Pis'ma v ZhETF, vol. 101, iss. 9, pp. 723-729
© 2015 May 10
ПО ИТОГАМ ПРОЕКТОВ РОССИЙСКОГО ФОНДА ФУНДАМЕНТАЛЬНЫХ ИССЛЕДОВАНИЙ Проект РФФИ # 12-02-00594
Classical integrable systems and Knizhnik-Zamolodchikov-Bernard
equations
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
723
5*
The zero curvature equations (l)-(4) keep their forms under the gauge transformations
D + L
dt + M
D
dt
gLg-1 -{Dg)g-\ -gMg-l-{dtg)g-'
(6)
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
L(z,S,r])=tr2[RUz)S2}
(8)
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
H
CM
1 2 v
2V
- I v -
2 V 2u
1
2 (2u)
(9)
LCM(z)
/ v v
1 V--Z- + -
2u z
v v
2u z
in the canonical coordinates {v,u} = 1. Its Lax matrix
\
(10)
2 u ' z )
can be gauged transformed to the following form:
1
\
v v
2u z
-v +
L(z, S) = - x
z
Sii - z2S12
Su
S21 - z2(Sn - S22) ~ z4S12 S22+z2Su
'(H)
The residue matrix is given by the following change of variables:
S
Su S12 S21 S22
(
1
-vu
1 v
"2 M
y — (« m3 — 2z/m2 ) ——vu + i/
(12)
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) = -
S12
0
(14)
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)} :=
i,j,k,l
= {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
02x2
L
. In our example the
rn{z)
L
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
(18)
(19)
(20)
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]
S12
0
(«Su - S22) + (rj2 +z2 + r/z)S12 -S
(22)
12
Sn(v,u) = -^{ev/c-e-v/c
with the change of variables
u "2 1
2M
u
S12(v,u)
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
(23)
and equation of motion
tr<S
S=[S,J\S% JV(S) = — 12X2-
V
rjS 12
0
■rfSi2 +r/(Sn -S22) -r/S12
(24)
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\(z) mw),r12(z - w)], (25)
with the rational r-matrix (18). The quadratic Poisson brackets for the matrix elements of S is the classical Sklyanin algebra:
{Sl,S2} = [J"(S')l<S2,Pl2].
(26)
The most important statement here is the following: while the non-relativistic top is described by the classical r-matrix (20) the relativistic top is related in the same way to the quantum R-matrix:
n
U>{z, S) = ]T Rlkl{z) Eit Slk = tr2 [R?2(z)S2] ,
i,j,k,l = 1
(27)
where the relativistic deformation parameter r] plays the role of the Planck constant. That is to say that R^2(z) satisfies the quantum Yang-Baxter equation:
Rv12(z - w)Rv13(z - y)B%3(w -y) =
= RU™-y)RVi2(z-™)RUz~y)- (28)
With a knowledge of the Lax matrix we know the quantum i?-matrix as well. For the case (22) we have:
R\z) = 0
\
h^ + z-1 00 0
-h-z h-1 z-1 0
-h-z z~l h-1 0
y~h3 - 2zh2 - 2hz2 - z3 h + z h + z h~1 + z~1j
(29)
The classical limit
Rv12(z) = h-11 (g) 1 + r12(z) + hml2+ 0(h2) (30)
provides the classical Yang-Baxter equation (19) from the quantum one (28) and corresponds to the non-relativistic limit at the level of mechanical systems:
R S
7] := z//c, c —> 00 : H = = -r]-1trS(v,u)=2+^HCM+o(\) (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),
(32)
where zj. are the inhomogeneities parameters.
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