Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations

We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard (KZB) equations with $\tilde N$ punctures by deformation of the corresponding quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is $\tau$. At the level of classical mechanics the deformation parameter $\tau$ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Next, we notice that the identities underlying generic (elliptic) KZB equations follow from some additional relations for the properly normalized $R$-matrices. The relations are noncommutative analogues of identities for (scalar) elliptic functions. The simplest one is the unitarity condition. The quadratic (in $R$ matrices) relations are generated by noncommutative Fay identities. In particular, one can derive the quantum Yang-Baxter equations from the Fay identities. The cubic relations provide identities for the KZB equations as well as quadratic relations for the classical $r$-matrices which can be halves of the classical Yang-Baxter equation. At last we discuss the $R$-matrix valued linear problems which provide ${\rm gl}_{\tilde N}$ Calogero-Moser (CM) models and Painleve equations via the above mentioned identities. The role of the spectral parameter plays the Planck constant of the quantum $R$-matrix. When the quantum ${\rm gl}_N$ $R$-matrix is scalar ($N=1$) the linear problem reproduces the Krichever's ansatz for the Lax matrices with spectral parameter for the ${\rm gl}_{\tilde N}$ CM models. The linear problems for the quantum CM models generalize the KZ equations in the same way as the Lax pairs with spectral parameter generalize those without it.


Introduction
Let V be a finite-dimensional module of the group GL N . The quantum R-matrix is an operator R : V ⊗ V → V ⊗ V satisfying the quantum Yang-Baxter equation [31]: where z, w -spectral parameters. We consider a special class of non-dynamical R-matrices which includes Belavin's elliptic gl N R-matrix and its (nontrivial) degenerations, i.e. z is a local coordinate on the (degenerated) elliptic curve. Let us fix the normalization of R in the way that the unitarity condition takes the form where Φ (z) is the function defined in the elliptic case 1 as Φ (z) = Nφ(N , z) , φ(z, u) = ϑ ′ (0)ϑ(u + z) ϑ(z)ϑ(u) , (1.3) where ϑ(z) = θ 11 (z|τ ) is the odd Riemann theta-function, τ -elliptic moduli.
We demonstrate here that starting with the R-matrix one can construct different families of classical and quantum integrable system. These constructions are based on two special features of the R-matrices. The first one is the quasi-classical expansion. With the normalization (1.2)-(1.3) it acquires the form: where r 12 (z) is the classical r-matrix. It leads to integrable Euler-Arnold gl N tops 2 and Gaudin systems.
The second is the property of Painlevé-Calogero correspondence, which is equivalent to the heat equation: The latter leads to the monodromy preserving equations (non-autonomous tops, Schlesinger systems) and the KZB systems.
At last, the main tool is the set of identities for the quantum R-matrices which we introduce below. R-matrix is an operator acting on the tensor product of vector spaces V . Consider a set of points z 1 , ..., zÑ (on the curve where z is a local coordinate). Let be the R-matrix acting on the a-th and b-th components of V ⊗Ñ . In our case R-matrices satisfy the following property: i.e. the terms of the expansion (1.4) are of definite parity: r ab = −r ba , m ab = m ba .
(1. 8) We show that the R-matrices satisfy a set of relations similar to identities for function φ(z, u) (1.3). In particular, φ(z, u) satisfies the Fay identity φ(x, z ab )φ(y, z bc ) = φ(x − y, z ab )φ(y, z ac ) + φ(y − x, z bc )φ(x, z ac ) , (1.9) where z ab = z a − z b . We notice that the following analogue of the Fay identity holds: It will be shown that one can derive the quantum Yang-Baxter equation (1.1) from (1.10).
While the quantum R-matrix is similar to φ( , z) the classical r-matrix is the analogue of function E 1 (z) = ∂ z log ϑ(z). For example, the following relation holds: (r ab + r bc + r ca ) 2 = 1 a ⊗ 1 b ⊗ 1 c N 2 (℘(z a − z b ) + ℘(z b − z c ) + ℘(z c − z a )) , (1.11) where ℘(z) is the Weierstrass ℘-function with moduli τ . It is the analogue of the identity leads to the following relations: r ab r ac − r bc r ab + r ac r bc = m ab + m bc + m ac . (1.14) Difference of (1.14) written for indices a, b, c and a, c, b gives (1.13).
Let us remark that the class of R-matrices we discuss here includes Baxter-Belavin's one [4,5] as the most general. Its trigonometric analogue was found in [8,3] (we are going to consider it in separate publications). At last the rational case is known from [8,32,23]. In the simplest cases one gets the ordinary XXZ and XXX Yang's R-matrices. In the rational case the Yang's R-matrix [33] (with normalization (1.2)) is of the form: where P ab is the permutation operator. We deal with non-trivial deformations of (1.15). In particular, they allow us to define not only KZ but also KZB equations. At the same time the rest of our construction works for ordinary XXX (and XXZ) R-matrices as well 3 .
The purpose of the paper is twofold. First, we construct the rational analogue of the (elliptic) KZB equations. For this purpose we find τ deformation of the quantum R-matrix suggested in [23]. Second, we show that integrable systems of Calogero-Moser type admit higher rank Lax representations which generalize the Krichever's one [17] in the same way as (1.10) generalize (1.9). The standard (non-diagonal) matrix elements φ(λ, z a − z b ) are replaced by the quantum R-matrices R λ ab , i.e. the spectral parameter is given by the Planck constant entering R-matrix. Our constructions are independent of specific form of the R-matrix, but based only on the set of identities (such as (1.10), (1.14), (1.5)) which can be verified separately.

Rational KZB equations
Besides the standard trigonometric and rational versions of the elliptic R-matrix there are more sophisticated degenerations. In this paper we consider one of them [23] and show that it leads to some modifications of the standard Gaudin and Schlesinger systems and the KZ (KZB) equations.
The Belavin's R-matrix depends on the moduli of the elliptic curve τ . We notice that it satisfies the heat equation (1.5) and treat this equation as Painevé-Calogero property. In [18] it was formulated in the following way: the Lax pair of the CM model satisfies also the monodromy preserving equations and describe the (higher rank) Painlevé equations. We refer to (1.5) as the heat equation because this equation for the function φ( , z) follows from the heat equation for ϑ-function 2∂ τ ϑ(z|τ ) = ∂ 2 z ϑ(z|τ ). The natural (noncommutative) analogue of ϑ-function is the modification of bundle Ξ(z, τ ). In the elliptic case it was found in [14] in the context of the IRF-Vertex transformation, and then described in [20] (see also [22,23]) as an example of the Symplectic Hecke Correspondence for integrable systems. Its rational analogue was suggested in [2] and was know to be free of τ dependence. Here we explain how to introduce the τ -dependence. We construct the τ deformation of the rational R-matrix based on the heat equation The solution provides possibility for construction of the rational analogue of the KZB equations where r and m are the terms of the expansion (1.4) and τ indicates the τ -deformation. The system of KZ of KZB equations is known to be related to the quantum (and classical) CM models by the Matsuo-Cherednik construction [26,9] (see also [27]). Recently relations between CM (and Ruijsenaars-Schneider (RS)) models to quantum spin chains were actively investigated [1,15].

R-matrix valued Lax pairs
The Fay type identities (1.10) for the quantum R-matrices allows to suggest extended version of the Krichever's ansatz for CM Lax pairs with spectral parameter [17]. Consider the following block matrix Lax operator L =Ñ a,b=1Ẽ ab ⊗ L ab (1.18) whereẼ ab is the standard basis of glÑ and 19) When N = 1 the gl N R-matrix reduces to its scalar analogue -function φ(z, ) and we reproduce the answer from [17] forÑ-body CM system. Notice that the Planck constant of gl N R-matrix plays here the role of the spectral parameter for glÑ CM model. The corresponding M-operator is given in (4.14). The Lax equation ∂ t L = [L, M] is equivalent to dynamics ofÑ CM particles (1.20) In the same way the monodromy preserving equation The corresponding linear problem has the form Let us also mention that the linear problem for the quantum version of CM model resembles very much the KZ connections from the first line of (1.17). Equation (1.23) (or (1.22) withL) generalizes the first line of (1.17) in the same way as the Lax pairs with spectral parameter generalize those without it. We hope to clarify exact relations between R-matrix valued linear problems and KZB equations in our future papers.
Choosing elliptic, trigonometric or the rational R-matrix we describe the CM models similarly to gl 1 case [17]. Notice that the gl N R-matrix itself describes gl N integrable systems such as integrable tops which are gauge equivalent to CM or RS models. Here we use gl N R-matrices as auxiliary spaces for derivation of glÑ models. The next natural step is to get similar result for the Ruijsenaars-Schneider (quantum) model. In this case we deal with two Planck constants. Our general idea is that the both Planck constants can play different roles, i.e. each of the constants can be either the spectral parameter in a "classical-quantum" glÑ system (of (1.19) type) or the Planck constant in a quantum gl N system or the relativistic deformation parameter in a classical relativistic gl N model (see [23]) 4 . We hope that this can shed light on numerous dualities in integrable systems mentioned in [28], [25], [34], [11].

From integrable tops to KZB equations
In this section we describe the sequence of steps which leads to the KZB equations [12] starting from integrable tops. As it was mentioned above, our consideration is independent on the choice of particular top model. The basic element is the underlying quantum R-matrix [23].
First, we briefly recall the structures underlying integrable tops and proceed to the nonautonomous dynamics. It is described by the monodromy preserving equations. In the same way the Schlesinger system is originated from the corresponding Gaudin model. At last, the KZB equations arise from the quantization of the Schlesinger system [29,19,16].

Integrable tops
In [23] we defined the relativistic integrable top by means of the quantum R-matrix. The gl N Lax matrix is given by E ij S ij is the gl N -valued dynamical variable 5 , and R η 12 (z) is the corresponding quantum non-dynamical R-matrix. It satisfies the quantum Yang-Baxter equation (1.1). The non-relativistic limit (η → 0) and satisfies the classical Yang-Baxter equation: As it was mentioned in [23] the matrices (2.3), (2.4) appear to be the Lax pair of the nonrelativistic top. It means that the Lax equation is equivalent to equations of motion where the inverse inertia tensor is given by the linear functional

Painlevé-Calogero correspondence and non-autonomous tops
The (classical) Painlevé-Calogero correspondence was suggested in [18]. It claims that the (Krichever's) Lax pair of the elliptic Calogero-Moser model can be also used for the monodromy preserving equations, which describe the higher rank Painlevé equations in the elliptic form.
Let us formulate here the Painlevé-Calogero correspondence in the form of the quantum non-dynamical R-matrix property.
Definition 1 Suppose that the quantum R-matrix entering (2.1) depends on some additional parameter τ : R ,τ (z) = R(z, , τ ). We say that the R-matrix satisfies the property of the "Painlevé-Calogero correspondence" if the following relation holds 6 : (2.12) Plugging the expansion (1.4) into (2.12) we get a set of relations. The first non-trivial is where r τ 12 (z) = r 12 (z, τ ) is the classical r-matrix. An example of the R-matrix with this property is given by the Baxter-Belavin's one [4] (see Appendix B). The parameter τ in this example equals τ ell /2πı, where τ ell is the module of the underlying elliptic curve, and the property (2.13) is due to the heat equation for the theta-functions (2.14)

From (2.13) and (2.3)-(2.4) it follows that
where L τ (z, S) = L(z, S, τ ), M τ (z, S) = M(z, S, τ ). Therefore, we can define the monodromy preserving equations in time τ as the non-autonomous version of the integrable top's equations of motion (2.8) 7 : Indeed, the total derivative d τ L τ (z, S) contains both -the partial derivatives by explicit and implicit dependence on τ : The first term is cancelled by ∂ z M τ (z, S) (2.15), and we get the same result as in (2.8) following from the Lax equations (2.7). But this time it contains explicit dependence on τ via Similarly to the autonomous case this system is Hamiltonian (see (2.10)) with (2.20) and the Poisson brackets are given by (2.11).

Gaudin models
The phase space of the Gaudin model [6] is the direct product of n coadjoint orbits, i.e.Ñ copies of S: S a ∈ gl N , a = 1, ...,Ñ with some fixed eigenvalues. Its Poisson structure is the direct sum of (2.11). The Lax matrix has n simple poles at {z a , a = 1, ...,Ñ} with residues S a . It is given in terms of the top Lax matrix (2.3): Here we imply the existence of the deformation parameter τ (2.14)-(2.20) from the very beginning in order not to repeat (almost) the same notations with τ and without τ as we made for the top and its non-autonomous version.

Schlesinger systems
Similarly to the description of Painlevé equation in the form of non-autonomous tops let us also represent the Schlesinger system [30] as the non-autonomous Gaudin model.
Therefore, the monodromy preserving equations (or compatibility conditions for isomonodromic deformations)

KZB equations
The relation between KZB equations and the quantum monodromy preserving equations was described in [29] (see also [19,16]). Let us formulate it using notations of (1.4) with the τdeformation satisfying (2.13). The KZB equations have form: Here r τ ac and m τ ac are the operators acting by a-th and c-th components of U(gl N ) ⊗Ñ (the tensor product ofÑ copies of the universal enveloping algebra). Recall that in classical integrable systems (as well as in the Schlesinger systems) we used the fundamental representation ρ N of gl N (see e.g. (2.3)-(2.4)): where {e a ij }: [e a ij , e a kl ] = δ ab (e a il δ kj − e a kj δ il ) is the standard basis in the a-th component of U(gl N ) ⊗Ñ . In this notation The fundamental representation is given by ρ N (e a ij ) = 1 ⊗ ... ⊗ 1 ⊗ E ij ⊗ 1 ⊗ ... ⊗ 1, where (E ij ) kl = δ ik δ jl is on the a-th place. Then r-matrix is an operator acting on the a-th and b-th components of an element of the tensor product V ⊗Ñ . The operator is represented by matrix of NÑ × NÑ size because it also contains (as factors) the product of identity operators for the rest of components c =a,b 1 c . The residue of r-matrix is (up to factor N in (B.11)) the permutation operator replacing a-th and b-th components of an element of the tensor product V ⊗Ñ to which ψ belongs.

Rational non-autonomous tops and KZB equations
The rational top was first studied for small rank cases in [32] by degenerating the elliptic Lax matrix [20]. Later it was constructed for gl N case using its relation to the rational Calogero-Moser model [2]. The idea was to compute the classical (skew-symmetric non-dynamical) rmatrix as follows: In [23] this relation was extended to the quantum R-matrix by proceeding to the relativistic top: where the classical Lax matrix L (z) depends on the constant playing the role of the relativistic deformation parameter. The Lax matrix was found using its relation to the Ruijsenaars-Schneider (RS) model. In the spinless case the gauge transformation relating two models can be written explicitly in terms of the RS particles coordinates q j : g(z, q) = Ξ(z, q)D −1 , where 11 Ξ(z, q) = (z + q j ) ̺(i) ,

τ -deformation of quantum rational R-matrix
Our aim is to construct τ -dependent R-matrix satisfying the Painlevé-Calogero property (2.12) starting from the τ -independent one (3.2). The answer follows from (3.8) (see below). It appears that the deformation of the Yang's rational R-matrix suggested in [23] admits this kind of deformation similarly to the elliptic case. The idea is to deform first Ξ(z) (3.4). Let us find Ξ(z, q| τ ) satisfying the heat equation with the boundary condition Ξ(z| 0) = Ξ(z) .

(3.6)
Then the R-matrices (3.1), (3.2) constructed by means of Ξ(z| τ ) satisfy the property (2.12) 12 . The solution of (3.5)-(3.6) is given by or Ξ(z| τ ) = exp τ 2 T Ξ(z) , (3.8) 11 The explicit from of L RS (z, η) as well as diagonal matrix D ij = δ ij k =i (q i − q k ) is not used in what follows. 12 It can be also proved directly by using explicit answer for the quantum R-matrix [23].
where T is the nilpotent operator representing the action of ∂ 2 z on the N-dimensional columnvector (1, z, z 2 , ..., z N −2 , z N ) T . It is N × N matrix with elements For example, for N = 2, 3, 4 we have: (3.10) i.e. Ξ(z| τ ) = T Ξ(z| 0). Then for N = 2, 3, 4 the operator T equals It follows from (3.1)-(3.3) and (3.8) that τ -deformation of R-matrix is given by the following gauge transformation: written in terms of (3.11). See Appendix A for explicit answer in gl 3 case.

Rational KZB equations
It follows from (3.13) that (3.14) Then the condition (2.13) is fulfilled as well as (2.51)  The proof is direct. Below we give explicit examples of τ -deformations in the rational case.

Example: gl 2 case
Quantum R-matrix (satisfying (2.12)): Classical r-matrix and m-matrix (the next term of expansion of (3.17) in ) satisfying (2.13): The following additional relation holds: Some similar formulae for gl 3 case are given in the Appendix A.
4 Planck constant as spectral parameter

R-matrix valued Fay identities
In this paragraph we show that the quantum R-matrices satisfy a set of relations which are similar to their scalar analogues -the functions Φ (1.2). It is convenient to discuss the elliptic case (B.5)-(B.14) because the trigonometric and rational versions are obtained by some (nontrivial) degenerations.
The function φ(x, z) (B.5) (or (B.14)) satisfies the Fay identity: where z ab = z a − z b . Let us formulate its noncommutative analogue.

2)
where R ab = R ab (z a − z b ).
Consider the derivative of (4.2) with respect to z b : where F ab (z) = ∂ z R ab (z). The function F ab (z) has no singularities at = 0. Therefore, we can put = ′ in (4.8). This gives The latter equation is analogue of the following identity underlying Lax equations (integrability) of the Calogero-Moser model [7,17].

R-matrix valued linear problem for Calogero-Moser model
Consider the eigenvalue problem LΨ = ΨΛ (4.11) for the following block matrix operator L =Ñ a,b=1Ẽ ab ⊗ L ab , (4.12) whereẼ ab is the standard basis of glÑ and It is worth mentioning that in gl 1 case (N = 1) this operator is the Krichever's Lax matrix with spectral parameter for the Calogero-Moser model [17]. The eigenvalue matrix consists of vectors ψ 1 , ..., ψÑ . In the case of quantum CM model (p a → ∂ za ) equation (4.11) should have well defined limit → 0 which gives the KZ equations for ψ 1 = ... = ψÑ = ψ. Alternatively, one can quantize the model as p a → ∇ a . At the level of classical mechanics and N = 1 the difference between ∂ za and ∇ a is given by the canonical map p a → p a + ν c =a The spectral parameter in (4.13) is -the Planck constant. The M-operator is defined as follows: M ab = νδ ab d a + ν(1 − δ ab )F ab + νδ ab F 0 , (4.14) where M-operator (4.14) is also straightforward generalization of the one proposed in [17] except the last term F 0 . The latter is not needed in N = 1 case because in this case it is proportional to the identity matrix. The term [p, F ] is cancelled by ∂ t R (due toż a = p a ).
Consider the off-diagonal block ac. It has three inputs from The sum of the inputs equals zero. We used that F 0 ab = F 0 ba (due to F 0 ab = ∂ za r ab (z a − z b )). On a diagonal block we get equations of motion: (4.20) It is natural to expect that the same receipt works for other root systems (not only gl N ) as well, i.e. one can replace the function φ(x, z) in the Lax matrix with the corresponding quantum R-matrix.
Denote the off-diagonal part of (4.13) by L 0 : L 0 ab = (1 − δ ab )R ab . We conjecture that 13 : 13 The proof will be given elsewhere.

Half of the classical Yang-Baxter equation
Consider the unitarity condition R ab R ba = Φ (z ab )Φ (z ba ). Its expansion in the 0 order gives Here r ab = r τ ab (z a − z b ), m ab = m τ ab (z a − z b ). Next, consider (4.23)-(4.25). In the 1 order it provides the following relation between r and m matrices: we can combine (4.27) and (4.28) into two "halves" of the classical Yang-Baxter equation: r ab r ac − r bc r ab + r ac r bc = m ab + m bc + m ac (4.29) and r ac r ab − r ab r bc + r bc r ac = m ab + m bc + m ac . (4.30) The difference of (4.29) and (4.30) gives (4.28) while the sum leads to (4.27).
Proof: Consider the Yang-Baxter equation R ca R cb R ab = R ab R cb R ca in the 0 order. It is given by the sum of (4.35) and (4.36). Consider also (4.23) in the 0 order. It is given by the difference of (4.35) and (4.36).

37)
for i = 1, ...,Ñ and 14 where r τ ij and m τ ij are the coefficients of the expansion (1.4) and ν is a free constant. Then the conformal block satisfies the following equation: where ∆ = i ∂ 2 i and m τ jj = m τ jj (0) are scalar operators depending on τ .
Proof: Let us omit the dependence on τ , i.e. r τ ij := r ij .

Painlevé equations
The block matrix Lax pair (4.13), (4.14) can be also used for description of the Painlevé equations likewise it was done in [18] in N = 1 case, i.e. the result of Proposition 4.3 is naturally generalized to the following one:   The proof repeats the one for the Proposition 4.3. Additionally one should use the property (2.12) of the Painlevé-Calogero correspondence.