to these subspaces, and there exist sets of commutative operators m j and m ′ j (117) such that |ψ j = m j |Ψ and ψ j | = Ψ| m ′ j . In particular, one may consider the linear spaces |Ψ and Ψ| in the basis of diagonal m j , m ′ j : where in the notations of (98 The A).
Turn at the first to the conditions for Ψ 0 in (136). Since the total Fock vacuum is the eigenstate of q-oscillator counterpart of J[A], one may consider directly Evidently, J 0 commutes with all φ (127) and therefore allows the projection W ⊗N M → W (128). Simply applying (131), Let further |t and t| be the eigenstates of t nm , (93). Analogously to (135) let It follows from the second lines of (136) where e.g. P t is a polynomial of µ v of a total power not more that the number of bosons in |t , its structure with respect to ν v is rather simple. In addition, since we consider the eigenstates in the Fock counterpart of A ⊗N M , polynomials P t and P ′ t must commute with all φ (127) and therefore must allow the projection W ⊗N M → W (128): where w 2 = −vu, w-factor comes from ν v factors; J and K are integers; P t and P ′ t are polynomials of a power not higher than the total number of bosons in the state | t . In the next section, considering the examples, we will see that J n and K m are the eigenvalues of J n and K m (72) and, moreover, the degrees of Q m and Q n are exactly K m and J n .
In the next section we will add q −Jn and q −Km to the definition of u n and v m . This allows one to cancel the half-integer pre-factors v −Jn/2 of Q n (v) and u −Km/2 of Q m (u).
The corresponding values of u n and v m may be obtained via conditions Q(0) = 1 and Q n (0) = 1.

Examples
Let us illustrate (97) for the six-vertex model first, and then for an arbitrary square lattice.
Condition Q(0) = 1 gives (see (149)) (153) v −1 + v n µ n,1 µ n,2 q h n,1 +h n,2 = n µ n,1 q h n,1 + n µ n,2 q h n,2 , which has two solutions corresponding to two Baxter's functions Q: the first one is , and corresponding Q = Q 2 (u) is the polynomial of the power n h n,2 = K 2 . In some sense, two functions Q correspond to two sheets of classical q = 1 spectral hyperelliptic curve v −1 φ(u) + vφ ′ (u) = t(u). Note, we consider now the Bethe Ansatz equations for U q ( sl 2 ) chain with arbitrary J n , this is the inhomogeneity of highest spin. Six-vertex case corresponds to J n = 1 for all n.
Equations (156) may be obtained in the other way -via Q(v) = v|Q . Condition u n = λ n,1 λ n,2 q h n,1 +h n,2 −1 , is a polynomial of the power J n = h n,1 + h n,2 . In the six-vertex case J n = 1, therefore Q n (v) = v − ζ n , and equating to zero each v-term of the right of equations (97), one comes to The second equality gives (156).

4.2.
U q ( sl M ) equations. Let the lattice has arbitrary N and M. We will consider (97) for Q|u = Q(u) and for u|Q = Q(u) -the last function was not mentioned before, but it is interesting to discuss what it is.
Equations Q|J(u, v)|u = 0 and u|J(u, v)|Q = 0 read correspondingly Suppose, Q(u) ∼ u K and Q(u) ∼ u K when u → ∞. Then (159) and the second line of (162) provides the following conditions for K and K: The charges J n and K m are given by (72). Equation (165)  In what follows, we will assume the generic set of µ nm , so that all v m are different. Let v = v m corresponds to Q(u) = Q m (u) and Q(u) = Q m (u). Then (166) and (167)  (v i − v j ), therefore W (u) = 0.
Using these "arithmetical" considerations, one may express the fundamental transfer matrices τ m (u) of sl M via determinants det ||Q j (q 2p i u)v p i j || i,j=1,...,M , where p i is a subset of (0, 1, . . . .M). Moreover, for a more general sets of p i , any transfer matrix of sl M may be expressed as such a determinant [5].

Conclusion
This paper has a modest aim just a to give a correct form of "T-Q" equations. We can say nothing about their solution. But we would like to note, that from the point of view of three dimensional models, the thermodynamical limit is the limit N, M → ∞ with non-singular ratio N : M. The nested Bethe Ansatz equations was never investigated in this limit since Q m (u) (174) with finite m has a finite number of roots. In addition, an excitation corresponds to a change of the structure of occupation numbers. From sl M point of view of the previous section, it corresponds not only to a change of K m related to the powers of Q-operators, but as well it corresponds to a change of J n which is a change of the sl M -structure of the nested Bethe Ansatz.
Let us better conclude this paper by a brief comparison of two exactly integrable models in 2 + 1 dimensional space-time. From the point of view of the algebra of observables, these models should be called "q-oscillator model" and "Weyl-algebra model". The last one is a quantum-mechanical reformulation of Zamolodchikov-Bazhanov-Baxter model of statistical mechanics, which has a long history [8][9][10][11][12]. Both models are based on two slightly different forms of local linear problem, the linear problem for Weyl-algebra model may be found in any of Refs. [13]. Solution of both classical models may be expressed in terms of algebraic geometry, [3] and [14]. Equations of motion may be understood as a canonical mapping conserving certain symplectic structure, [1] and [15]. Poisson structure allows an immediate quantization, [1] and [13]. Quantum-mechanical integrals of motion may be combined into a direct sum of transfer matrices for fundamental representations of either sl N or sl M , [1,2] and [16]. And finally, the solvability of the models is based on quantized auxiliary linear problem and remarkable features of their operator-valued matrices of coefficients, this paper and [17].
The continuous limit of both classical models may be illustrative. Equations of motion for six fields q j , q * j , j = 1, 2, 3, follow from the action where for the classical continuous limit of q-oscillator model V = q * 1 q * 2 q * 3 − q 1 q 2 q 3 , this is nothing but the model of three-wave resonant interaction [18]. For the classical continuous limit of Weyl-algebra model the potential is V = (q * 1 − q 2 )(q * 2 − q 3 )(q * 3 − q 1 ) [19].