Abstract
In this review we explain interrelations between the Elliptic CalogeroMoser model, integrable Elliptic EulerArnold top, monodromy preserving equations and the KnizhnikZamolodchikovBernard equation on a torus.
ITEPTH40/04
Universality of CalogeroMoser model
M.A.Olshanetsky
RIMS, Kyoto University, Japan;
Institute of Theoretical and Experimental Physics, Moscow, Russia,
Dedicated to 70th birthday of Francesco Calogero
Contents
1 Introduction
The Calogero Model first proposed by Francesco Calogero as a model of exactly solvable onedimensional nuclei [5, 6]. Later different generalizations of the model on the classical and quantum level were introduced in Refs. [23, 27, 35] (see also reviews [28, 29]). Nowadays these models, that we will call for brevity the Calogero model (CM), play a fundamental role in the contemporary theoretical physics. We shortly remind some of them. The first indication of this role came from the papers [1, 18] where interrelations between classical solutions of the rational and elliptic CM and special solutions of the KdV and KP equations were established. Last fifteen years a wide range of applications was discovered. Among them are interrelations between the CalogeroSutherland model [35] and the Fractional Quantum Hall Effect [2], integrable onedimensional spin models with longrange interactions [16]. Important role plays the classical CM in the SUSY YangMills theory [11] and in the string theory [36].
Most likely, the fundamental character of CM can be explained by their grouptheoretical and geometrical nature. In the very beginning of seventies during Francesco Calogero visit to ITEP Ascold Perelomov and I have realized that the CalogeroSutherland Hamiltonians up to a conjugation coincide with the radial parts of the second Casimir operators on and . This observation was a starting point of our investigations of classical and quantum integrable systems, related to Lie algebras. According to this approach it was established that solutions of the classical rational and the trigonometric models come from a free motion on Lie algebras and Lie groups [17, 29, 30]. In this way their quantum counterparts are related to the representation theory of simple Lie algebras [31]. It imlies, in particular, that the wavefunctions are just some special matrix elements in irreducible representations.
In the elliptic case the situation is more elaborate. The classical elliptic CalogeroMoser model (ECMM) is a particular example of the Hitchin systems [13]. It is a wide class of classical integrable systems that come from a topological 3d gauge theory. The inclusion of CM in the Hitchin theory was observed independently in Refs. [8, 12, 25, 26].
In this brief review we touch another facets of the classical and quantum ECCM. In Sect.1 we discuss equivalence of the classical ECMM and the socalled elliptic top (ET). The later describes the classical degrees of freedom that located on a vertex of the generalization of the XYZ lattice model. For the twoparticle case it leads to the equivalence between the twodimensional version of ECMM [19, 22] and the LandauLifshitz model. This section is based on Ref. [22]. The correspondence between the classical ECMM for two particle case and the VI equation [21] is discussed in Sect.2. Finally, in Sect.3 we present the interpretation of the equation corresponding to the quantum ECMM and KnizhnikZamolodchikovBernard equation that arises in the WessZuminoWitten model on a torus.
2 CalogeroMoser model and Integrable tops
2.1 ECMM with spin
Description of the system
The ECMM system is defined by the Hamiltonian
(2.1) 
on the phase space with the canonical brackets
(2.2) 
Here are coordinates and are their momenta. In what follows we assume that , .
Let be a torus endowed with a complex structure with parameter . The doubleperiodicity of the Weierstrass function implies that the particles lie on the torus , while . In fact, in the potential in (2.1) we will consider another doubleperiodic function
where is the second Eisenstein function and . The additional constant becomes essential only on the quantum level.
The system has the ”spin” generalization [10]. Let be an order matrix. We consider as an element of the Lie algebra . The linear (LiePoisson) brackets on for the matrix elements assume the form
(2.3) 
Let be a coadjoint orbit
(2.4) 
where is the diagonal subgroup of . The phase space of the ECMM with spin is
(2.5) 
where is the symplectic quotient
with respect to the action of . It implies i) the moment constraint
,
ii) the gauge fixing, for example, as .
Note that
(2.6) 
The spin ECMM Hamiltonian has the form
(2.7) 
The case (2.1) corresponds to the most degenerate nontrivial orbit when eigenvalues coincide. In this case . The coupling constant is proportional to .
The equations of motion can be read off from (2.2), (2.3) and (2.7)
(2.8) 
(2.9) 
(2.10) 
where the operator acts on as
Lax representation
The system has the Lax representation
Introduce an auxiliary elliptic curve with the same modular parameter as above. It plays the role of the basic spectral curve with the spectral parameter . The Lax matrix depends on and has the form
(2.11) 
where
(2.12) 
and is the odd thetafunction
The matrix corresponding to the flow (2.8)–(2.10) takes the form
(2.13) 
The Lax matrix is a quasiperiodic meromorphic functions on the spectral curve taking values in the Lie algebra with a simple pole at
(2.14) 
(2.15) 
where . These conditions uniquely characterized the nondiagonal part of .
The Lax equation is equivalent to the linear problem
(2.16) 
(2.17) 
The additional equation
(2.18) 
implies that is also meromorphic on .
2.2 Elliptic top on
Description of the top
Consider the EulerArnold top (EAT) on the group . Its phase space is embedded in the Lie coalgebra as a coadjoint orbit. It is endowed with LiePoisson brackets (2.3).
The EAT is determined by a symmetric operator , that is called the inverse inertia operator. The Hamiltonian of the system is , where . A special choice of leads to an integrable system. The elliptic top (ET) is an example of an integrable EAT.
To define the inverse inertia operator for ET we choose another basis in . Define two type of matrices
Consider a twodimensional lattice of order . The matrices
generate a basis in . The commutation relations in this basis assume the form
where
(2.19) 
The the Poisson structure on the dual space is given by the linear LiePoisson brackets coming from (2.19)
(2.20) 
Let , be a regular lattice of order on . Introduce the following constant on : . Then the operator for the ET is defined as
(2.21) 
Let . The Hamiltonian has the form
(2.22) 
It defines the equations of motion
(2.23) 
or
(2.24) 
The phase space of ET is a coadjoint orbit of
(2.25) 
Note that it dimension coincides with .
2.3 The map of the ECMM system to the ET system
The map is defined as the conjugation of by some matrix :
(2.30) 
The matrix is a meromorphic quasiperiodic map . It is uniquely defined by its quasiperiodicity and the pole at . The latter means that can be considered as a singular gauge transformation. Assume that an eigenvector of the residue of at belongs to the kernel of . Then it can be proved that (2.30) preserves the order of the pole.
The matrix has the following form. The quasiperiodicity of and leads to the following relations
(2.31) 
(2.32) 
Let be the diagonal matrix defining the coadjoint orbit (2.4) in the ECMM
(2.33) 
Then depends on a choice of the eigenvector of the orbit matrix , that belongs to the kernel of and corresponds to the eigenvalue (2.33). It has the form , where stands on the th place and is defined up to a maximal parabolic subgroup.
We construct first  matrix that satisfies (2.31) and (2.32) but has a special kernel:
(2.34) 
where is the theta function with a characteristic
It can be proved that the kernel of at is generated by the following columnvector :
Then the matrix assumes the form
(2.35) 
It leads to the map .
This transformation means that the particle degrees of freedom of the ECMM along with the spin variables boil down to the orbits variables . For the most degenerate orbit in the standard ECMM, defined by the coupling constant this transformation leads to the degenerate orbit of the ET with the same value of Casimir. Note that equation for ECMM with spin (2.9) remind the equation of motion for the EAT with the timedependent operator (2.7), (2.8). The only difference is the structure of the phase spaces (2.5) and (2.25). The gauge transform carries out the pass from to . It depends only on the part of variablis on , namely on and through the eigenvector .
Consider in detail the case , when the system has the one degree of freedom. Let the eigenvector of has the form and put , where denote the sigma matrices. Then the transformation has the form
(2.36) 
3 CalogeroMoser model and Isomonodromic deformations
The famous Painlevé VI equation depends on four free parameters (PVI) and has the form
(3.1) 
It can be transformed to the elliptic form [3, 23, 33] that we will use. Let , are the half periods of the elliptic curve and
Then (3.1) takes the form
(3.2) 
wherethe variables are replaced as
We will not consider here the general case and restrict ourselves to the case ^{1}^{1}1the general case was investigated in [37].. Then PVI assumes the form
It is a nonautonomous Hamiltonian system with the same Hamiltonian as for the twobody ECMM
but now the module plays the role of the time.
Let us introduce the new parameter and consider the equation
(3.3) 
It can be achieved by the rescaling the dynamical variables and the halfperiods
(3.4) 
The equation (3.3) has the Lax representation
Let , , , where corresponds to some fixed module and is deined by (2.12), . The Lax matrices assume the form
(3.5) 
The Lax equation can be considered as the consistency condition for the linear system
(3.6) 
(3.7) 
(3.8) 
where (3.8) implies the holomorphity of the BakerAkhiezer function in the coordinates deformed by : , .
We will prove that the linear problem for the twobody ECMM (2.16)–(2.18) coincides with (3.6)–(3.7) in the limit . The constant plays the role of the Planck constant and (2.16)–(2.18) is the result of the quasiclassical limit. Define the time corresponding to the twobody ECMM Hamiltonian as , and represent the BakerAkhiezer function in the WKB approximation form
(3.9) 
where is a group valued function and , are diagonal matrices. Substitute (3.9) in the linear system (3.6),(3.8),(3.7). If and , there are no terms of order . In the quasiclassical limit we put . In the zero order approximation we come to the linear system of the twobody ECMM (2.16)–(2.18). The BakerAkhiezer function takes the form
This passage from the autonomous twobody ECCM to the Painlevé VI equation is an example of the Whitham quantization. The quasiclassical limit of the full PVI yields the generalization of ECMM [15].
4 CalogeroMoser model and KnizhnikZamolodchikovBernard equation
The KnizhnikZamolodchikovBernard equation (KZB) is the generalization on a torus of the KnizhnikZamolodchikov equation [20] obtained by D.Bernard [4]. Its solutions are correlation functions of the WessZuminoWitten model on a the torus with marked points. The KZB equation has the form of the nonstationer Schrödinger equation where the role of times is played by the module and the position of points. The classical limit of the KZB equations in general case was considered in [14]. We consider the case . The correlation function depends on a finitedimensional representation attributing to the marked point. The KZB equation has the form
(4.1) 
where are generators of the matrix elements in and . To pass to the classical limit in the KZB equations we replace the conformal block by its quasiclassical expression
(4.2) 
where . Consider the classical limit and assume that values of the Casimirs corresponding to the irreducible representations also go to infinity. Let all values are finite. It allows to fix the coadjoint orbits in the marked point. In the classical limit (4.1) is transformed to the HamiltonJacobi equation for the action
In this way we come to isomonodromy preseving case.
On the critical level we come to the eigenvalue problem for the quantum ECMM for the zero eigenvalue. It allows to describe the wavefunctions of the quantum ECMM [7, 9].
We summarize the result of last two sections in the following diagram.
Here before going from PVI to the classical ECMM we renormalize the variables and the halfperiods according with (3.4).
Acknowledgments.
I thank
the Research Institute for Mathematical Science of Kyoto University for the hospitality
where this work was done.
The work is supported by the grants NSh19992003.2 of the scientific
schools, RFBR030217554 and CRDF RM12545.
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