On R-Matrix Valued Lax Pairs for Calogero-Moser Models

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In this paper we consider the Calogero-Moser models [11] and their generalizations of diﬀerent types. The Hamiltonian of the elliptic classical slN model

N p2 N H = i ν2 ℘(q q ) (1.1) 2 − i − j i=1 i>j X X together with the canonical Poisson brackets

p , q = δ , p ,p = q , q =0 . (1.2) { i j} ij { i j} { i j} provides equations of motion for N-particle dynamics:

N 2 ′ q˙i = pi , q¨i = ν ℘ (qik) . (1.3) 6 kX:k=i All variables and the coupling constant ν are assumed to be complex numbers. Equations (1.3) can be written in the Lax form. The Krichever’s Lax pair with spectral parameter [30] reads as follows3:

N L(z)= E L (z) , L (z)= δ p + ν(1 δ )φ(z, q ) , q = q q , (1.4) ij ij ij ij i − ij ij ij i − j i,j=1 X N N M (z)= νd δ + ν(1 δ )f(z, q ) , d = E (q )= f(0, q ) , (1.5) ij i ij − ij ij i 2 ik − ik 6 6 kX:k=i kX:k=i i.e. the Lax equations L˙ (z) H, L(z) = [L(z), M(z)] (1.6) ≡{ } are equivalent to (1.3) identically in z. The deﬁnitions and properties of elliptic functions entering (1.1)-(1.5) are given in the Appendix. The proof is based on the identity (A.14) written as φ(z, q )f(z, q ) f(z, q )φ(z, q )= φ(z, q )(f(0, q ) f(0, q )) . (1.7) ab bc − ab bc ac bc − ab and φ(z, q )f(z, q ) f(z, q )φ(z, q )= ℘′(q ) . (1.8) ab ba − ab ba ab These are particular cases of the genus one Fay identity (A.13)

φ(z, q )φ(w, q )= φ(w, q )φ(z w, q )+ φ(w z, q )φ(z, q ) . (1.9) ab bc ac − ab − bc ac The model (1.1)-(1.3) is included into a wide class of Calogero-Moser models associated with root systems [38]. The corresponding Lax pairs with spectral parameter were found in [15, 9]. In particular, for the BCN root system described by the Hamiltonian

1 N N N N H = p2 ν2 (℘(q q )+ ℘(q + q )) µ2 ℘(2q ) g2 ℘(q ) (1.10) 2 a − a − b a b − a − a a=1 a=1 a=1 X Xa

The Lax pair (1.4)-(1.5) of the slN model (1.1)-(1.3) has the following generalization [32] (the R-matrix-valued Lax pair):4

N (z)= E (z) , (z)=1⊗N δ p + ν(1 δ )R z (q ) (1.12) L ij ⊗Lij Lij N˜ ij i − ij ij ij i,j=1 X N (z)= νd δ + ν(1 δ )F z(q )+ νδ 0 , d = F 0(q ) , (1.13) Mij i ij − ij ij ij ij F i − ik ik 6 kX:k=i where F z(q)= ∂ R z (q), F 0(q)= F z(q) = F 0( q) (A.21) and ij q ij ij ij |z=0 ji − N 1 N 0 = F 0 (q )= F 0 (q ) . (1.14) F km km 2 km km k>mX k,mX=1 It has block-matrix structure5. The blocks are enumerated by i, j = 1...N as matrix elements in (1.4). Each block of (z) is some GL(N˜)-valued R-matrix in fundamental representation, L ˜ acting on the N-th tensor power of N˜-dimensional vector space = (CN )⊗N . So the size of H ⊗N each block is dim dim , and dim = N˜ N , i.e. (z) Mat Mat . We will refer to H × H H L ∈ N ⊗ N˜ Mat component as auxiliary space, and to Mat⊗N = ⊗2 – as ”quantum” space since is the N N˜ ∼ H H Hilbert space of GL(N˜) spin chain (in fundamental representation) on N sites.

An R-matrix Rij acts trivially in all tensor components except i, j. It is normalized in a way that for N˜ = 1 it is reduced to the Kronecker function φ(z, qij) (A.6) [48]. For instance, in one of the simplest examples Rij is the Yang’s R-matrix [50]:

1 1 NP˜ Rη (q)= ⊗ + 12 , (1.15) 12 η q

where P12 is the permutation operator (A.5). In general (and as a default) case Rij is the Baxter-Belavin [4, 5] elliptic R-matrix (A.18). The properties of this R-matrix are very similar to those of the function φ(z, q). The key equation for Rij (which is needed for existence of the R-matrix-valued Lax pair) is the associative Yang-Baxter equation [19]

Rz Rw = Rw Rz−w + Rw−zRz , Rz = Rz (q q ) . (1.16) ab bc ac ab bc ac ab ab a − b It is a matrix generalization of the Fay identity (1.9), and it is fulﬁlled by the Baxter-Belavin R-matrix [40]. The degeneration of (1.16) similar to (1.7) is of the form:

Rz F z F z Rz = F 0 Rz Rz F 0 . (1.17) ab bc − ab bc bc ac − ac ab 4Equations of motion following from (1.12)-(1.13) contain the coupling constant Nν˜ instead of ν in (1.1), (1.3). 5The operator-valued Lax pairs with a similar structure are known [27, 23, 24, 6, 28]. We discuss it below.

3 It underlies the Lax equations for the Lax pair (1.12)-(1.13). The last term 0 in (1.13) is not needed in (1.5) since for N˜ = 1 it is proportional to the identity N N matrix.F But it is important for N˜ > 1 since it changes the order of R and F 0 in the r.h.s.× of (1.17). Namely,

[Rz , 0]+ Rz F z F z Rz = Rz F 0 F 0 Rz , a = c. ac F ab bc − ab bc ac bc − ab ac ∀ 6 (1.18) 6 6 6 bX=a,c Xb=c Xb=a This identity provides cancellation of non-diagonal blocks in the Lax equations. See [40, 41, 33, 34, 52] for diﬀerent properties and applications of R-matrices of the described type. Here we need two more important properties. These are the unitarity

Rz (q )Rz (q )=1 1 N˜ 2(℘(Nz˜ ) ℘(q )) (1.19) 12 12 21 21 ⊗ − 12 and the skew-symmetry Rz (q)= R−z( q) . (1.20) ab − ba − On the one hand these properties are needed for the Lax equations since they lead to F 0 (q)= F 0 ( q) (1.21) ab ba − and to the analogue of (1.8) (obtained by diﬀerentiating the identity (1.19))

Rz F z F z Rz = N˜ 2℘′(q ) , (1.22) ab ba − ab ba ab which provides equations of motion in each diagonal block in the Lax equations. On the other hand, together with (1.19) and (1.20) the associative Yang-Baxter equation η η η η η η leads to the quantum Yang-Baxter equation RabRacRbc = RbcRacRab. In this respect we deal with the quantum R-matrices satisfying (1.16), (1.19), (1.20), and the Planck constant of R- matrix plays the role of the spectral parameter for the Lax pair (1.12)-(1.13). In trigonometric case the R-matrices satisfying the requirements include the standard GL(N˜) XXZ R-matrix [31] and its deformation [13, 1] (GL(N˜) extension of the 7-vertex R-matrix). In the rational case the set of the R-matrices includes the Yang’s one (1.15) and its deformations [13, 46] (GL(N˜) extension of the 11-vertex R-matrix). The aim of the paper is to clarify the origin of the R-matrix-valued Lax pairs and examine some known constructions, which work for the ordinary Lax pairs. First, we study extensions of (1.12)-(1.13) to other root systems. More precisely, we propose R-matrix-valued extensions of the D’Hoker-Phong Lax pairs for (untwisted) Calogero-Moser models associated with classical root systems and BCN (1.10). The auxiliary space in these cases is given by Mat2N or Mat2N+1 because such root systems are obtained from sl2N or sl2N+1 cases by discrete reduction. There are two natural possibilities for arranging tensor components of the quantum spaces. The ﬁrst one is to keep 2N +1 (or 2N) components of the quantum spaces in the reduced root system. The second – is to leave only N (or N + 1) components. We study both cases. Next, we proceed to quantum Calogero-Moser models [11, 39, 12]. To some extent they are described by quantum analogue of the Lax equations (1.6) [47, 10]:

[Hˆ , Lˆ(z)] = ~ [Lˆ(z), M(z)] , (1.23) where Hˆ is the quantum Hamiltonian (it is scalar in the auxiliary space), Lˆ(z) is the quantum Lax matrix and ~ is the Planck constant. The operators Hˆ and Lˆ(z) are obtained from the

4 ~ classical (1.1) and (1.4) by replacing momenta pi with ∂qi , and the coupling constant in the Hamiltonian acquires the quantum correction. We verify if the obtained R-matrix-valued Lax pairs are generalized to quantum case in a similar way. It appears that (besides the slN case) only models associated with SO type root systems are generalized. As a result we show6

Proposition 1.1 The D’Hoker-Phong Lax pairs for (untwisted) classical Calogero-Moser models associated with classical root systems and BCN admit R-matrix-valued extensions with additional constraints:

– for the coupling constants in CN and BCN cases;

– for the size of R-matrix (N˜ =2) in BN and DN cases.

The latter cases are directly generalized to quantum Lax equations, while the CN and BCN cases are not. The AN Lax pair is generalized to the quantum case straightforwardly without any restrictions.

The Calogero-Moser models [11, 39] possess also spin generalizations [20]. Its Lax description is known at classical [8] and quantum levels [27, 23, 24, 6, 28]. It is important to note that for the quantum Calogero-Moser models with spin the quantum Lax pairs have the same operator- valued (tensor) structure as in (1.12)-(1.13). The term analogues to 0 (1.14) is treated as a part of the quantum Hamiltonian, describing interaction of spins. WeF explain (in Section 3) how the R-matrix-valued Lax pairs generalize (and reproduce) the previously known results.

Finally, we discuss an origin of the R-matrix-valued Lax pairs (for slN case (1.12)-(1.13) with ˜ GLN˜ R-matrices) by relating them to Hitchin systems on SL(NN)-bundles over elliptic curve. Originally systems of this type were derived by A. Polychronakos from matrix models [43] and later were described as Hitchin systems with nontrivial characteristic classes [51, 35]. It is also known as the model of interacting tops since it is Hamiltonian (or equations of motion) are treated as interaction of N SL(N˜)-valued elliptic tops. The relation between the R-matrix-valued Lax pairs and the interacting tops comes from rewriting the Lax equation for (1.12)-(1.13) in the form H, + [ν 0, (z)] = [ (z), ¯ (z)] , (1.24) { L} F L L M where in contrast to (1.13) ¯ does not include the 0 term (1.14). In this respect the R- matrix-valued Lax pair is ”half-quantum”:M the spin variablesF are quantized in the fundamental representation, while the positions and momenta of particles remain classical. The 0 term in this treatment is the (anisotropic) spin exchange operator. We will show that theF classical analogue for such spin exchange operator appear in the above mentioned Hitchin systems. Alternatively, the result is formulated as follows.

Proposition 1.2 The quantum Hamiltonian Hˆ tops of the model of N interacting SL(N˜) elliptic tops (with spin variables being quantized in the fundamental representation) coincides with the sum of the quantum Calogero-Moser Hamiltonian (1.23) and 0-term (1.14) F Hˆ tops = Hˆ CM + ~ν 0 +1⊗N const (1.25) F N˜ up to a constant proportional to identity matrix in End( ) and redeﬁnition of the coupling constants. H

We prove this Proposition in the end of Section 4. 6Some more details are given in the Conclusion.

5 2 Classical root systems

In [15] the following Lax pair was found for the model (1.10):

′ ′ ′ P + A B1 C1 A + d B1 C1 T ′ ′T ′ L = B2 P + A C2 M = B2 A + d C2 (2.1)  T − T   ′T ′T  C2 C1 0 C2 C1 d0     where A, B, P, D Mat , C ,C are columns of length N and ∈ N 1 2 P = δ p , A = ν(1 δ )φ(z, q q ) , ab ab a ab − ab a − b (B ) = ν(1 δ )φ(z, q + q )+ µδ φ(z, 2q ) , 1 ab − ab a b ab a (2.2) (B ) = ν(1 δ )φ(z, q q )+ µδ φ(z, 2q ) , 2 ab − ab − a − b ab − a

(C ) = gφ(z, q ) , (C ) = gφ(z, q ) , 1 a a 2 a − a g2 dab = δabda , da = ℘(qa)+ µ℘(2qa)+ ν (℘(qa qb)+ ℘(qa + qb)) , (2.3) ν − 6 Xb=a d0 =2ν ℘(qc) . c X The superscript T stands for transposition, and the prime means the derivative with respect to the second argument of functions φ(z, q), i.e. φ(z, q) are replaced by f(z, q) as in (1.5). The Lax pair (2.1), (2.2), (2.3) satisﬁes the Lax equations (1.6) with the Hamiltonian (1.10) if the additional constraint (1.11) for the coupling constants ν, µ and g holds true. In the following particular cases the classical root systems arise: 2 2 – BN (so2N+1): µ = 0, g =2ν ;

– CN (sp2N ): g = 0;

–DN (so2N ): µ = 0, g = 0. In order to deal with 2N 2N matrices in C and D cases one may subtract d 1 from × N N 0 2N+1 the M-matrix. Our purpose is to generalize (2.1), (2.2), (2.3) to R-matrix-valued Lax pair of (1.12), (1.13) z type. The problem is that Rij(q) carries the indices, which enumerate tensor components in quantum space. Hence we should deﬁne a number of these components and arrange them. There are two natural possibilities to do it7:

the ﬁrst one is to keep the number of components to be equal to the Lax matrix size, i.e. • to 2N +1 (or, to 2N if g = 0).

7There is also a kind of intermediate case in the so-called universal Lax pairs [28]. The number of quantum spaces (sites) is equal to the Lax matrix size, and the R-matrix is proportional to the permutation operator deﬁned by Weyl group elements of the corresponding root system. For example, in DN case the permutation of i-th and j-th sites (1 i, j N) includes also permutation of i + N-th and j + N-th sites. Elliptic R-matrix exists for GL case only.≤ For≤ this reason we do not know generalization for these operators to (elliptic) R-matrix satisfying the necessary conditions.

6 the second possibility is to leave only half of this set (coming from sl2N+1) likewise it is • performed for spin chains with boundaries [45, 7]. Put it diﬀerently, the number of ”spin sites” in quantum space is equal to the rank of the root systems.

The ﬁrst possibility leads to µ = ν, and we are left with BCN and CN cases. They are 8 described by straightforward reductions from sl2N+1 and sl2N respectively , so in this case g = ν as well. At the same time the case g = ν for BCN root system also works. See the explanation below (2.19). − The second possibility implies µ = 0 by the following reason. The diagonal elements in the blocks B1 and B2 (standing behind Ea,N+a and EN+a,a, a =1, ...N) should have elements with z R-matrices restricted to a single quantum space, i.e. Raa( 2qa). But such terms can not be involved into ansatz based on (1.16), which is identity in Mat±⊗3. Thus we are left with B and N˜ N D cases. Moreover, we will see that only N˜ = 2 is possible and g = √2ν. N ±

2.1 CN case

In this case g = 0 and µ = ν, the auxiliary space is Mat , the quantum space is Mat⊗2N , and 2N N˜ N˜ is arbitrary. The Lax pair:

′ ′ P + A1 B1 A1 + D1 + B1 = = ′ F ′ (2.4) L B2 P + A2 M B A + D2 + − 2 2 F where P = p E 1⊗2N , a aa N˜ a ⊗ z z A1 = ν Eab Rab(qa qb) ,P A2 = ν Eab Ra+N,b+N ( qa + qb) , a,b ⊗ − a,b ⊗ − z z B1 = νP Eab Ra,b+N (qa + qb) , B2 P= ν Eab Ra+N,b( qa qb) , (2.5) a,b ⊗ a,b ⊗ − − ′ z ′ z A1 = ν PEab Fab(qa qb) , A2 = ν EPab Fa+N,b+N ( qa + qb) , a,b ⊗ − a,b ⊗ − ′ z ′ z B1 = ν PEab Fa,b+N (qa + qb) , B2 =Pν Eab Fa+N,b( qa qb) , a,b ⊗ a,b ⊗ − − P P 0 0 D1 = ν Eaa da , da = Fac(qa qc) Fa,c+N (qa + qc) , a ⊗ − c: c=6 a − − c 0 0 (2.6) D2 = ν EaaP da+N , da+N = PFa+N,c(qa + qc) P Fa+N,c+N (qa qc) a ⊗ − c − c: c=6 a − P P P and = ν1 0 with F N ⊗F 0 1 0 0 1 0 0 = 2 (Fcd(qc qb)+ Fc+N,d+N (qc qd)) + 2 (Fc,d+N (qc + qd)+ Fc+N,d(qc + qd)) . (2.7) F c=6 d − − c,d 8 For example,P the reduction from sl2N with positions of particlesP ui i =1...2N to CN is achieved by identifying ui = qi and ui+N = qi, i =1...N (and the same for momenta). −

7 2.2 BCN case

Here g = µ = ν, the auxiliary space is Mat , the quantum space is Mat⊗(2N+1), and N˜ is 2N+1 N˜ arbitrary.± P + A B C A′ + D + B′ C′ 1 1 1 1 1 F 1 1 = B P + A C = B′ A′ + D + C′ (2.8)  2 2 2   2 2 2 2  L CT − CT 0 M C′T C′T F D + 2 1 2 1 3 F     where the blocks A, B and P are the same as in CN case and (C ) = νRz (q ) , (C ) = νRz ( q ) , 1 a ± a,2N+1 a 2 a ± a+N,2N+1 − a CT = νRz (q ) , CT = νRz ( q ) , 1 a ± 2N+1,a+N a 2 a ± 2N+1,a − a (C′ ) = νF z (q ) , (C′ ) = νF z ( q ) , 1 a ± a,2N+1 a 2 a ± a+N,2N+1 − a (2.9) C′T = νF z (q ) , C′T = νF z ( q ) , 1 a ± 2N+1,a+N a 2 a ± 2N+1,a − a 0 0 D3 = νd2N+1 , d2N+1 = Fc,2N+1(qc) Fc+N,2N+1(qc) . − c − c P P 0 The blocks D1 = ν Eaa da, D2 = ν Eaa da+N and = ν1N are given by: a ⊗ a ⊗ F ⊗F P 0 P 0 0 da = Fac(qa qc) Fa,c+N (qa + qc) Fa,2N+1(qa) , − c: c=6 a − − c − P P (2.10) 0 0 0 da+N = Fa+N,c(qa + qc) Fa+N,c+N (qa qc) Fa+N,2N+1(qa) − c − c: c=6 a − − and P P 1 0 = (F 0 (q q )+ F 0 (q q ))+ F 2 cd c − b c+N,d+N c − d c=6 d 1 X (2.11) + (F 0 (q + q )+ F 0 (q + q )) + F 0 (q )+ F 0 (q ) . 2 c,d+N c d c+N,d c d c,2N+1 c c+N,2N+1 c c c Xc,d X X For shortness we can also write (2.10)-(2.11) as

BC(N) Sp(2N) 0 BC(N) Sp(2N) 0 da = da Fa,2N+1(qa) , da+N = da+N Fa+N,2N+1(qa) , − − (2.12) 0 = 0 d . FBC(N) FSp(2N) − 2N+1 Let us also comment on the necessity of g = µ = ν. Consider, for example, the block 13 ± { } of the Lax equation: [ , ]13 = PC′ + AC′ + B C′ + C D + [C , ] A′C D C B′ C . (2.13) L M 1 1 1 2 1 3 1 F − 1 − 1 1 − 1 2 ′ The term PC1 gives the equation of motion. Next, (AC′ A′C ) = νg Rz (q q )F z (q ) F z (q q )Rz (q )= 1 − 1 a ab a − b b,2N+1 b − ab a − b b,2N+1 b b X (2.14) = νg F 0 (q q )Rz (q ) Rz (q )F 0 (q q ) . ab a − b b,2N+1 b − b,2N+1 b ab a − b Xb 8 The rest of the terms (after applying the unitarity condition) yield: [ , ]13 = eq. of motion+ L M a +νg [F 0 (q )Rz (q ) Rz (q )F 0 (q q )]+ b: b6=a b,2N+1 b a,2N+1 a − a,2N+1 a ab a − b P +νg [F 0 (q )Rz (q ) Rz (q )F 0 (q + q )]+ (2.15) b: b6=a b,2N+1 b a,2N+1 a − a,2N+1 a a,b+N a b P +µg(F 0 (q )Rz (q ) Rz (q )F 0 (2q ))+ a,2N+1 a a,2N+1 a − a,2N+1 a a,b+N a +νgRz (q )d gD Rz (q )+ νg[Rz (q ), 0] . a,2N+1 a 2N+1 − 1a a,2N+1 a a,2N+1 a F Here D and 0 have the following form: 1 F 2 D = g F 0 µF 0 ν F 0 (q q )+ F 0 (q + q ) , (2.16) 1a − ν a,2N+1 − a,a+N − b: b6=a ab a − b a,b+N a b 1 P 0 = F 0 (q q )+ F 0 (q q ) + F 2 cd c − b c+N,d+N c − d 6 Xc=d

1 0 0 µ 0 0 + F (qc + qd)+ F (qc + qd)) + (F (2qc)+ F (2qc) + (2.17) 2 c,d+N c+N,d ν c,c+N c+N,c 6 c Xc=d X 

0 0 + Fc,2N+1(qc)+ Fc+N,2N+1(qc) . c c X X After rearranging the summands we obtain:

νg Rz (q ), 0 (F 0 (q q )+ F 0 (q + q )) a,2N+1 a F − ab a − b a,b+N a b − 6 h bX: b=a µ F 0 (2q ) F 0 (q ) + −ν a,a+N a − c,2N+1 c c X i (2.18) g3 + F 0 (q ) νg F 0 (q )+ µg F 0 (q ) νg F 0 (q ) ν a,2N+1 a − a,2N+1 a a+N,2N+1 a − a+N,2N+1 a × Rz (q ) . × a,2N+1 a The commutator vanishes because all terms, which act non-trivially in a and 2N + 1 quantum spaces are subtracted from 0, and we are left with the following expression: F g3 νg F 0 (q )+(µg νg)F 0 (q ) (2.19) ν − a,2N+1 a − a+N,2N+1 a z (multiplied by Ra,2N+1(qa)), which must be equal to zero for validity of the Lax equations. In ˜ 0 0 the scalar case (N = 1) Fb,2N+1 is proportional to Fb+N,2N+1, and this requirement leads only to the condition (1.11). But in the case N˜ = 1 it totally ﬁxates the constants (up to a sign). 6 Calculations for blocks 23 , 31 and 32 are absolutely analogous. And all other blocks in [ , ] depend only on square{ } { of }g, consequently{ } are not sensitive to its sign. The natural way L M to solve such kind of problem is to reduce the number of quantum spaces. This is what we do in the next paragraph.

9 2.3 DN case with N quantum spaces

R-matrix-valued SO2N Lax pair with N quantum spaces has the same block structure as the one for Sp(2N) with 2N quantum spaces (2.4):

′ ′ P + A1 B1 A1 + D1 + B1 = = ′ F ′ (2.20) L B2 P + A2 M B A + D2 + − 2 2 F But now the corresponding blocks have the form: (A ) = ν(1 δ ) R z (q q ) , (A ) = ν(1 δ ) R z ( q + q ) , 1 ab − ab ab a − b 2 ab − ab ab − a b (B ) = ν(1 δ ) R z (q + q ) , (B ) = ν(1 δ ) R z ( q q ) , 1 ab − ab ab a b 2 ab − ab ab − a − b (A′ ) = ν(1 δ ) F z (q q ) , (A′ ) = ν(1 δ ) F z ( q + q ) , 1 ab − ab ab a − b 2 ab − ab ab − a b (2.21) (B′ ) = ν(1 δ ) F z (q + q ) , (B′ ) = ν(1 δ ) F z ( q q ) , 1 ab − ab ab a b 2 ab − ab ab − a − b 0 0 (D1)a =(D2)a = Da = ν (Fac(qa qc)+ Fac(qa + qc)) , − c: c=6 a − P = νδ 0 , Fab abF 1 (2.22) 0 = (F 0 (q q )+ F 0 (q + q )) . F 2 cd c − d cd c d 6 Xc=d Notice that blocks B1 and B2 are oﬀ-diagonal here. Representing the Lax pair schematically as = + R, = D + + F , where consists of momenta part, R includes A , A , B , B and L P M F P 1 2 1 2 similarly for , we can rewrite the r.h.s. of the Lax equations in the form: M [ , ] = [ + R,D + + F ] = [ , F ] + [R,D] + [R, ] + [R, F ] . (2.23) L M P F P F The calculations are performed for N N ( Mat⊗N ) blocks 11 , 12 , 21 , 22 separately. × ⊗ N˜ { } { } { } { }

Proof for Block {11}

The ﬁrst summand in (2.23):

z z [P, F ]= νpc[Ecc 1, Eab Fab(qa qb)] = ν(q ˙a q˙b)Eab Fab(qa qb) . (2.24) c,a=6 b ⊗ ⊗ − a=6 b − ⊗ − P P The second summand in (2.23):

2 z z [R,D]= ν Eab (Rab(qa qb)Db DaRab(qa qb)) = a=6 b ⊗ − − − 2 P 0 z 0 z ν Eab Fac(qa qc)Rab(qa qb)+ Fac(qa + qc)Rab(qa qb) a=6 b ⊗ c: c=6 a − − − − P zP 0 z 0 Rab(qa qb)Fbc(qb qc)+ Rab(qa qb)Fbc(qb + qc) = − c: c=6 b − − − (2.25) 2 P 0 z 0 z = ν Eab Fac(qa qc)Rab(qa qb)+ Fac(qa + qc)Rab(qa qb) a=6 b ⊗ c: c=6 a,b − − − − R z (Pq q )F 0 (q P q) R z (q q )F 0 (q + q )) + F 0 (q q )R z (q q )+ − ab a − b bc b − c − ab a − b bc b c ab a − b ab a − b +F 0 (q + q )R z (q q ) R z (q q )F 0 (q q ) R z (q q )F 0 (q + q ) . ab a b ab a − b − ab a − b ba b − a − ab a − b ba b a 10 The last summand in (2.23): [R, F ](oﬀ-diagonal) =

2 z z z z = ν Eab Rac(qa qc)Fcb(qc qb) Fac(qa qc)Rcb(qc qb)+ a=6 b ⊗ c: c=6 a,b − − − − − P z P z z z (1.17) +Rac(qa + qc)Fcb( qc qb) Fac(qa + qc)Rcb( qc qb) = (2.26) − − − − − 2 0 z z 0 = ν Eab Fcb(qc qb)Rab(qa qb) Rab(qa qb)Fac(qa qc)+ a=6 b ⊗ c: c=6 a,b − − − − − P +F 0 ( Pq q )R z (q q ) R z (q q )F 0 (q + q ) . cb − c − b ab a − b − ab a − b ac a c [R, F ](diagonal) =

2 z z z z = ν Eaa Rac(qa qc)Fca(qc qa) Fac(qa qc)Rca(qc qa)+ a ⊗ c: c=6 a − − − − − P P (1.22) (2.27) +R z (q + q )F z ( q q ) F z (q + q )R z ( q q ) = ac a c ca − c − a − ac a c ca − c − a 2 2 ′ ′ = ν N˜ Eaa (℘ (qa qb)+ ℘ (qa + qb)) . a ⊗ c: c=6 a − Hence P P ˙ = [P, F ] + [R, F ](diagonal) (2.28) L provides equations of motion (with ν replaced by Nν˜ ). Let us verify that the sum of the remaining terms in the r.h.s. of the Lax equations is equal to zero. Indeed, [R, F ](oﬀ-diagonal) + [R, ] + [R,D]= F

2 z 0 0 0 = ν Eab Rab(qa qb), Fab(qa qb) Fab(qa + qb) (2.29) a=6 b ⊗ − F − − − − P h 0 0 0 0 (Fac(qa qc)+ Fac(qa + qc)+ Fbc(qb qc)+ Fbc(qb + qc)) . − c: c=6 a,b − − P i This equals zero, since there are no terms in the right part of the commutator acting non-trivially in a-th and b-th quantum spaces. All such terms are cancelled by 0. F

Proof for block {12}

The r.h.s. of the Lax equations: [L, M] = [ + R,D + + F ] = a,b+N P F a,b+N (2.30) = [ , F ] + [R,D] + [R, ] + [R, F ] , P a,b+N a,b+N F a,b+N a,b+N As in the case of block 12 the term [ , F ] is cancelled by the oﬀ-diagonal part in the { } P a,b+N block 12 of the l.h.s. of the Lax equation. But 12 block of the Lax operator has no diagonal part in{ this} new case. So the sum of the remaining{ terms} in (2.30) should vanish:

[R,D] + [R, ] + [R, F ] =0 (2.31) a,b+N F a,b+N a,b+N 11 Let us verify it. First, consider the last term in (2.31):

[R, F ]a,b+N (oﬀ-diagonal) =

2 z z z z = ν Rac(qa qc)Fcb(qc + qb) Fac(qa qc)Rcb(qc + qb)+ c: c=6 a,b − − − z P z z z (1.17) +Rac(qa + qc)F ( qc + qb) Fac(qa + qc)R ( qc + qb) = (2.32) cb − − cb − 2 0 z z 0 = ν Fcb(qc + qb)Rab(qa + qb) Rab(qa + qb)Fac(qa qc)+ c: c=6 a,b − − +FP0 ( q + q )R z (q + q ) R z (q + q )F 0 (q + q ) . cb − c b ab a b − ab a b ac a c For the diagonal part there is no appropriate identity, and we leave it as it is:

[R, F ]a,a+N (diagonal) =

2 z z z z = ν Rac(qa qc)Fca(qc + qa) Fac(qa qc)Rca(qc + qa)+ (2.33) c: c=6 a − − − +RPz (q + q )F z ( q + q ) F z (q + q )R z ( q + q ) . ac a c ca − c a − ac a c ca − c a The ﬁrst term in (2.31) is again simpliﬁed through (1.17):

[R,D] =(R z (q q )D D R z (q q )) = a,b+N ab a − b b − a ab a − b

2 0 z 0 z = ν Fac(qa qc)Rab(qa + qb)+ Fac(qa + qc)Rab(qa + qb) c: c=6 a,b − − P R z (q + q )F 0 (q q ) R z (q + q )F 0 (q + q )+ (2.34) − ab a b bc b − c − ab a b bc b c +F 0 (q q )R z (q + q )+ F 0 (q + q )R z (q + q ) ab a − b ab a b ab a b ab a b −

R z (q + q )F 0 (q q ) R z (q + q )F 0 (q + q ) − ab a b ba b − a − ab a b ba b a Gathering all the oﬀ-diagonal terms in (2.31) we get the commutator

[R, F ] (oﬀ-diagonal) + [R,D] + [R, ] = a,b+N a,b+N F a,b+N

= ν2 R z (q + q ), 0 F 0 (q q ) F 0 (q + q ) ab a b F − ab a − b − ab a b − (2.35) h 0 0 0 0 Fac(qa qc)+ Fac(qa + qc)+ Fbc(qb qc)+ Fbc(qb + qc) , − c: c=6 a,b − − P i which equals zero by the same reason as for block 11 (2.29). Therefore, we are left with the { } diagonal term [ , ] = [R, F ] , (2.36) L M a,a+N a,a+N and it vanishes if R z (u)F z (v) F z (v)R z (u)=0 . (2.37) ab ba − ab ba 12 The latter is of course not true in general case, but is true in some particular cases. For example, it is obviously holds true for the Yang’s case (1.15). What is more important for us is that (2.37) holds true in N˜ = 2 case with the Baxter’s R-matrix since it is of the form 3 z z R12(u)= ϕα(u) σα σα. It is easy to check that for such R-matrices α=0 ⊗ P z z [Rab(u), Rab(v)]=0 (2.38) due to the properties of the Pauli matrices. By diﬀerentiating (2.38) with respect to v (and z z using the additional symmetry Rab(u)= Rba(u)) one ﬁnds (2.37). With this property we have R z (q q )F z (q + q ) F z (q q )R z (q + q )+ ac a − c ca c a − ac a − c ca c a (2.39) +R z (q + q )F z ( q + q ) F z (q + q )R z ( q + q )=0 , ac a c ca − c a − ac a c ca − c a

and this is the expression, standing under the sum in [R, F ]a,a+N . The proofs for blocks 21 and 22 are performed in a similar way. { } { }

2.4 BN case with N +1 quantum spaces

Here R-matrix-valued Lax pair with N + 1 quantum spaces has the same block structure as the one for BC(N) with 2N + 1 quantum spaces (2.8)

P + A B C A′ + D + B′ C′ 1 1 1 1 1 F 1 1 = B P + A C = B′ A′ + D + C′ (2.40)  2 − 2 2   2 2 2 F 2  L CT CT 0 M C′T C′T D + 2 1 2 1 3 F     but now the corresponding blocks have the following form. The blocks A, B and P are the same as in the previous case (2.21), and the rest are:

(C ) = √2ν R z (q ) , (C ) = √2ν R z ( q ) , 1 a ± a,N+1 a 2 a ± a,N+1 − a (CT ) = √2ν R z (q ) , (CT ) = √2ν R z ( q ) , 1 a ± N+1,a a 2 a ± N+1,a − a (C′ ) = √2ν F z (q ) , (C′ ) = √2ν F z ( q ) , 1 a ± a,N+1 a 2 a ± a,N+1 − a (2.41) (C′T ) = √2ν F z (q ) , (C′T ) = √2ν F z ( q ) , 1 a ± N+1,a a 2 a ± N+1,a − a 0 D3 = 2ν Fc,N+1(qc) . − c P The D and terms are given by: F 0 0 0 (D1)a =(D2)a = ν Fac(qa qc)+ Fac(qa + qc) 2ν Fa,N+1(qa) , (2.42) − c: c=6 a − − P and 1 = νδ 0 , 0 = F 0 (q q )+ F 0 (q + q ) +2 F 0 (q ) . Fab abF F 2 cd c − d cd c d c,N+1 c (2.43) 6 c Xc=d X 13 Or, equivalently BC(N) BC(N) SO(2N) 0 (D1)a =(D2)a = Da 2ν Fa,N+1(qa) , − (2.44) ν 0 = ν 0 D . FBC(N) FSO(2N) − 3 As in the previous case the calculations are performed separately for N N ( Mat⊗N ) blocks × ⊗ N˜ 11 , 12 , 21 , 22 ; for N 1 ( Mat⊗N ) blocks 13 , 23 ; for 1 N ( Mat⊗N ) blocks { } { } { } { } × ⊗ N˜ { } { } × ⊗ N˜ 31 , 32 and for 1 1 ( Mat⊗N ) block 33 . Evaluations for the blocks 11 , 12 , 21 , N˜ {22} are{ similar.} In particular,× ⊗ we again come{ to} condition (2.37). The calculation{ } for{ the} block{ } { } 33 is as follows: { } [L, M] =(CT C′ C′T C + CT C′ C′T C )= 2N+1,2N+1 2 1 − 2 1 1 2 − 1 2

2 z z z z =2ν RN+1,a( qa)Fa,N+1(qa) FN+1,a( qa)Ra,N+1(qa)+ a − − − (1.22) (2.45) +R zP (q )F z ( q ) F z (q )R z ( q ) = N+1,a a a,N+1 − a − N+1,a a a,N+1 − a 2 2 ′ 2 ′ =2ν (N˜ ℘ ( qa)+ N˜ ℘ (qa))=0 . a − P In the end of the Section let us remark that the Hamiltonian (1.10) is a particular case of the most general model [26]

1 N N 3 N H = p2 ν2 (℘(q q )+ ℘(q + q )) ν2℘(q + ω ) , (2.46) 2 a − a − b a b − γ a γ a=1 γ=0 a=1 X Xa

3 Quantum Lax pairs and spin models

Spinless case. The classical Lax pairs for Calogero-Moser models can be quantized through the quantum Lax equation [47, 10]

[H,ˆ Lˆ(z)] = ~ [Lˆ(z), M(z)] . (3.1)

Before discussing the R-matrix-valued Lax pairs consider the ordinary case N˜ = 1. As mentioned before, in this case the Lax pair (1.12)-(1.13) diﬀers from (1.4)-(1.5) by the scalar (in the auxiliary space) part 0 (1.14) of M-matrix. For N˜ =1 F 2 ′′′ 0 N N ϑ (0) ˜ = ν ℘(q ) + const , const = − . F |N=1 − ij 3 ϑ′(0) (3.2) i>j X

14 This term is cancelled out from the classical Lax equations. But it becomes important in quantum case because it is treated as a part of the quantum Hamiltonian in (3.1). With this term the coupling constant in

N N ˆ 1 2 ~ ~ H = pˆ ν(ν + ) ℘(q q ) , pˆ = ∂ i (3.3) 2 i − i − j i q i=1 i>j X X acquires the quantum correction. Another reason for treating (3.2) as a part of the Hamiltonian is as follows. After subtracting 0 from M we are left (in N˜ = 1 case) with the original M- F matrix (1.5). In rational and trigonometric cases (A.8) this M-matrix satisﬁes the so-called sum up to zero condition: Mij = Mji = 0 for all i . (3.4) 6 6 jX:j=i jX:j=i It can be proved [47] that with this condition the total sums

Iˆ = ts(Lˆk)= (Lˆk) , k Z k ij ∈ + (3.5) i,j X

are the quantum integrals of motion, i.e. [H,ˆ Iˆk] = 0 (while in the classical case the integrals of k motion can be deﬁned as tr(L )). The second Hamiltonian Iˆ2 yields Hˆ (3.3). In the elliptic case the sum up to zero condition (3.4) is not fulﬁlled. At the same time the quantum Lax equation holds true.

Spin case. Similar construction works for the spin Calogero-Moser model [20]. The operator valued Lax pair [27, 23, 6, 28] is of the form

N Lˆspin(z)= E Lˆspin(z) , Lˆspin(z)= δ p + ν(1 δ )φ(z, q )P , (3.6) ij ⊗ ij ij ij i − ij ij ij i,j=1 X N N Mˆ spin(z)= νd δ + ν(1 δ )f(z, q )P , d = E (q )P = f(0, q )P , (3.7) ij i ij − ij ij ij i 2 ik ik − ik ik 6 6 kX:k=i kX:k=i where P Mat⊗N is the permutation operator (of i-th and j-th tensor components) in quantum ij ∈ N˜ space = (CN˜ )⊗N , or the spin exchange operator. The tensor structure of (3.6)-(3.7) is the same asH for the R-matrix-valued Lax pair (1.12)-(1.13). The term 0 (1.13) in this case F 0 = ν E (q q )P F − 2 i − j ij (3.8) i>j X is treated as a part of the quantum Hamiltonian

1 N N Hˆ spin = pˆ2 ν(ν + ~ P )E (q q ) (3.9) 2 i − ij 2 i − j i=1 i>j X X likewise it was performed in the spinless case (3.3). It describes the spin exchange interaction. Again, in the rational and trigonometric cases the sum up to zero condition is fulﬁlled, and

15 ts(Lˆk) provides the higher Hamiltonians including the one (3.9) for k = 2. Another recipe for constructing the higher Hamiltonians comes from the underlying Yangian structure [6]. From the point of view of R-matrix-valued case (1.12)-(1.13) the Lax pair Lˆspin(z) and Mˆ spin(z)+1 0 corresponds to the special case when Rz (q ) is replaced by φ(z, q )P . N ⊗F ij ij ij ij The associative Yang-Baxter equation (1.16) holds true in this case since the matrix-valued expressions PabPbc = PacPab = PbcPac are cancelled out, and we are left with the scalar identity (1.9).

Anisotropic spin case. In [24] the anisotropic extension of the (trigonometric) spin Calogero- Moser-Sutherland model was suggested. It is based on the su(2) XXZ (trigonometric) classical r-matrix 1 rXXZ(q)=(1 1+ σ σ ) cot(q)+(σ σ + σ σ ) , (3.10) 12 ⊗ 3 ⊗ 3 1 ⊗ 1 2 ⊗ 2 sin(q) z which is used in the Lax pair in the same way as Rij(q) in (1.12):

N LˆXXZ(z)= E LˆXXZ(z) , LˆXXZ(z)= δ p + ν(1 δ )rXXZ(q ) . (3.11) ij ⊗ ij ij ij i − ij ij ij i,j=1 X In this respect the Lax matrix (1.12) with the elliptic Baxter-Belavin R-matrix generalizes the case (3.10) in the same way as the elliptic Lax matrix (1.4) generalizes the one without spectral parameter for the classical (trigonometric) Sutherland model. Let us slightly modify (3.11) to have exactly the form (1.12). Consider (1.12)-(1.13) deﬁned through the XXZ quantum R-matrix

3 1 R z (q)= ϕz (q) σ σ , ϕz(q)= ϕz(q)= , 12 α α ⊗ α 1 2 sin q α=0 (3.12) X 1 1 ϕz(q) = (cot z + cot q + ) , ϕz(q) = (cot z + cot q ) . 0 sin z 3 − sin z This results in the quantum Hamiltonian with the 0 term F 0 1 a b a b cos qab a b a b = 2 (σ0 σ0 + σ3 σ3)+ 2 (σ1 σ1 + σ2 σ2) , (3.13) F sin qab ⊗ ⊗ sin qab ⊗ ⊗ Xa

Proposition 3.1 For the Lax pairs (1.12)-(1.13) for AN−1, (2.20) for DN and (2.40) for BN root systems one can deﬁne the quantum Lax pair by simple replacing momenta pi with pˆi = ~ ˆ ∂qi . The quantum Lax equations hold true with the quantum Hamiltonian H obtained from the classical H(p, q) as H(ˆp, q).

The proof of this statement is direct. It is similar to the classical case and (3.6). The main idea, needed for the proof is that in the above mentioned cases

[P,D + 0]=0 , (3.14) F 16 where D is a diagonal part of the M-operator. This happens because the only terms in D depending on some qa are those, which simultaneously act non-trivially on the tensor component number a in quantum space. We already know from the classical case that Da subtracts exactly 0 0 these terms from . For CN and BCN cases this statement is not true. For example, D1a contains the termsFF 0 ( q + q ), which do not commute withp ˆ . −F a+N,c+N − a c a From what has been said it seems also natural to remove the 0 terms from the corresponding M-matrices and treat them as a part of the quantum Hamiltonians,F which then take the form

Hˆ = H(ˆp, q)+ ~ ν 0 , (3.15) F where the terms 0 are given by (1.14) for A , (2.20) for D and (2.43) for B root systems. F N−1 N N Let us emphasize that 0 term in the elliptic quantum Hamiltonian (3.15) does not follow F from the Lax matrix. The construction (3.5) used in the rational and trigonometric cases does not work here since the sum up to zero condition (3.4) is not valid. Moreover, in the classical case the corresponding Lax matrices provide the Hamiltonians without 0. However, we are F going to argue that (3.15) is meaningful even in the classical case (see the next Section). The quantum Lax matrices (3.6), are closely related to the Knizhnik-Zamolodchikov equations [29]. As was mentioned in [23, 24] the KZ connections (3.11) are obtained as

N N ˆXXZ ~ = L (z)= ∂ i + ν r (q ) , (3.16) ∇i ij q ij ij j=1 6 X jX: j=i and the ground state for this model solves the KZ equations

ψ =0 , i =1...N . (3.17) ∇i The interrelation between the quantum Calogero models and the KZ equations was observed by Matsuo and Cherednik [36, 12] (see also[17]). Their construction, in fact, underlies the construction of the operator valued Lax pairs.9 The relation to KZ equations is modiﬁed in the elliptic case as follows. To the R-matrix-

valued Lax pair (1.12)-(1.13) let us associate the glN˜ KZB equations on elliptic curve with N punctures at qi: ψ =0 , = ~ ∂ + ν r (q q ) , ∇i ∇i i ij i − j j:j=6 i ν X (3.18) ψ =0 , = ~ ∂ + m (q q ) , ∇τ ∇τ τ 2 jk j − k 6 Xj=k where rij is the classical r-matrix, mij is the next term in the classical limit (A.19), and i = 1, ..., N. The commutativity of the connections follows from the classical Yang-Baxter equation

[rab,rac] + [rab,rbc] + [rac,rbc]=0 (3.19) and the identities [rab, mac + mbc] + [rac, mab + mbc]=0 , (3.20)

9 It should be also mentioned that diﬀerent types of spinless and spin Calogero-Moser models can be described in the framework of the Cherednik-Dunkl and/or exchange operator formalism [14, 42]. See e.g. [6, 7, 49, 18]. In our approach we do not use the particles exchange operators.

17 which are obtained from the associative Yang-Baxter equation (1.16) and (1.19)-(1.20). It was shown in [32] that a solution of (3.18) satisﬁes also

1 N Nν~ ∂ ψ = ~2∂2 ν2N˜ 2 ℘(q q )+ ~ ν ∂ r (q q )+ const ψ (3.21) − τ 2 qi − i − j i ij i − j i=1 i

4 Spin exchange operators from Hitchin systems

The purpose of this Section is to explain the classical origin of the 0 term (related to the elliptic R-matrix) in the Hamiltonian (3.15) and/or (1.24). We start withF a brief description of the classical spin Calogero-Moser model, and then proceed to the model of N interacting SLN˜ tops, which provides the classical analogue of the Hamiltonian with the (elliptic) anisotropic spin exchange operator.

Spin Calogero-Moser model. In classical mechanics the slN spin Calogero-Moser models are described as follows [8] (see also [3, 6]). The phase space is a direct product C2N : the 2N × O particles degree of freedom pi, qi, i = 1...N parameterize C and the coadjoint orbit of GLN group is parameterized by S , i, j =1...N, i.e. the classical spin variables are arranged into O ij matrix S = i,j EijSij MatN with ﬁxed eigenvalues. The Poisson structure is a direct sum of the canonical brackets (1.2)∈ and the Lie-Poisson structure P S ,S = S δ + S δ . (4.1) { ij kl} − il kj kj il The latter is realized by embedding S into a (larger) space with canonical variables:

N˜ S = ξaηa , ξa, ηb = δ δ . (4.2) ij i j { i j } ij ab a=1 X The Lax pair Lspin(z)= δ (p + S E (z))+(1 δ )S φ(z, q ) , (4.3) ij ij i ii 1 − ij ij ij M spin(z)=(1 δ )S f(z, q q ) (4.4) ij − ij ij i − j and the Hamiltonian N p2 N H spin = i S S E (q q ) . (4.5) 2 − ij ji 2 i − j i=1 i>j X X 18 satisfy the Lax equations (1.6) and deﬁne integrable system if the additional constraints hold:10

Sii = const , for all i . (4.6)

The are generated by the action of the Cartan subgroup of SLN .

The appearance of the spin exchange operator (3.8) comes from rewriting SijSji in the

Hamiltonian in terms of N MatN˜ -valued matrices:

N˜ N˜ i j i i i a a b b a b SijSji = ξ η ξ η = tr(BB) , B= E˜ab Bab Mat ˜ , Bab= ξ η . (4.7) i j j i ∈ N i i a,bX=1 a,bX=1 i Due to (4.6) tr B= const for all i =1, ..., N. Hence, we come to N minimal coadjoint orbits of i i ˜ GLN˜ . The quantization of Bab in the fundamental representation of glN˜ is given by Eab. This N˜ i j i j N˜ i j provides the permutation operator Pij = EabEba for quantization of tr(BB)= BabBba. a,b=1 a,b=1 Let us remark that the classical r-matrixP structure for (4.3) is deﬁned by dynamicalPr-matrix. Its quantization is performed in terms of quantum dynamical R-matrix [3], while the approach using the quantum Lax pairs of (3.6)-(3.7) type is more like the Matsuo-Cherednik construction.

It is dual to the direct one since it is based on glN˜ KZ equations, and the positions of particles are the N punctures on the base spectral curve.

The classical version of the 0 term (3.15) coming from the elliptic R-matrix (A.18) is as follows: F N N 0 1 0 (A.21) 1 = F (q ) = ∂ i r (q ) , (4.8) F 2 ij ij 2 q ij ij 6 6 Xi=j Xi=j where using (A.20) and (A.10) we have

∂qi rij(qij)=

i j i j (4.9) E2(qij) T 0 T 0 + ϕγ(qij,ωγ)(E1(qij + ωγ) E1(qij)+2πı∂τ ωγ) T γ T −γ . − − ∈ Z ×Z 6 γ N˜XN˜ ,γ=0 It follows from the previous discussion that the classical analogue of (4.8)-(4.9) is

0 class = N F 1 i j i j (4.10) E2(qij)B0 B0 + ϕγ(qij,ωγ)(E1(qij + ωγ ) E1(qij)+2πı∂τ ωγ) Bγ B−γ , 2 − − 6 6 Xi=j Xγ=0 i i where B are the N MatN˜ -valued matrices (classical spin variables), and Bα – their components in the basis (A.1).

10The expressions for the M-matrix (4.4) is valid only before the Poisson reduction is performed, which may led to additional (Dirac) terms. For instance, if N˜ = 1 in (4.2) (the minimal orbit) then this reduction kills all spin variables, and the model is reduced to the spinless Calogero-Moser model (1.4). As a result the diagonal terms in (1.5) appear.

19 Hitchin systems on SLNN˜ -bundles. In the Hitchin approach [25] to elliptic models [37] the positions of particles are treated as coordinates on the moduli space of Higgs bundles over (punctured) elliptic curve. The GLn-bundles are classified according to Atiyah [2]. Dimensions of the moduli spaces are obtained as the greatest common divisors of the rank n and degrees of underlying holomorphic vector bundles. By changing the degrees we describe different type models. This leads to integrable systems called interacting elliptic tops [51]. Similar construction exists for arbitrary complex simple Lie group [35]. The classification is based on characteristic classes, which are in one to one correspondence with the center of the structure group. Here we consider the model related to the SL(NN,˜ C)-bundle and the characteristic class is N (or exp(2πı N )). It means that the model contains N particles degrees of freedom, and the NN˜ Lax matrix is MatNN˜ -valued:

N ij ij L(z)= E L (z) , L (z) Mat ˜ , (4.11) ij ⊗ ∈ N i,j=1 X where Lij(z)=

q (4.12) = δ p + ii T E (z)+ ii T ϕ (z,ω ) + (1 δ ) ij T ϕ (z,ω ij ) , ij i S0 0 1 Sα α α α − ij Sα α α α − ˜ 6 α N Xα=0 X ij 2 ij where Tα is the basis (A.1) and α – the components in this basis of N matrices MatN˜ . ij S S ∈ The variables α are dual to the basis Eij Tα in MatNN˜ . Therefore, the Poisson brackets are deﬁned in theS same way as the commutation⊗ relations for the set E T , i.e. { ij ⊗ α} ij, kl = δ κ kj δ κ il . (4.13) {Sα Sβ } il α,β Sα+β − kj β,α Sα+β ij Of course can be decomposed in the standard basis in MatN˜ as well. Then the brackets (4.13) acquireS the form:

N˜ ij, kl = δ δ kj δ δ il , ij = E˜ ij . (4.14) {Sab Scd } il ad Scb − kj cb Sad S ab Sab a,b,X=1 The analogue for constraints (4.6) is again generated by the coadjoint action of the Cartan subgroup of GL GL ˜ : N ⊂ NN ii = const for all i =1, ..., N . (4.15) S0 The quadratic Hamiltonian is evaluated from trL2(z):

1 N 1 N 1 N q Htops = p2 ii ii E (ω ) ij ji E (ω ij ) . (4.16) 2 i − 2 Sα S−α 2 α − 2 Sα S−α 2 α − ˜ i=1 i=1 6 6 α N X X Xα=0 Xi=j X

Consider the case of the minimal coadjoint orbit of GLNN˜ . Then the matrix

N ij ij = E , Mat ˜ , i, j =1, ..., N (4.17) S ij ⊗ S S ∈ N i,j=1 X 20 is of rank 1, that is N˜ ij = E˜ ξa ηb (4.18) S ab i j a,b,X=1 a b with ξi and ηj are as in (4.2). This parametrization is in agreement with the Poisson brackets (4.14) or (4.13). From (4.7) we also have

i ii =B . (4.19) S Due to (A.4) ij = tr( ij T )/N˜. Therefore, using the N˜-dimensional columns ξi and the Sα S −α N˜-dimensional rows ηj we have

i j j i i j j i i j ij ji tr(η T−α ξ )tr(η Tα ξ ) tr(η T−α ξ η Tα ξ ) tr(B Tα B T−α) (4.20) − = = = . Sα S α N˜ 2 N˜ 2 N˜ 2 Plugging this expression into (4.16) we get

N N N i j i i tops 1 2 1 1 tr(B Tα B T−α) qij H = p Bα B−α E2(ωα) E2(ωα ) . (4.21) 2 i − 2 − 2 ˜ 2 − ˜ i=1 i=1 6 6 α N N X X Xα=0 Xi=j X The Hamiltonians of this type were originally found in [43] from the study of matrix models. In [51] the model (4.21) was called the interacting elliptic tops, since the second term is the sum of Hamiltonians for SLN˜ elliptic integrable tops, and the last term describes their interaction. i i Let us write the last term in (4.21) more explicitly. For this purpose plug B= γ Tγ Bγ and j j B= µ Tµ Bµ into (4.21): P

PN N i j i j qij 2 qij tr(B Tα B T−α)E2(ωα )= κ BγBµ E2(ωα )tr(Tγ Tµ)= − ˜ α,µ − ˜ 6 α N 6 α,µ,γ N Xi=j X Xi=j X (4.22) N N i j i j (A.4) 2 qij 2 qij = N˜ κ B−µBµ E2(ωα )= N˜ κ BµB−µ E2(ωα + ) , α,µ − ˜ α,µ ˜ 6 α,µ N 6 α,µ N Xi=j X Xi=j X 2 where we used TαTµT−α = κα,µTµ coming from (A.3), and changed the indices i, j in the last line. Finally, we sum up over the index α by applying the Fourier transformation formulae (A.24)-(A.25). This yields

N i j 1 qij ˜ 3 0 tr(B Tα B T−α)E2(ωα )= N class , (4.23) 2 − ˜ − F 6 α N Xi=j X where 0 was deﬁned in (4.10). The answer for the Hamiltonian (4.21) or (4.16) is as follows Fclass N N i i tops 1 2 1 ˜ 0 H = p Bα B−α E2(ωα)+ N class . (4.24) 2 i − 2 F i=1 i=1 6 X X Xα=0

21 Hence, we showed that the classical analogue of the anisotropic elliptic spin exchange operator (see (3.15), (3.22), (4.8)) arises in the model (4.11)-(4.16) likewise the classical version of the isotropic spin exchange operator (4.7) comes from the spin Calogero model (4.3), (4.5). The proof of Proposition 1.2 comes from quantization of (4.24). First, notice that the second term turns into a constant term proportional to identity matrix (in Mat⊗N ) when quantized in N˜ i i the fundamental representation since T α T −α=1N˜ . It could be added to the original definition of 0 (1.14) as the terms corresponding to m = k under the sum. The potential of the Calogero- F Moser model comes from the scalar part (the first term in (4.9)) of the 0 term (4.8). The F difference between E2 and ℘-functions (A.8) is also included into the constant term proportional to identity matrix. In this way come to the statement (1.25). Similarly, in classical mechanics i the Calogero-Moser potential comes from the first term of (4.10) since B0 are equal to the same constant (4.15) for all i.

5 Conclusion

We considered R-matrix-valued Lax pairs for N-body Calogero-Moser models. The one for AN−1 root system was previously known [32]. We proposed their extensions to other root systems. Namely, we studied generalizations of the D’Hoker-Phong Lax pairs [15] for the classical roots systems in the untwisted case. These Lax pairs are block-matrices of 2N 2N or (2N + 1) (2N + 1) size, and each block is of the size N˜ r N˜ r, where r – is the number× of quantum spaces× × (spin sites). Two possibilities were considered. The ﬁrst one is to keep all 2N (or 2N + 1) quantum spaces in R-matrices. This leads to the Lax pairs for CN and BCN cases. The second possibility is to leave only half (N or N + 1) quantum spaces. It results in constructing BN and DN models with GL2 (N˜ = 2) Baxter’s R-matrix. The summary of admissible values of the coupling constants and the number of quantum spaces in R-matrices is presented in the table below (horizontally are the numbers of quantum spaces).

N N+1 2N 2N+1 g = 0, µ =0 SO(2N) N˜ =2 g = √2ν, µ =0 ± SO(2N+1) N˜ =2 g = 0, µ = ν Sp(2N) N˜ = any g = ν, µ = ν ± BC(N) N˜ = any Number of spin quantum spaces and values of coupling constants.

Recall that the ordinary Lax pairs (2.1)-(2.3) were deﬁned for the following values of the coupling constants:

– SO2N : µ = 0, g = 0; 2 2 – SO2N+1: µ = 0, g =2ν ;

22 – Sp2N : g = 0; – BC : g(g2 2ν2 + νµ) = 0. N − In this respect our results are as follows: the R-matrix-valued ansatz generalizing (2.1)-(2.3) works with additional constraints. For SO cases the additional condition is N˜ = 2, while for CN and BCN cases there is no restriction on N˜ but the constants should satisfy µ = ν together with g =0 or g = ν for C or BC root systems respectively. ± N N Then we proceed to the quantum Lax pairs. A short summary is that the classical R- matrix-valued Lax pairs are generalized to quantum Lax pairs only for SO cases from the above table. The quantum Lax pairs are naturally related to the spin Calogero-Moser models. The corresponding spin exchange operators 0 appear as a scalar parts of the R-matrix-valued F M-matrices. On the other hand the same operators can be derived from KZ or KZB equations. We demonstrate these relations for slN R-matrix-valued Lax pair. The link between the operator-valued Lax pairs and KZ equations comes from the Matsuo-Cherednik duality. Its quasi-classical version provides the so-called quantum-classical duality between the quantum spin chains (Gaudin models) and the classical many-body systems of Ruijsenaars-Schneider (Calogero-Moser) type [22]. In this paper we deal with another example of quantum-classical relation. We treat the Lax equations for the classical Calogero-Moser model (1.12)-(1.14) with R-matrix-valued Lax pairs as half-quantum model (1.24), which quantum part is described by the spin exchange operator known previously as the ”noncommutative spin interactions” [43]. The spin variables are quantized in the fundamental representation, while the particles degrees of freedom remain classical. We show that the classical counterpart of the elliptic anisotropic

spin exchange operator comes from the Hitchin type system on SLNN˜ -bundle with nontrivial characteristic class over elliptic curve. See the Proposition 1.2. It was shown in [44] that the spin exchange operator 0 (4.9) for N˜ = 2 being reduced to F the equilibrium position qj = j/N provides the Hamiltonian for anisotropic extension of the Inozemtsev elliptic long-range chain. In view of the relation of the R-matrix-valued Lax pairs and the Hitchin systems on SL(NN˜)-bundles we expect that these type long-range integrable spin chains admit Lax representations of size NN˜ NN˜ at both - classical and quantum levels. × They are obtained from the one for interacting tops (4.12) by the substitution pj = 0, qj = j/N. Such Lax pair allows to calculate the higher Hamiltonians. These questions will be discussed in our next publication [21].

6 Appendix

In addition to the standard basis in MatN˜ we use the one [5]

πı a1 a2 T = T = exp a a Q Λ , a =(a , a ) Z ˜ Z ˜ (A.1) a a1a2 ˜ 1 2 1 2 ∈ N × N N constructed by means of the ﬁnite dimensional representation of Heisenberg group

2πı N˜ N˜ Qkl = δkl exp( k) , Λkl = δ ˜ , Q =Λ =1 ˜ ˜ . (A.2) N˜ k−l+1=0 modN N×N

23 The following relations hold πı T T = κ T , κ = exp (β α β α ) , (A.3) α β α,β α+β α,β ˜ 1 2 − 2 1 N 

tr(TαTβ)= Nδ˜ α,−β , (A.4) where α + β =(α1 + β1,α2 + β2). The permutation operator takes the form

˜ N 1 P = E˜ E˜ = T T , (A.5) 12 ij ⊗ ji ˜ α ⊗ −α i,j=1 N ∈ Z ×Z X α XN˜ N˜ ˜ where Eij is the standard basis in MatN˜ .

The Kronecker function is deﬁned in the rational, trigonometric (hyperbolic) and elliptic case as follows: 1/η +1/z rational case , − φ(η, z)= coth(η) + coth(z) trigonometric case , (A.6)  ϑ′(0)ϑ(η+z) − elliptic case .  ϑ(η)ϑ(z) − In the latter case the theta-function is the odd one

1 2 1 1 ϑ(z)= exp πıτ(k + ) +2πı(z + )(k + ) . (A.7) 2 2 2 ∈Z Xk Similarly, the ﬁrst Eisenstein (odd) function and the Weierstrass (even) ℘-function:

1/z , 1/z2 , coth(z) , 1/ sinh2(z) , E1(z)= ℘(z)= ′′′ (A.8)  ′  1 ϑ (0) ϑ (z)/ϑ(z) , ∂ E (z)+ ′ .   − z 1 3 ϑ (0) The derivative   E (z)= ∂ E (z) (A.9) 2 − z 1 is the second Eisenstein function. The derivative of the Kronecker function:

f(z, q) ∂ φ(z, q)= φ(z, q)(E (z + q) E (q)) . (A.10) ≡ q 1 − 1 Due to the following behavior of φ(z, q) near z =0

φ(z, q)= z−1 + E (q)+ z (E2(q) ℘(q))/2+ O(z2) . (A.11) 1 1 − we also have f(0, q)= E (q) . (A.12) − 2 The Fay trisecant identity:

φ(z, q)φ(w,u)= φ(z w, q)φ(w, q + u)+ φ(w z,u)φ(z, q + u). (A.13) − − For the Lax equations the following degenerations of (A.13) are needed

φ(z, x)f(z,y) φ(z,y)f(z, x)= φ(z, x + y)(℘(x) ℘(y)) , (A.14) − − 24 φ(η, z)φ(η, z)= ℘(η) ℘(z)= E (η) E (z) . (A.15) − − 2 − 2 Also φ(z, q)φ(w, q)= φ(z + w, q)(E (z)+ E (w)+ E (q) E (z + w + q)) = 1 1 1 − 1 (A.16) = φ(z + w, q)(E (z)+ E (w)) f(z + w, q) . 1 1 − The set of N˜ 2 functions

η a2 a1 + a2τ ϕ (z) = exp(2πı z) φ(z, η + ) , a =(a1, a2) Z ˜ Z ˜ (A.17) a N˜ N˜ ∈ N × N is used in the deﬁnition of the Baxter-Belavin’s [4, 5] elliptic R-matrix

Rη (z)= T T ϕ (z,ω + η) . 12 α ⊗ −α α α (A.18) α X The classical limit (behavior near η = 0) 1 1 Rη (z)= ⊗ + r (z)+ η m (z)+ O(η2) (A.19) 12 η 12 12 is similar to (A.11). The classical r-matrix

r (z)=1 1E (z)+ T T ϕ (z,ω ) 12 ⊗ 1 α ⊗ −α α α (A.20) 6 Xα=0 is skew-symmetric due to (1.20) or (1.19). From (A.19) we conclude that F 0 (q)= ∂ Rη (q) = ∂ r (q)= F 0 ( q) . (A.21) 12 q 12 |η=0 q 12 21 − The ﬁnite Fourier transformation for the set of functions (A.17) is as follows (see e.g. [53]): 1 z κ2 ϕ (Nη,ω˜ + )= ϕ (z,ω + η) , γ . ˜ α,γ α α ˜ γ γ ∀ (A.22) N α N X It is generated by the arguments symmetry (similarly to φ(z, q)= φ(q, z))

z q/N˜ ˜ R12(q)P12 = R12 (Nz) . (A.23) In particular, (A.22) leads to

˜ 2 ˜ ˜ 2 ˜ E2(ωα + η)= N E2(Nη) or ℘(ωα + η)= N ℘(Nη) (A.24) α α X X and for γ =0 6 κ2 E (ω + η)= N˜ 2ϕ (Nη,ω˜ )(E (Nη˜ + ω ) E (Nη˜ )+2πı∂ ω ) . α,γ 2 α − γ γ 1 γ − 1 τ γ (A.25) α X Acknowledgments. We are grateful to A. Levin, M. Olshanetsky and I. Sechin for helpful discussions. The work was performed at the Steklov Mathematical Institute of Russian Academy of Sciences, Moscow. This work is supported by the Russian Science Foundation under grant 14-50-00005.

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