Pis'ma v ZhETF, vol. 101, iss. 9, pp. 723-729

© 2015 May 10


Classical integrable systems and Knizhnik-Zamolodchikov-Bernard


G. Aminov+*1\ A.Levin+x 1), M. 01shanetsky+* V, A. Zotov+*° ^ + Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia

* Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia Згпгп x Department of Mathematics, National Research University Higher School of Economics, 101000 Moscow, Russia

°Steklov Mathematical Institute of the RAS, 119991 Moscow, Russia

Поступила в редакцию 6 April 2015

This paper is a short review of results obtained as part of The Russian Foundation for Basic Research project 12-02-00594. We mainly focus on interrelations between classical integrable systems, Painlevé-Schlesinger equations and related algebraic structures such as classical and quantum Д-matrices. The constructions are explained in terms of simplest examples.

DOI: 10.7868/S0370274X15090131

1. Zero curvature equations. We consider integrable equations in classical and quantum Hamiltonian mechanics. In classical mechanics they are described usually by the Lax equations

dtL=[L,M}. (1)

Here L and M are matrices (operators), depending on the phase variables u = (u\,..., un), v = (v\,..., vn), S and additional (spectral) parameter z, L = L(u, v, S; z), M = M(u,v,S;z). We assume that z e E where E is a torus or its degenerations. The variables u, v are the canonical Darboux variable {vj,Uk} = Sjk■ The Lax equations can be derived from the d = 4 (super) Yang-Mills theories with the gauge group G compactified on E. In this case L is identified with a scalar field taking values in the adjoint representations (the Higgs field) restricted on E, while M is an element of the Lie algebra of the gauge transformations. The variables S are elements of the Lie algebra Lie(G). They Poisson commute with (u, v) and their brackets are the Poisson-Lie brackets on Lie(G). In terms of the Lax operators the Poisson brackets are defined by means of the classical r-matrices (see examples below).

-^e-mails: aminov@itep.ru; alevin@hse.ru; olshanet@itep.ru; zotov@mi.ras.ru

Eq. (1) describes an autonomous Hamiltonian integrable mechanics. To come to the non-autonomous Hamiltonian system we replace (1) by

dtL - ndzM = [L, M], (2)

where k is a parameter. This equation is the monodromy preserving condition for the linear equation

(«¿>Z + L)V = 0, (3)

and L now plays the role of connection. In particular, (2) describes the Painlevé equation, Schlesinger system and their generalizations. In the limit k —> 0 we come to (1).

Another generalizations of (1) are the Zakharov-Shabat equations for 1+1 integrable field theories which possess the soliton type solutions:

dtL — kdxM = [L, M], (4)

One can also consider a generalization of (4) and (2) given by

dtL - ndzM - kdxM = [L, M]. (5)

We refer to the models described by this equation as the Painlevé field theories.

Письма в ЖЭТФ том 101 вып. 9-10 2015



The zero curvature equations (l)-(4) keep their forms under the gauge transformations

D + L

dt + M



gLg-1 -{Dg)g-\ -gMg-l-{dtg)g-'


where the differential operator D is given by

D = 0, A = ndz, D = kdx, D = kdx + ndz (7)

for the Eq. (1), (2), (4), (5) respectively.

The purpose of the paper is to show that all types of zero-curvature equations can be described in a similar way, i.e. there exist a universal type Lax pairs which can be used for all the cases. They describe a wide class of integrable systems and related problems. We start from the classical integrable mechanics, which deals with two types of models - many-body systems (interacting particles) and integrable cases of motion of solid body in multidimensional space. First, we demonstrate that many-body systems of Calogero-Ruijsenaars type can be formulated as integrable tops of Euler-Arnold type. Using special gauge transformation the Ruijsenaars-Schneider model is represented in the form



where the relativistic deformation parameter r¡ enters the Lax matrix as the Planck constant of a certain quantum fi-matrix. Being formulated as tops the many-body systems are then naturally included into a more general class of integrable models, which consists of spin chains and Gaudin models. The top-like description also makes it easy to pass to 1 + 1 integrable equations including one-dimensional Landau-Lifshitz type magnetics, principal chiral models and their generalizations. At the same time the Gaudin models can be considered as autonomous version of the Schlesinger systems - the monodromy preserving equations which can be reduced to Painlevé equations. Finally, we come to the quantum version of the Schlesinger models described by the Knizhnik-Zamolodchikov-Bernard (KZB) equations well-known in studies of conformai field theories. The consistency condition for the KZB connections is guarantied by identities for the initial fi-matrix entering (8). In the end we briefly discuss that the equations of Painlevé-Schlesinger type can be generalized to the so-called Painlevé field theories.

2. Calogero-Moser model as integrable top. Let us start with the most simple example (more complicated and general cases can be found in [1-3]) - 2-body Calogero-Moser model. The Hamiltonian is given as



1 2 v


- I v -

2 V 2u


2 (2u)



/ v v

1 V--Z- + -

2u z

v v

2u z

in the canonical coordinates {v,u} = 1. Its Lax matrix



2 u ' z )

can be gauged transformed to the following form:



v v

2u z

-v +

L(z, S) = - x


Sii - z2S12


S21 - z2(Sn - S22) ~ z4S12 S22+z2Su


The residue matrix is given by the following change of variables:


Su S12 S21 S22




1 v

"2 M

y — (« m3 — 2z/m2 ) ——vu + i/


i.e. the canonical variables v, u are transformed into the generators of the Lie algebra gl2 with the Poisson-Lie brackets:

{Sij,Ski} = SuSkj — SkjSu. (13)

The Hamiltonian (9) acquires the form

H= -S12(Su -S22) = ^tr [SJ(S)],

J(S) = -




Su — S 22 —S 12

of the integrable (rational) top of Euler-Arnold type with the inverse inertia tensor J(S). Equations of motion

S={H,S} = [S,J(S)], (15)

can be written in the Lax form (1) with the Lax matrix (11) and the M-matrix

M(z,S) = -í Sl2 ° V (16)

y 5*11 — 5*22 + 2z S12 —S12 J

The top form of the Calogero-Moser model allows us to relate to it the non-dynamical r-matrix. Indeed, the classical r-matrix provides the Poisson brackets between matrix elements of the Lax matrix in the form:

Eij <g> Ekl {Lij(z), Lki(w)} :=


= {Li(z), L2(w)} = [Li(z) + L2{w),r12{z ~ «;)], (17)

where for gl2 : L\ = L eg) 1 =

L2 = 1 ® L =

L11 1-2x2 ¿12 1-2x2 ¿21 12x2 ¿22 12x2



. In our example the



V 02X2

classical r-matrix equals

0 \

0 0

l/z )

It satisfies the classical Yang-Baxter equation [7-12(2 - w),r13(z)} + [r12(z - w),r23(w)} + + [ri3(z),r23(u>)} = 0 and is simply related to the Lax matrix: L(z) = tr2 [r12(z)S2].

( 1 /z 0 0

—z 0 l/z

—z l/z 0

V-3 z z




3. Relativistic models and quantum R-matrices. As it was shown in [3] the construction of the top model can be generalized to the relativistic deformation of integrable systems. The simplest example here is the rational 2-body Ruijsenaars-Schneider model. It is described by the Hamiltonian

rs = 2U-1 / 2U + T1 ,

2u 2u ' V '

where r¡ is the coupling constant and c is the light speed. As in the previous case the Ruijsenaars-Schneider model can be rewritten in the form of the (relativistic) top. The Lax matrix has the following form

L\z,S) = -S2y2 + Í^l2y2-{z + r1) x 2 T]



(«Su - S22) + (rj2 +z2 + r/z)S12 -S



Sn(v,u) = -^{ev/c-e-v/c

with the change of variables

u "2 1




ev/c _ e-v/c

S21(v,u) = ~ \evlc{u-ri)2 -e-v/c(u + r])2

S22(v, u) = ±- \ev/c(u - V)2 - e-v'c(u + V)2 ¿u L


and equation of motion


S=[S,J\S% JV(S) = — 12X2-


rjS 12


■rfSi2 +r/(Sn -S22) -r/S12


generated by the Lax equations (1) with (22) and M(z,S) = —L(z,S) (i.e. the M-matrix here is of the same form as the L-matrix (11) in the non-relativistic case up to the sign).

The Poisson brackets are defined by the quadratic r-matrix structure

{L\{z), L»} = [L\(z) mw),r12(z - w)], (25)

with the rational r-matrix (18). The quadratic Poisson brackets for the matrix elements of S is the classical Sklyanin algebra:

{Sl,S2} = [J"(S')l<S2,Pl2].


The most important statement here is the following: while the non-relativistic top is described by the classical r-matrix (20) the relativistic top is related in the same way to the quantum R-matrix:


U>{z, S) = ]T Rlkl{z) Eit Slk = tr2 [R?2(z)S2] ,

i,j,k,l = 1


where the relativistic deformation parameter r] plays the role of the Planck constant. That is to say that R^2(z) satisfies the quantum Yang-Baxter equation:

Rv12(z - w)Rv13(z - y)B%3(w -y) =

= RU™-y)RVi2(z-™)RUz~y)- (28)

With a knowledge of the Lax matrix we know the quantum i?-matrix as well. For the case (22) we have:

R\z) = 0


h^ + z-1 00 0

-h-z h-1 z-1 0

-h-z z~l h-1 0

y~h3 - 2zh2 - 2hz2 - z3 h + z h + z h~1 + z~1j


The classical limit

Rv12(z) = h-11 (g) 1 + r12(z) + hml2+ 0(h2) (30)

provides the classical Yang-Baxter equation (19) from the quantum one (28) and corresponds to the non-relativistic limit at the level of mechanical systems:


7] := z//c, c —> 00 : H = = -r]-1trS(v,u)=2+^HCM+o(\) (31)

and S(v,u) = —■j lim cS(v,u).

4. Spin chains and Gaudin models. After integrable many-body systems are included into the class of integrable tops we can proceed to more complicated models. Having the quadratic Poisson structure (25) the classical periodic spin chain with n sites is naturally defined via the monodromy matrix

T(z,S1,...,Sn) = T(z) = Lr>(z-z1,S1)...Lr>(z-zn,Sn),


where zj. are the inhomogeneities parameters.

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