Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations
aa r X i v : . [ h e p - t h ] O c t ITEP-TH-27/14
Planck Constant as Spectral Parameterin Integrable Systems and KZB Equations
A. Levin ♭ ♯
M. Olshanetsky ♯ ♮
A. Zotov ♦ ♯ ♮ ♭ – NRU HSE, Department of Mathematics, Myasnitskaya str. 20, Moscow, 101000, Russia ♯ – ITEP, B. Cheremushkinskaya str. 25, Moscow, 117218, Russia ♮ – MIPT, Inststitutskii per. 9, Dolgoprudny, Moscow region, 141700, Russia ♦ – Steklov Mathematical Institute RAS, Gubkina str. 8, Moscow, 119991, Russia
E-mails: [email protected], [email protected], [email protected]
Abstract
We construct special rational gl N Knizhnik-Zamolodchikov-Bernard (KZB) equationswith ˜ N punctures by deformation of the corresponding quantum gl N rational R -matrix.They have two parameters. The limit of the first one brings the model to the ordinaryrational KZ equation. Another one is τ . At the level of classical mechanics the defor-mation parameter τ allows to extend the previously obtained modified Gaudin models tothe modified Schlesinger systems. Next, we notice that the identities underlying generic(elliptic) KZB equations follow from some additional relations for the properly normal-ized R -matrices. The relations are noncommutative analogues of identities for (scalar)elliptic functions. The simplest one is the unitarity condition. The quadratic (in R ma-trices) relations are generated by noncommutative Fay identities. In particular, one canderive the quantum Yang-Baxter equations from the Fay identities. The cubic relationsprovide 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 discussthe R -matrix valued linear problems which provide gl ˜ N Calogero-Moser (CM) models andPainlev´e equations via the above mentioned identities. The role of the spectral parameterplays the Planck constant of the quantum R -matrix. When the quantum gl N R -matrix isscalar ( N = 1) the linear problem reproduces the Krichever’s ansatz for the Lax matriceswith spectral parameter for the gl ˜ N CM models. The linear problems for the quantumCM models generalize the KZ equations in the same way as the Lax pairs with spectralparameter generalize those without it. ontents τ -deformation of quantum rational R -matrix . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Rational KZB equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Example: gl case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 R -matrix valued Fay identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 R -matrix valued linear problem for Calogero-Moser model . . . . . . . . . . . . . . . . . 174.3 Half of the classical Yang-Baxter equation . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Identities for KZB equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Painlev´e equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 gl (rational) case 226 Appendix B: Belavin’s R -matrix 23 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]: R ~ ( z − w ) R ~ ( z ) R ~ ( w ) = R ~ ( w ) R ~ ( z ) R ~ ( z − w ) , (1.1)where z, w - spectral parameters. We consider a special class of non-dynamical R -matriceswhich includes Belavin’s elliptic gl N R -matrix and its (nontrivial) degenerations, i.e. z is a localcoordinate on the (degenerated) elliptic curve. Let us fix the normalization of R ~ in the waythat the unitarity condition takes the form R ~ ( z ) R ~ ( z ) = 1 ⊗ ~ ( z )Φ ~ ( − z ) , (1.2)2here Φ ~ ( z ) is the function defined in the elliptic case asΦ ~ ( z ) = N φ ( N ~ , z ) , φ ( z, u ) = ϑ ′ (0) ϑ ( u + z ) ϑ ( z ) ϑ ( u ) , (1.3)where ϑ ( z ) = θ ( 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 ofclassical and quantum integrable system. These constructions are based on two special featuresof the R -matrices. The first one is the quasi-classical expansion. With the normalization (1.2)-(1.3) it acquires the form: R ~ ( z ) = 1 ~ ⊗ r ( z ) + ~ m ( z ) + O ( ~ ) , (1.4)where r ( z ) is the classical r -matrix. It leads to integrable Euler-Arnold gl N tops and Gaudinsystems.The second is the property of Painlev´e-Calogero correspondence, which is equivalent to theheat equation: ∂ τ R ~ ( z ) = ∂ z ∂ ~ R ~ ( z ) (1.5)The latter leads to the monodromy preserving equations (non-autonomous tops, Schlesingersystems) and the KZB systems.At last, the main tool is the set of identities for the quantum R -matrices which we introducebelow. R -matrix is an operator acting on the tensor product of vector spaces V . Consider a setof points z , ..., z ˜ N (on the curve where z is a local coordinate). Let R ~ ab = R ~ ( z a − z b ) , (1.6)be the R -matrix acting on the a -th and b -th components of V ⊗ ˜ N . In our case R -matrices satisfythe following property: R ~ ab ( z a − z b ) = − R − ~ ba ( z b − z a ) , (1.7)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: R ~ ab R ~ ′ bc = R ~ ′ ac R ~ − ~ ′ ab + R ~ ′ − ~ bc R ~ ac (1.10) In the rational case we use Φ ~ ( z ) = z − + ~ − . The trigonometric case will be considered separately. The integrable tops were previously proved to be related (equivalent) to the (spin) Calogero-Ruijsenaarsmodels by the symplectic Hecke transformations. See. e.g. [20, 22, 23]
3t 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 offunction E ( z ) = ∂ z log ϑ ( z ). For example, the following relation holds:( r ab + r bc + r ca ) = 1 a ⊗ b ⊗ c N ( ℘ ( 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( E ( z a − z b ) + E ( z b − z c ) + E ( z c − z a )) = ℘ ( z a − z b ) + ℘ ( z b − z c ) + ℘ ( z c − z a ) . (1.12)Together with (1.11) the classical Yang-Baxter equation[ r ab , r ac ] + [ r ac , r bc ] + [ r ab , r bc ] = 0 (1.13)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 toconsider it in separate publications). At last the rational case is known from [8, 32, 23]. In thesimplest cases one gets the ordinary XXZ and XXX Yang’s R -matrices. In the rational case theYang’s R -matrix [33] (with normalization (1.2)) is of the form: R ~ , Yang ab = 1 a ⊗ b ~ + P ab z a − z b , (1.15)where P ab is the permutation operator. We deal with non-trivial deformations of (1.15). Inparticular, they allow us to define not only KZ but also KZB equations. At the same time therest of our construction works for ordinary XXX (and XXZ) R -matrices as well . 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 -matrixsuggested in [23]. Second, we show that integrable systems of Calogero-Moser type admit higherrank 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 thequantum R -matrices R λab , 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 onlyon the set of identities (such as (1.10), (1.14), (1.5)) which can be verified separately.
1. Rational KZB equations
Besides the standard trigonometric and rational versions of the elliptic R -matrix there aremore sophisticated degenerations. In this paper we consider one of them [23] and show that it It is interesting if similar construction works for Toda-like models which can be obtained from the ellipticsystems by nontrivial (Inozemtsev) degenerations. R -matrix depends on the moduli of the elliptic curve τ . We notice that itsatisfies the heat equation (1.5) and treat this equation as Painev´e-Calogero property. In [18] itwas formulated in the following way: the Lax pair of the CM model satisfies also the monodromypreserving equations and describe the (higher rank) Painlev´e equations. We refer to (1.5) as theheat equation because this equation for the function φ ( ~ , z ) follows from the heat equation for ϑ -function 2 ∂ τ ϑ ( z | τ ) = ∂ 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, andthen described in [20] (see also [22, 23]) as an example of the Symplectic Hecke Correspondencefor integrable systems. Its rational analogue was suggested in [2] and was know to be freeof 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 equation2 ∂ τ Ξ = ∂ z Ξ . (1.16)The solution provides possibility for construction of the rational analogue of the KZB equations ˆ ∇ a ψ = 0 , ∇ a = ∂ z a + X c = a r τac ( z a − z c ) , ˆ ∇ τ ψ = 0 , ∇ τ = ∂ τ + 12 X b,c m τbc ( z b − z c ) , (1.17)where r and m are the terms of the expansion (1.4) and τ indicates the τ -deformation. Thesystem of KZ of KZB equations is known to be related to the quantum (and classical) CMmodels by the Matsuo-Cherednik construction [26, 9] (see also [27]). Recently relations betweenCM (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 versionof the Krichever’s ansatz for CM Lax pairs with spectral parameter [17]. Consider the followingblock matrix Lax operator L = ˜ N X a,b =1 ˜E ab ⊗ L ab (1.18)where ˜E ab is the standard basis of gl ˜ N and L ab = δ ab p a a ⊗ b + ν (1 − δ ab ) R ~ ab , R ~ ab = R ~ ab ( z a − z b ) . (1.19)When N = 1 the gl N R -matrix reduces to its scalar analogue – function φ ( z, ~ ) and we reproducethe answer from [17] for ˜ N -body CM system. Notice that the Planck constant of gl N R -matrix5lays here the role of the spectral parameter for gl ˜ N CM model. The corresponding M -operatoris given in (4.14). The Lax equation ∂ t L = [ L , M ] is equivalent to dynamics of ˜ N CM particles¨ z a = N ν X b = a ℘ ′ ( z a − z b ) . (1.20)In the same way the monodromy preserving equation ∂ τ L − ∂ ~ M = [ L , M ] leads to the Painlev´eequations ∂ τ z a = N ν X b = a ℘ ′ ( z a − z b ) . (1.21)The corresponding linear problem has the form( ∂ ~ + L )Ψ = 0 . (1.22)Let us also mention that the linear problem for the quantum version of CM modelˆ L Ψ = ΨΛ , ˆ L ab = δ ab ∂ z a a ⊗ b + ν (1 − δ ab ) R ~ ab (1.23)resembles very much the KZ connections from the first line of (1.17). Equation (1.23) (or(1.22) with ˆ L ) generalizes the first line of (1.17) in the same way as the Lax pairs with spectralparameter generalize those without it. We hope to clarify exact relations between R -matrixvalued linear problems and KZB equations in our future papers.Choosing elliptic, trigonometric or the rational R -matrix we describe the CM models similarlyto gl case [17]. Notice that the gl N R -matrix itself describes gl N integrable systems such asintegrable tops which are gauge equivalent to CM or RS models. Here we use gl N R -matrices asauxiliary spaces for derivation of gl ˜ N models. The next natural step is to get similar result forthe 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 theconstants can be either the spectral parameter in a ”classical-quantum” gl ˜ N system (of (1.19)type) or the Planck constant in a quantum gl N system or the relativistic deformation parameterin a classical relativistic gl N model (see [23]) . We hope that this can shed light on numerousdualities in integrable systems mentioned in [28], [25], [34], [11]. Acknowledgments.
The work was supported by RFBR grants 12-02-00594 (A.L. and M.O.) and 12-01-00482 (A.Z.). The work of A.L. was partially supported by AG Laboratory GU-HSE, RF governmentgrant, ag. 11 11.G34.31.0023 and by the Simons Foundation. The work of A.Z. was partially supportedby the D. Zimin’s fund ”Dynasty”, by the Program of RAS ”Basic Problems of the Nonlinear Dynamicsin Mathematical and Physical Sciences” Π19 and by grant RSCF 14-50-00005.
In this section we describe the sequence of steps which leads to the KZB equations [12] startingfrom integrable tops. As it was mentioned above, our consideration is independent on the choiceof particular top model. The basic element is the underlying quantum R -matrix [23]. Let us also remark that in [24] we have already found an R -matrix intermediate between the Belavin’s andthe Felders’ one. Her we use a different description. Presumably, the interrelation between different descriptionsis given by the Fourier-Mukai type transformation. In [23] we defined the relativistic integrable top by means of the quantum R -matrix. The gl N Lax matrix is given by L η ( z, S ) = tr ( R η ( z ) S ) , S = Res z =0 L η ( z, S ) , (2.1)where S = N P i,j =1 E ij S ij is the gl N -valued dynamical variable , and R η ( z ) is the correspondingquantum non-dynamical R -matrix. It satisfies the quantum Yang-Baxter equation (1.1). Thenon-relativistic limit ( η → L η ( z, S ) = η − tr SN N × N + L ( z, S ) + η M ( z, S ) + O ( η ) (2.2)is related to the classical limit ( ~ →
0) (1.4) via (2.1): L ( z, S ) = tr ( r ( z ) S ) , S = Res z =0 L ( z, S ) , (2.3) M ( z, S ) = tr ( m ( z ) S ) . (2.4)The quantity r ( z ) in (1.4), (2.3) is the classical r -matrix. It is skew-symmetric (1.8) r ( z ) = − r ( − z ) (2.5)and satisfies the classical Yang-Baxter equation:[ r ( z − w ) , r ( z )] + [ r ( z − w ) , r ( w )] + [ r ( z ) , r ( w )] = 0 . (2.6)As it was mentioned in [23] the matrices (2.3), (2.4) appear to be the Lax pair of the non-relativistic top. It means that the Lax equation ∂ t L ( z, S ) = [ L ( z, S ) , M ( z, S )] (2.7)is equivalent to equations of motion ∂ t S = [ S, J ( S )] , (2.8)where the inverse inertia tensor is given by the linear functional J ( S ) = M (0 , S ) . (2.9)The equations (2.8) are Hamiltonian with the Hamiltonian function H top ( S ) = 12 tr( S J ( S )) (2.10)and the Poisson-Lie brackets on gl ∗ N { S , S } = [ S , P ] (2.11)or { S ij , S kl } = δ il S kj − δ kj S il . { E ij , i, j = 1 ...N } is the standard basis in the fundamental representation of gl N : (E ij ) kl = δ ik δ jl . .2 Painlev´e–Calogero correspondence and non-autonomous tops The (classical) Painlev´e–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 monodromypreserving equations, which describe the higher rank Painlev´e equations in the elliptic form.Let us formulate here the Painlev´e–Calogero correspondence in the form of the quantumnon-dynamical R -matrix property. Definition 1
Suppose that the quantum R -matrix entering (2.1) depends on some additionalparameter τ : R ~ ,τ ( z ) = R ( z, ~ , τ ) . We say that the R -matrix satisfies the property of the”Painlev´e–Calogero correspondence” if the following relation holds : ∂ τ R ~ ,τ ( z ) = ∂ z ∂ ~ R ~ ,τ ( z ) . (2.12)Plugging the expansion (1.4) into (2.12) we get a set of relations. The first non-trivial is ∂ τ r τ ( z ) = ∂ z m τ ( z ) , (2.13)where r τ ( z ) = r ( z, τ ) is the classical r -matrix. An example of the R -matrix with this propertyis given by the Baxter-Belavin’s one [4] (see Appendix B). The parameter τ in this example equals τ ell / πı , where τ ell is the module of the underlying elliptic curve, and the property (2.13) is dueto the heat equation for the theta-functions2 ∂ τ ϑ ( z | τ ) = ∂ z ϑ ( z | τ ) . (2.14)From (2.13) and (2.3)-(2.4) it follows that ∂∂τ L τ ( z, S ) = ∂∂z M τ ( z, S ) , (2.15)where L τ ( z, S ) = L ( z, S, τ ), M τ ( z, S ) = M ( z, S, τ ). Therefore, we can define the monodromypreserving equations in time τd τ L τ ( z, S ) − ∂ z M τ ( z, S ) = [ L τ ( z, S ) , M τ ( z, S )] , S = S ( τ ) (2.16)( d τ = ddτ ) as the non-autonomous version of the integrable top’s equations of motion (2.8) : ∂ τ S = [ S, J τ ( S )] . (2.17)Indeed, the total derivative d τ L τ ( z, S ) contains both – the partial derivatives by explicit andimplicit dependence on τ : d τ L τ ( z, S ( τ )) = d τ tr ( r τ ( z ) S ) = tr (cid:16) ( ∂ τ r τ ( z )) S (cid:17) + tr (cid:16) r τ ( z ) ( ∂ τ S ) (cid:17) . (2.18)The first term is cancelled by ∂ z M τ ( z, S ) (2.15), and we get the same result as in (2.8) followingfrom the Lax equations (2.7). But this time it contains explicit dependence on τ via J τ ( S ) = M τ (0 , S ) . (2.19) Notice that the definition depends on the gauge choice. These models are no more integrable but can be treated as alternative description of (higher) Painlev´eequations. See [21] for the example of Painlev´e VI. H τ ( S ) = 12 tr( S J τ ( S )) (2.20)and the Poisson brackets are given by (2.11).Let us keep the notation ∂∂τ (but not ∂ τ ) for the partial derivative by only explicit dependenceon τ , i.e. ∂∂τ L τ ( z, S ( τ )) = tr (cid:16) ( ∂ τ r τ ( z )) S ( τ ) (cid:17) . (2.21) The phase space of the Gaudin model [6] is the direct product of n coadjoint orbits, i.e. ˜ N copies of S : S a ∈ gl N , a = 1 , ..., ˜ N with some fixed eigenvalues. Its Poisson structure { S a , S b } = δ ab [ S a , P ] (2.22)is the direct sum of (2.11). The Lax matrix has n simple poles at { z a , a = 1 , ..., ˜ N } with residues S a . It is given in terms of the top Lax matrix (2.3): L G ( z ) = ˜ N X a =1 L τ ( z − z a , S a ) = ˜ N X a =1 tr (cid:16) r τ ( z − z a ) S a (cid:17) . (2.23)Here we imply the existence of the deformation parameter τ (2.14)-(2.20) from the very beginningin order not to repeat (almost) the same notations with τ and without τ as we made for the topand its non-autonomous version.We consider the flows corresponding to Hamiltonians h a = − ˜ N X c = a tr ( S a L τ ( z a − z c , S c )) = − ˜ N X c = a tr (cid:16) r τ ( z a − z c ) S a S c (cid:17) (2.24)for a = 1 , ..., ˜ N and H = 12 ˜ N X b,c =1 tr (cid:0) S b M τ ( z b − z c , S c ) (cid:1) = 12 ˜ N X b,c =1 tr (cid:16) m τ ( z a − z c ) S b S c (cid:17) . (2.25)Notice that the terms coming from b = c in (2.25) are the top Hamiltonians H τ ( S c ) (2.20). Thefunctions (2.24)-(2.25) Poisson commute because (2.22) is equivalent to the classical exchangerelations { L G ( z ) , L G ( w ) } = [ L G ( z ) + L G ( w ) , r τ ( z − w )] . (2.26)The dynamics generated by (2.24)-(2.25) ∂ t a S b = − [ S b , L τ ( z a − z b , S a )] , b = a∂ t a S a = n X c = a [ S a , L τ ( z c − z a , S c )] (2.27)9or a = 1 , ..., ˜ N and ∂ t S a = [ S a , J τ ( S a )] + X c = a [ S a , M τ ( z a − z c , S c )] (2.28)possesses the Lax representations ∂ t d L G ( z ) = [ L G ( z ) , M G , d ] , d = 0 , ..., ˜ N (2.29)where M G , a ( z ) = − L τ ( z − z a , S a ) , a = 1 , ..., ˜ N (2.30)and M G , ( z ) = ˜ N X c =0 M τ ( z − z c , S c ) . (2.31) Similarly to the description of Painlev´e equation in the form of non-autonomous tops let us alsorepresent the Schlesinger system [30] as the non-autonomous Gaudin model.First, it follows from (2.23) and (2.30) that ∂∂z a L G ( z ) = ∂∂z M G , a ( z ) . (2.32)Secondly, it follows from (2.23), (2.31) and (2.15) that ∂∂τ L G ( z ) = ∂∂z M G , ( z ) . (2.33)Therefore, the monodromy preserving equations (or compatibility conditions for isomonodromicdeformations) ∂ z a L G ( z ) − ∂ z M G , a ( z ) = [ L G ( z ) , M G , a ( z )] (2.34)and ∂ τ L G ( z ) − ∂ z M G , ( z ) = [ L G ( z ) , M G , ( z )] (2.35)generate dynamics in time variables z a and τ . They have form form of non-autonomous versionsof the Gaudin’s one (2.27)-(2.28): ∂ z a S b = − [ S b , L τ ( z a − z b , S a )] , b = a∂ z a S a = ˜ N X c = a [ S a , L τ ( z c − z a , S c )] (2.36)for a = 1 , ..., ˜ N and ∂ τ S a = [ S a , J τ ( S a )] + X c = a [ S a , M τ ( z a − z c , S c )] . (2.37)The Hamiltonians (2.24)-(2.25) and the Poisson structure (2.22) are of the same form . In (2.32) and (2.33) the partial derivatives are taken with respect to explicit dependence on τ or z a (2.21). The elliptic case was considered in [19, 16, 10, 22]. .5 KZB equations The relation between KZB equations and the quantum monodromy preserving equations wasdescribed in [29] (see also [19, 16]). Let us formulate it using notations of (1.4) with the τ -deformation satisfying (2.13). The KZB equations have form: (cid:26) ˆ ∇ a ψ = 0 , ˆ ∇ τ ψ = 0 , (2.38)where ∇ a = ∂ z a + X c = a r τac ( z a − z c ) , (2.39) ∇ τ = ∂ τ + 12 X b,c m τbc ( z b − z c ) . (2.40)Here r τac and m τac are the operators acting by a -th and c -th components of U(gl N ) ⊗ ˜ N (the tensorproduct of ˜ N copies of the universal enveloping algebra). Recall that in classical integrablesystems (as well as in the Schlesinger systems) we used the fundamental representation ρ N ofgl N (see e.g. (2.3)-(2.4)): r τ ( z ) = ρ N ( r τ ( z )) = X i,j,k,l r τij,kl E ij ⊗ E kl ,m τ ( z ) = ρ N ( m τ ( z )) = X i,j,k,l m τij,kl E ij ⊗ E kl , (2.41)The algebra U(gl N ) ⊗ ˜ N can be considered as a quantization of the classical phase space with thePoisson structure (2.22). Indeed, let S a → ˆ S a : ˆ S aij := e aji , (2.42)where { e aij } : [ e aij , e akl ] = δ ab ( e ail δ kj − e akj δ il ) is the standard basis in the a -th component ofU(gl N ) ⊗ ˜ N . In this notation r τab = X i,j,k,l r τij,kl ( z a − z b ) e aij e bkl = X i,j,k,l r τij,kl ( z a − z b ) ˆ S aji ˆ S blk , (2.43) m τab = X i,j,k,l m τij,kl ( z a − z b ) e aij e bkl = X i,j,k,l m τij,kl ( z a − z b ) ˆ S aji ˆ S blk . (2.44)The fundamental representation is given by ρ N ( e aij ) = 1 ⊗ ... ⊗ ⊗ E ij ⊗ ⊗ ... ⊗
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 -thcomponents of an element of the tensor product V ⊗ ˜ N . The operator is represented by matrix of N ˜ N × N ˜ N size because it also contains (as factors) the product of identity operators for the restof components N c = a,b c . The residue of r -matrix is (up to factor N in (B.11)) the permutationoperator replacing a -th and b -th components of an element of the tensor product V ⊗ ˜ N to which ψ belongs. 11hen [ ˆ S a , ˆ S b ′ ] = δ ab [ ˆ S a ′ , P ′ ] , ˆ S a = N X i,j =1 ˆ S aij ρ N ( e aij ) (2.45)or [ ˆ S aij , ˆ S bkl ] = δ ab (cid:16) ˆ S akj δ il − ˆ S ail δ kj (cid:17) . The indices 0 , ′ in (2.45) are the notations for the compo-nents of (cid:16) ρ N (U(gl N ) ⊗ ˜ N ) (cid:17) ⊗ – tensor product of auxiliary spaces. To quantize the Hamiltonian(2.25) we also need to fix the ordering. Consider the symmetric (Weyl) ordering \ S aij S bkl = 12 (cid:16) ˆ S aij ˆ S bkl + ˆ S bkl ˆ S aij (cid:17) . (2.46)Then the KZB connections (2.39)-(2.40) are written in terms of the quantum versions of theclassical Hamiltonians h a and H (2.24)-(2.25):ˆ ∇ a = ∂ z a − ˆ h a , ˆ ∇ τ = ∂ τ + ˆ H . (2.47)In the same time the KZB equations (2.38) acquire the form of the non-stationary Schr¨odingerequations in times z , ..., z ˜ N and τ .The compatibility conditions of KZB equations (2.38)[ ˆ ∇ a , ˆ ∇ b ] = 0 (2.48)[ ˆ ∇ a , ˆ ∇ τ ] = 0 (2.49)are fulfilled identically . The first one (2.48) follows from the classical Yang-Baxter equation[ r ab , r bc ] + [ r bc , r ac ] + [ r ab , r ac ] = 0 , (2.50)where r ab = r τab ( z a − z b ). The set of identities underlying (2.48) consists of the property (2.13) ∂ τ r ab = ∂ z a m ab , (2.51)where m ab = m τab ( z a − z b ) and 12 [ r ab , m aa + m bb ] + [ r ab , m ab ] = 0 , (2.52)[ r ab , m bc ] + [ r ab , m ac ] + [ r ac , m ab ] + [ r ac , m bc ] = 0 . (2.53) Remark 1
One can get more identities relating r ab and m ab and higher order terms of expansion(1.4) from the Yang-Baxter equation (1.1) R ~ ,τab R ~ ,τac R ~ ,τbc = R ~ ,τbc R ~ ,τac R ~ ,τab . The first non-trivialidentity is (2.50). The next one is [ r ab , m ac ] + [ m ab , r ac ] + [ r ab , m bc ] + [ m ab , r bc ] + [ r ac , m bc ] + [ m ac , r bc ]++ r ab r ac r bc − r bc r ac r ab = 0 , (2.54) where r ab = r τab ( z a − z b ) , m ab = m τab ( z a − z b ) . This statement was verified directly in different cases. See [13, 16] for elliptic examples. Rational non-autonomous tops and KZB equations
The rational top was first studied for small rank cases in [32] by degenerating the elliptic Laxmatrix [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) r -matrix as follows: r ( z ) = ∂L ( z, S ) ∂S , S = Res z =0 L ( z ) . (3.1)In [23] this relation was extended to the quantum R -matrix by proceeding to the relativistictop: R ~ ( z ) = ∂L ~ ( z, S ) ∂S , S = Res z =0 L ~ ( z ) , (3.2)where the classical Lax matrix L ~ ( z ) depends on the constant ~ playing the role of the rela-tivistic 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 L η ( z, S ) = g ( z ) L RS ( z, η ) g − ( z ) (3.3)can be written explicitly in terms of the RS particles coordinates q j : g ( z, q ) = Ξ( z, q ) D − ,where Ξ( z, q ) = ( z + q j ) ̺ ( i ) ,̺ ( i ) = i − i ≤ N − ̺ ( N ) = N . (3.4) τ -deformation of quantum rational R -matrix Our aim is to construct τ -dependent R -matrix satisfying the Painlev´e-Calogero property (2.12)starting from the τ -independent one (3.2). The answer follows from (3.8) (see below). Itappears that the deformation of the Yang’s rational R -matrix suggested in [23] admits this kindof deformation similarly to the elliptic case. The idea is to deform first Ξ( z ) (3.4). Let us findΞ( z, q | τ ) satisfying the heat equation2 ∂ τ Ξ( z | τ ) = ∂ z Ξ( z | τ ) (3.5)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) .The solution of (3.5)-(3.6) is given byΞ( z | τ ) = exp (cid:16) τ ∂ z (cid:17) Ξ( z ) (3.7)or Ξ( z | τ ) = exp (cid:16) τ T (cid:17) Ξ( z ) , (3.8) The explicit from of L RS ( z, η ) as well as diagonal matrix D ij = δ ij Q k = i ( q i − q k ) is not used in what follows. It can be also proved directly by using explicit answer for the quantum R -matrix [23]. T is the nilpotent operator representing the action of ∂ z on the N -dimensional column-vector (1 , z, z , ..., z N − , z N ) T . It is N × N matrix with elements T ij = (cid:26) j ( j + 1) δ i − ,j , i < N ,j ( j + 1) δ i − ,j , i = N . (3.9)For example, for N = 2 , , T N =2 = ! , T N =3 = , T N =4 = . (3.10)Denote T := exp (cid:16) τ T (cid:17) , (3.11)i.e. Ξ( z | τ ) = T Ξ( z | N = 2 , , T equals T N =2 = τ ! , T N =3 = τ , T N =4 = τ τ τ . (3.12)It follows from (3.1)-(3.3) and (3.8) that τ -deformation of R -matrix is given by the followinggauge transformation: R ~ ( z | τ ) = T T R ~ ( z | T − T − (3.13)written in terms of (3.11). See Appendix A for explicit answer in gl case. It follows from (3.13) that r τab ( z a − z b ) = T a T b r ab ( z a − z b ) T − a T − b ,m τab ( z a − z b ) = T a T b m ab ( z a − z b ) T − a T − b . (3.14)Then the condition (2.13) is fulfilled as well as (2.51) for (2.43)-(2.44).The Lax pair (2.3)-(2.4) is transformed by not only the gauge transformation since the residue S also changes. From (2.3)-(2.4) and (3.14) we have L ( z, S, τ ) = T L ( z, T − S T , T − , (3.15) M ( z, S, τ ) = T M ( z, T − S T , T − . (3.16)Let us summarize the results: Proposition 3.1
The τ -deformed quantum R -matrix (3.13) satisfies the Painlev´e-Calogero pro-perty (2.12). Proposition 3.2
The τ -deformed quantum r and m -matrices (3.14) define the KZB equations(2.38), i.e. the corresponding KZB connections ∇ a (2.39) and ∇ τ (2.40) are compatible (2.48),(2.49). The proof is direct. Below we give explicit examples of τ -deformations in the rational case.14 .3 Example: gl case Quantum R -matrix (satisfying (2.12)): R ~ ,τ ( z ) = ~ − + z − − ~ − z ~ − z − − ~ − z z − ~ − − ( z + ~ )( z + z ~ + ~ + 4 τ ) ~ + z ~ + z ~ − + z − (3.17)Classical r -matrix r τ ( z ) = z − − z z − − z z − − z − zτ z z z − (3.18)and m -matrix (the next term of expansion of (3.17) in ~ ) satisfying (2.13): m τ ( z ) = − − − z − τ (3.19)The following additional relation holds: − ∂ z r τ ( z ) = P z − m τ ( z ) + 12 m τ (0) . (3.20)Non-autonomous top Lax pair and Hamiltonian: L ( z, S | τ ) = 1 z S − z S S S − z ( S − S ) − z S − z τ S S + z S (3.21) M ( z, S | τ ) = − S S − S + 2 z S + 4 τ S − S (3.22) H ( S, τ ) = − S ( S − S ) − τ S . (3.23)The Gaudin (or Schlesinger) Hamiltonians: h a = ˜ N X c = a h a,c , h a,c = − tr ( r τ ( z a − z c ) S a S c ) = (3.24) − tr( S a S c ) z a − z c + ( z a − z c ) (cid:16) S a ( S c − S c ) + S c ( S a − S a ) + 4 τ S a S c (cid:17) + ( z a − z c ) S a S c ,h = 12 ˜ N X b,c =1 tr (cid:0) S b M ( z b − z c , S c ) (cid:1) = − n X b,c =1 S b ( S c − S c ) + S b S c (cid:2) ( z b − z c ) + 2 τ (cid:3) . (3.25)Some similar formulae for gl case are given in the Appendix A.15 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 sim-ilar 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: 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 ) , (4.1)where z ab = z a − z b . Let us formulate its noncommutative analogue. Proposition 4.1
The Belavin’s R -matrix (B.8) satisfies the following relation: R ~ ab R ~ ′ bc = R ~ ′ ac R ~ − ~ ′ ab + R ~ ′ − ~ bc R ~ ac , (4.2) where R ~ ab = R ~ ab ( z a − z b ) .Proof: Denote by T aα the basis element T α (B.1) standing on the a -th place in the tensor product1 ⊗ ... ⊗ ⊗ T α ⊗ ⊗ ... ⊗
1. It follows from the definition (B.8) and the multiplication rule (B.3)that R ~ ab R ~ ′ bc = X α,β T aα T bβ − α T c − β κ − α,β ϕ ~ α ( z a − z b ) ϕ ~ ′ β ( z b − z c ) , (4.3) R ~ ′ ac R ~ − ~ ′ ab = X α,β T aα T bβ − α T c − β κ β,α − β ϕ ~ ′ β ( z a − z c ) ϕ ~ − ~ ′ α − β ( z a − z b ) , (4.4) R ~ ′ − ~ bc R ~ ac = X α,β T aα T bβ − α T c − β κ β − α,α ϕ ~ ′ − ~ β − α ( z b − z c ) ϕ ~ α ( z a − z c ) , (4.5)Notice that κ − α,β = κ β,α − β = κ β − α,α due to (B.4). Then the statement (4.2) follows from (4.1),where x = ~ + ω α and y = ~ ′ + ω β . (cid:4) Proposition 4.2
The quantum Yang-Baxter equation (1.1) follows from (4.2), the property(1.7) and unitarity condition (1.2).Proof:
Consider (4.2) for a, b, c = 1 , , ~ ′ = ~ / R ~ R ~ / = R ~ / R ~ / + R − ~ / R ~ Replace ~ → ~ and multiply this relation by R ~ from the left: R ~ R ~ R ~ = R ~ R ~ R ~ − R ~ R − ~ R ~ . (4.6)Similarly, consider (4.2) for a, b, c = 1 , , ~ ′ = ~ /
2, replace ~ → ~ and multiply theobtained relation by R ~ from the right: R ~ R ~ R ~ = R ~ R ~ R ~ − R − ~ R ~ R ~ . (4.7)16he r.h.s of (4.6) equals r.h.s of (4.7) due to the property (1.7) and unitarity condition (1.2). (cid:4) Consider the derivative of (4.2) with respect to z b : R ~ ab F ~ ′ bc − F ~ ab R ~ ′ bc = F ~ ′ − ~ bc R ~ ac − R ~ ′ ac F ~ − ~ ′ ab , (4.8)where F ~ ab ( z ) = ∂ z R ~ ab ( z ). The function F ~ ab ( z ) has no singularities at ~ = 0. Therefore, we canput ~ = ~ ′ in (4.8). This gives R ~ ab F ~ bc − F ~ ab R ~ bc = F bc R ~ ac − R ~ ac F ab , (4.9)The latter equation is analogue of the following identity φ ( x, z ab ) f ( x, z bc ) − f ( x, z ab ) φ ( x, z bc ) = φ ( x, z ac )( ℘ ( z ab ) − ℘ ( z bc )) ,f ( x, z ab ) = ∂ z a φ ( x, z ab ) (4.10)underlying Lax equations (integrability) of the Calogero-Moser model [7, 17]. R -matrix valued linear problem for Calogero-Moser model Consider the eigenvalue problem L Ψ = ΨΛ (4.11)for the following block matrix operator L = ˜ N X a,b =1 ˜E ab ⊗ L ab , (4.12)where ˜E ab is the standard basis of gl ˜ N and L ab = δ ab p a a ⊗ b + ν (1 − δ ab ) R ~ ab , R ~ ab = R ~ ab ( z a − z b ) . (4.13)It is worth mentioning that in gl case ( N = 1) this operator is the Krichever’s Lax matrixwith spectral parameter for the Calogero-Moser model [17]. The eigenvalue matrix consists ofvectors ψ , ..., ψ ˜ N . In the case of quantum CM model ( p a → ∂ z a ) equation (4.11) should havewell defined limit ~ → ψ = ... = ψ ˜ N = ψ . Alternatively,one can quantize the model as p a → ∇ a . At the level of classical mechanics and N = 1 thedifference between ∂ z a and ∇ a is given by the canonical map p a → p a + ν P c = a E ( z a − z c ).The spectral parameter in (4.13) is ~ - the Planck constant. The M -operator is defined asfollows: M ab = νδ ab d a + ν (1 − δ ab ) F ~ ab + νδ ab F , (4.14)where F ~ ab = ∂ z a R ~ ab ( z a − z b ) , (4.15) d a = − ˜ N X c : c = a F ac , F ac = F ~ ac | ~ =0 , (4.16)17 = 12 ˜ N X b,c : b = c F bc = ˜ N X b,c : b>c F bc . (4.17) M -operator (4.14) is also straightforward generalization of the one proposed in [17] except thelast term F . The latter is not needed in N = 1 case because in this case it is proportional tothe identity matrix. Proposition 4.3
The linear problem ( ∂ t + M )Ψ = 0 , M = ˜ N X a,b =1 ˜E ab ⊗ M ab (4.18) is compatible with (4.11). The compatibility condition is equivalent to dynamics of gl ˜ N Calogero-Moser model.Proof:
The compatibility condition is the Lax equation ∂ t L = [ L , M ]. For brevity sake let usdenote L = p + R , M = d + F + F . The commutator equals[ L , M ] = [ p, F ] + [ R, d ] + [
R, F ] + [ R, F ] . (4.19)The term [ p, F ] is cancelled by ∂ t R (due to ˙ z a = p a ).Consider the off-diagonal block ac . It has three inputs from1. from [ R, F ]: P b = a,c R ~ ab F ~ bc − F ~ ab R ~ bc ( . ) = P b = a,c F bc R ~ ac − R ~ ac F ab ;2. from [ R, d ]: − R ~ ac P b = c F cb + P b = a F ab R ~ ac ;3. from [ R, F ]: [ L ac , F ].The sum of the inputs equals zero. We used that F ab = F ba (due to F ab = ∂ z a r ab ( z a − z b )).On a diagonal block we get equations of motion:˙ p a = ν X b = a R ~ ab F ~ ba − F ~ ab R ~ ba ( . ) = N ν X b = a ℘ ′ ( z a − z b ) . (4.20) (cid:4) It is natural to expect that the same receipt works for other root systems (not only gl N ) aswell, 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 : L ab = (1 − δ ab ) R ~ ab . We conjecture that :˜tr(( L ) k +1 ) aa = ˜ N X b ,...,b k =1 R ~ ab ... R ~ b k a = 1 ⊗ ... ⊗ ˜ N ˜ N X b ,...,b k =1 Φ ~ ( z a − z b ) ... Φ ~ ( z b k − z a ) , (4.21)where ˜tr denotes the trace over g l ˜ N component of L and the sums do not contain zero arguments(i.e. b = a , b = b , ... , b k = a ). Relation (4.21) means that traces of L (4.12)-(4.13) providesthe Hamiltonians of the gl ˜ N Calogero-Moser model (where z a are coordinates of particles). The proof will be given elsewhere. k = 1 (4.21) follows from the unitarity condition: X b R ~ ab R ~ ba = 1 a ⊗ b X b Φ ~ ( z a − z b )Φ ~ ( z b − z a ) = N ℘ ( N ~ ) − N ℘ ( z a − z b ) . (4.22)For k = 2 and ˜ N = 3 we have R ~ ab R ~ bc R ~ ca + R ~ ac R ~ cb R ~ ba = 1 a ⊗ b ⊗ c (cid:0) Φ ~ ( z ab )Φ ~ ( z bc )Φ ~ ( z ca ) + Φ ~ ( z ac )Φ ~ ( z cb )Φ ~ ( z ba ) (cid:1) (4.23)( z ab = z a − z b ) or, in particular R ~ R ~ R ~ + R ~ R ~ R ~ = 1 ⊗ ⊗ (cid:0) Φ ~ ( z )Φ ~ ( z )Φ ~ ( z ) + Φ ~ ( z )Φ ~ ( z )Φ ~ ( z ) (cid:1) (4.24)The function in the r.h.s. of (4.24) equalsΦ ~ ( z )Φ ~ ( z )Φ ~ ( z ) + Φ ~ ( z )Φ ~ ( z )Φ ~ ( z ) = − N ℘ ′ ( ~ ) in elliptic case , / ~ in rational case . (4.25) Consider the unitarity condition R ~ ab R ~ ba = Φ ~ ( z ab )Φ ~ ( z ba ). Its expansion in the ~ order gives r ab − m ab = 1 a ⊗ b N ℘ ( z a − z b ) . (4.26)Here r ab = r τab ( z a − z b ), m ab = m τab ( z a − z b ). Next, consider (4.23)-(4.25). In the ~ order itprovides the following relation between r and m matrices:[ r ab , r bc ] + + [ r bc , r ca ] + + [ r ab , r ca ] + + 2( m ab + m bc + m ac ) = 0 , (4.27)where [ ∗ , ∗ ] + is the anticommutator [ A, B ] + := AB + BA . Using the classical Yang-Baxterequation [ r ab , r ac ] + [ r ac , r bc ] + [ r ab , r bc ] = 0 (4.28)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).In the light of (4.26) the expansion R ~ ( z ) = ~ − + r ( z ) + ~ m ( z ) is similar to the expansion(B.9). Indeed, using (4.26) we have R ~ ab ( z ) = 1 ~ a ⊗ b + r ab + ~ m ab + ... = 1 ~ a ⊗ b + r ab + ~ (cid:0) r ab − N ℘ ( z ab ) (cid:1) + ... . (4.31)In the same time (4.27) can be re-written as( r ab + r bc + r ca ) = 1 a ⊗ b ⊗ c N ( ℘ ( z a − z b ) + ℘ ( z b − z c ) + ℘ ( z c − z a )) (4.32)using (4.26). It is an analogue of the elliptic functions identity( E ( z a − z b ) + E ( z b − z c ) + E ( z c − z a )) = ℘ ( z a − z b ) + ℘ ( z b − z c ) + ℘ ( z c − z a ) . (4.33)19 .4 Identities for KZB equations It follows from (4.26) that [ r ab , m ab ] = 0 . (4.34)This is equation (2.52) written in the fundamental representation (in this case m aa are somescalar operators). Equation (2.53) keeps its form in the fundamental representation. Let usprove it. Proposition 4.4
The following identities holds true: [ r ab , m ac + m bc ] + [ r ac , m ab + m bc ] = 0 , (4.35)[ r bc , m ab − m ac ] + r ab r bc r ac − r ac r bc r ab = 0 . (4.36) The first one underlies the compatibility of KZB equations. See (2.53).Proof:
Consider the Yang-Baxter equation R ~ ca R ~ cb R ~ ab = R ~ ab R ~ cb R ~ ca in the ~ order. It is given bythe sum of (4.35) and (4.36). Consider also (4.23) in the ~ order. It is given by the differenceof (4.35) and (4.36). (cid:4) The identities (4.26)-(4.27) allow also to get the following Matsuo-Cherednik’s like [26, 9]statement:
Proposition 4.5
Consider the gl N KZB equations for ˜ N punctures: ∇ i ψ = 0 , ∇ i = ∂ i + ν X j : j = i r τij ( z i − z j ) , (4.37) for i = 1 , ..., ˜ N and ∇ τ ψ = 0 , ∇ i = ∂ τ + ν X j = k m τjk ( z j − z k ) , (4.38) where r τij and m τij are the coefficients of the expansion (1.4) and ν is a free constant. Then theconformal block satisfies the following equation: (cid:18) ˜ N ν∂ τ + 12 ∆ (cid:19) ψ = − ν X i 12 ˜ N ν X j m τjj + ν N X i Consider the linear problem ( ∂ ~ + L )Ψ = 0 , ( ∂ τ + M )Ψ = 0 , (4.44) where L and M are defined by (4.13), (4.14). The compatibility condition ∂ τ L − ∂ ~ M = [ L , M ] (4.45) is equivalent to gl ˜ N Painlev´e equations ∂ τ z a = N ν X b = a ℘ ′ ( z a − z b | τ ) . (4.46)The proof repeats the one for the Proposition 4.3. Additionally one should use the property(2.12) of the Painlev´e-Calogero correspondence.21 Appendix A: gl (rational) case Undeformed gl quantum R -matrix: R ~ ( z ) = (A.1) ~ − + z − ~ − ~ + 3 z ~ + 2 z − ~ − z ~ − − z − ~ + 2 z z + 3 z ~ + 2 ~ + 3 z ~ − ~ − z ~ − z − ~ − z ~ − z − ~ − z z − z + 3 z ~ + 2 ~ + 3 z ~ z + 3 z ~ + ~ − ~ + 3 z ~ + 3 z ~ + 2 z + 3 z ~ + 3 z ~ z − ~ − z ~ + 3 z ~ − z + ~ z − − ~ − z z − ~ − ~ − + z − − z ~ − z − ~ ~ − z − − ~ − z − ~ − z + 3 ~ + 3 z ~ z − − ~ − z ~ + 3 ~ − z ~ − z − ~ − z − z ~ − z ~ z + 3 ~ − ~ + z z + 3 ~ ~ − + z − The τ -deformation generated by (3.13) with T N =3 from (3.12) yields R ~ ( z | τ ) = R ~ ( z | 0) + δR ~ ,τ ( z ) ,δR ~ ,τ ( z ) = (A.2) = 3 τ × ~ + 2 z − − − ~ + 2 z z + ~ )(2 z + z ~ + 2 ~ + 3 τ ) 3 z − ~ − z + 3 ~ − z − ~ tau -deformed r and m -matrix: r τ ( z ) = (A.3) z − z − τ + 2 z − z − z z − − z − z z − z + 6 τ z − z − τ − z − τ z − − τ − z − z z − − z − z + 6 τ z z + 3 τ − z + 3 τ z − − τ z + 12 τ z + 2 z τ z + 3 z − z − τ z − z − z − τ z z z z z − m τ ( z ) = z − − z + 6 τ − z − z − z − − z + 6 τ z z τ + 18 τ z + 3 z z − z − z − τ The Lax pair for τ -deformed (autonomous or non-autonomous) rational top can be found from(2.3)-(2.4). It describes dynamics generated by the following Hamiltonian: H = S − S S + 3 S S − S S + 6 τ S S − τ S + 9 τ S . (A.4) R -matrix Consider the following basis in gl N (some details can be found in [22]): T a = T a a = exp (cid:16) πıN a a (cid:17) Q a Λ a , (B.1)where a , a ∈ Z N and Q kl = δ kl exp( 2 πiN k ) , Λ kl = δ k − l +1=0 mod N , k, l = 1 , ..., N . (B.2)The multiplication is defied by the following relation: T a a T b b = κ a,b T a + b ,a + b , (B.3)23here κ a,b = exp (cid:16) πıN ( b a − b a ) (cid:17) . (B.4)For the odd Riemann theta function ϑ ( z ) = ϑ ( z | τ ) φ ( z, u ) = ϑ ′ (0) ϑ ( u + z ) ϑ ( z ) ϑ ( u ) , (B.5) ϕ a ( z ) = exp(2 πız∂ τ ω a ) φ ( z, ω a ) , ω a = a + a τN , (B.6) ϕ ~ a ( z ) = exp(2 πız∂ τ ω a ) φ ( z, ω a + ~ ) . (B.7)The Belavin’s R -matrix [5] can be defined as R ~ ( z ) = X α ∈ Z N × Z N ϕ ~ α ( z ) T α ⊗ T − α . (B.8)The local behavior of φ ( ~ , z ) (B.5) near ~ = 0 is give by φ ( ~ , z ) = 1 ~ + E ( z ) + ~ (cid:0) E ( z ) − ℘ ( z ) (cid:1) + ... , (B.9)where E ( z ) = ∂ z log ϑ ( z ) (B.10)and ℘ ( z ) is the Weierstrass ℘ -function. Therefore, expansion (1.4) of (B.8) gives r ( z ) = E ( z ) 1 ⊗ X α =0 ϕ α ( z ) T α ⊗ T − α , (B.11) m ( z ) = E ( z ) − ℘ ( z )2 1 ⊗ X α =0 f α ( z ) T α ⊗ T − α , (B.12)where f a ( z ) = exp(2 πız∂ τ ω a ) ∂ u φ ( z, u ) | u = ω α . (B.13)The function Φ entering the unitarity condition (1.2) equalsΦ ~ ( z ) = N φ ( N ~ , z ) . (B.14)Notice that the residue of the R -matrix (B.8) at z = 0 equals N P , where P = N − P a T a ⊗ T − a is the permutation operator.It follows from the heat equation for function (B.7) ∂ τ ϕ ~ a ( z ) = ∂ z ∂ ~ ϕ ~ a ( z ) (B.15)that the R -matrix (B.8) satisfies the property (2.12): ∂ τ R ~ ab = ∂ z ∂ ~ R ~ ab . (B.16)24 eferences [1] A. Alexandrov, S. Leurent, Z. Tsuboi, A. Zabrodin, Nucl. Phys. 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