EulerCalogeroMoser system from YangMills theory
Abstract
The relation between YangMills mechanics, originated from the 4dimensional
YangMills theory under the supposition of spatial homogeneity of the gauge
fields, and the EulerCalogeroMoser model is discussed in the framework of Hamiltonian
reduction.
Two kinds of reductions of the degrees of freedom are considered:
due to the gauge invariance and
due to the discrete symmetry.
In the former case, it is shown that after elimination of the gauge degrees of freedom
from the YangMills mechanics the resulting unconstrained system
represents the EulerCalogeroMoser model
with an external fourthorder potential.
Whereas in the latter, the EulerCalogeroMoser model
embedded in an external potential is derived
whose projection onto the invariant submanifold through the discrete
symmetry coincides again with the YangMills mechanics.
Based on this connection, the equations of motion
of the YangMills mechanics in the limit of the zero coupling constant
are presented in the Lax form.
pacs:
PACS: 03.20.+i, 11.10.Ef, 11.15.TkI Introduction
The present note is devoted to the discussion of the correspondence between the dynamics of particles with internal degrees interacting by pairwise forces on a line (EulerCalogeroMoser system [1, 2]) and YangMills theory with spatially constant gauge fields ( YangMills mechanics [3] (see also [4, 5] and references therein)).
The EulerCalogeroMoser model is the extension of the famous CalogeroSutherlandMoser models [6, 7, 8] (for its generalizations see [9] and reviews [10, 11]) with additional dynamical internal degrees of freedom included. It is interesting that these types of models arises in various areas of theoretical physics like the 2dimensional YangMills theory [12], black hole physics [13], spin chain systems [14], generalized statistics [15], higher spin theories [16], level dynamics for quantum systems [17], quantum Hall effect [18] and many others. An attractive feature of these generalizations is that it maintains the integrability property of the original CalogeroSutherlandMoser system. For the general elliptic version of the EulerCalogeroMoser system, the actionangle type variables have been constructed and the equations of motion have been solved in terms of Riemannian thetafunctions [19], the canonical symplectic form of this model is represented in terms of algebrogeometric data [20] using the general construction of Krichever and Phong [21].
During the past years a remarkable relation between the CalogeroMoser systems and the exact solutions of dimensional supersymmetric gauge theories has been found [22]. It has been recognized that the socalled SeibergWitten spectral curves are identical to the spectral curves of the elliptic CalogeroMoser system [23]. Furthermore the generalization of these relations to the supersymmetric gauge theories with general Lie algebras and an adjoint representation of matter hypermultiplet have been derived in [24] (for review of the recent results see, e.g., [25]).
Despite the existence of such a correspondence established on very general grounds, relations between gauge theories and integrable models are far from being understood. In the present note, we would like to point out a simple direct correspondence between the YangMills theory and the EulerCalogeroMoser model. This correspondence follows from the sequence of reductions of degrees of freedom thanks to different kinds of symmetries. At first, supposing the spatial homogeneity of gauge fields, the field theory is reduced to the 9dimensional degenerate Lagrangian model. Then the pure gauge variables are eliminated by applying the method of Hamiltonian reduction. Finally, rewriting the derived unconstrained matrix model in terms of special coordinates adapted to the action of rigid symmetry, one can arrive at the conventional form of the EulerCalogeroMoser Hamiltonian. More precisely, we shall demonstrate that the unconstrained YangMills mechanics represents the EulerCalogeroMoser system of type , i.e., the inversesquare interacting 3particle system with internal degrees of freedom related to the root system of simple Lie algebra [10, 11], and is embedded in a fourth order external potential written in the superpotential form.
Besides this reduction due to the continuous symmetry of the system, we discuss another possibility of relating the YangMills mechanics to higher order matrix models using the discrete symmetries. We shall explore the method of constructing generalizations of the CalogeroSutherlandMoser models elaborated recently by A. Polychronakos [26]. This method consists in the usage of the appropriate reduction of the original Calogero model by a subset of its discrete symmetries to an invariant submanifold of the phase space. Representing the EulerCalogeroMoser system with a special external potential as a symmetric matrix model, we shall show that such a matrix model, after projection onto the invariant submanifold of the phase space using a certain subset of discrete symmetries, is equivalent to the unconstrained YangMills mechanics. Finally, we give a Lax pair representation for the equations of motion of the YangMills mechanics in the limit of the zero coupling constant.
Ii Hamiltonian reduction of the YangMills mechanics
ii.1 The equivalent unconstrained matrix model
The dynamics of the YangMills 1form in 4dimensional Minkowski spacetime is governed by the conventional local functional
(1) 
defined in terms of the curvature 2form with the coupling constant . After the supposition of the spatial homogeneity of the connection
(2) 
the action (1) reduces to the action for a finite dimensional model, the socalled YangMills mechanics (YMM) described by the degenerate matrix Lagrangian
(3) 
The entries of the matrix are nine spatial components of the connection where with the Pauli matrices and denotes the covariant derivative Due to the spatial homogeneity condition (2), all dynamical variables and are functions of time only. The part of the Lagrangian corresponding to the selfinteraction of the gauge fields is gathered in the potential
(4) 
To express the YangMills mechanics in a Hamiltonian form, let us define the phase space endowed with the canonical symplectic structure and spanned by the canonical variables and where
(5) 
According to these definitions of the canonical momenta (5), the phase space is restricted by the three primary constraints
(6) 
and the evolution of the system is governed by the total Hamiltonian where the canonical Hamiltonian is given by
(7) 
and the matrix is defined by . The conservation of constraints (6) in time entails the further condition on the canonical variables
(8) 
that reproduces the homogeneous part of the conventional nonAbelian Gauss law constraints. They are the first class constraints obeying the Poisson brackets algebra
(9) 
In order to project onto the reduced phase space, we use the wellknown polar decomposition for an arbitrary matrix
(10) 
where is a positive definite symmetric matrix and is an orthogonal matrix . Assuming the nondegenerate character of the matrix , we can treat the polar decomposition as uniquely invertible transformation from the configuration variables to a new set of six Lagrangian coordinates and three coordinates . As it follows from further consideration, the variables parameterizing the elements of the group (Euler angles ) are the pure gauge degrees of freedom.
The field strength in terms of the new canonical variables is
(11) 
where are three leftinvariant vector fields on
(12)  
(13)  
(14) 
Here is the spin vector of the gauge field and
(15) 
Reformulation of the theory in terms of these variables allows one to easily achieve the Abelianization of the secondary Gauss law constraints. Using the representations (10) and (11), one can convince oneself that the variables and make no contribution to the secondary constraints (8)
(16) 
Hence, assuming nondegenerate character of the matrix
(17) 
we find the set of Abelian constraints equivalent to the Gauss law (8)
(18) 
After having rewritten the model in this form, we are able to reduce the theory to physical phase space by a straightforward projection onto the constraint shell . The resulting unconstrained Hamiltonian, defined as a projection of the total Hamiltonian onto the constraint shell
(19) 
can be written in terms of and as
(20) 
where denotes the gauge field spin tensor.
ii.2 Unconstrained model as particle motion on stratified manifold
In the previous section, the unconstrained dynamics of the SU(2) YangMills mechanics was identified with the dynamics of the nondegenerate matrix model (20). The configuration space of the real symmetric matrices can be endowed with the flat Riemannian metric
(21) 
whose group of isometry is formed by orthogonal transformations
(22) 
Since the unconstrained Hamiltonian system (20) is invariant under the action of this rigid group, we are interested in the structure of the orbit space given as a quotient . The important information on the stratification of the space of orbits can be obtained from the socalled isotropy group of points of configuration space which is defined as a subgroup of leaving point invariant . Orbits with the same isotropy group are collected into classes, called by strata. So, as for the case of symmetric matrix, the orbits are uniquely parameterized by the set of ordered eigenvalues of the matrix . One can classify the orbits according to the isotropy groups which are determined by the degeneracies of the matrix eigenvalues:

Principal orbittype strata, when all eigenvalues are unequal with the smallest isotropy group .

Singular orbit type strata forming the boundaries of orbit space with

two coinciding eigenvalues or , the isotropy group is .

all three eigenvalues are equal , here the isotropy group coinciding with the isometry group .

In the subsequent sections, we shall demonstrate that the dynamics of the YangMills mechanics, which takes place on the principal orbits is governed by the EulerCalogero model Hamiltonian with the external potential , while for singular orbits the corresponding system is either the Calogero model with the external potential for singular orbits of type (a) or one dimensional system with quartic potential for singular orbits of type (b).
ii.2.1 Hamiltonian on principal orbit strata
To write down the Hamiltonian describing the motion on the principal orbit strata, we introduce the coordinates along the slices and along the orbits . Namely, we decompose the nondegenerate symmetric matrix as
(23) 
with the matrix parameterized by the three Euler angles and the diagonal matrix and consider it as point transformation from the physical coordinates and to and . The Jacobian of this transformation is the relative volume of orbits
(24) 
and is regular for this stratum .
By using the generating function
(25) 
the canonical conjugate momenta can be found in the form
(26) 
where are the diagonal members of the orthogonal basis for the symmetric matrices under the scalar product
The original physical momenta can then be expressed in terms of the new canonical variables as
(27) 
with ,
(28) 
and the rightinvariant Killing vectors
(29)  
(30)  
(31) 
They satisfy the Poisson bracket algebra
(32) 
Thus, finally, we get the following physical Hamiltonian defined on the unconstrained phase space
(33) 
where
(34) 
and
(35) 
Note that the potential term in (35) has symmetry beyond the cyclic one. This fact allows us to write in the form
(36) 
with the superpotential .
This completes our reduction of the spatially homogeneous YangMills system to the equivalent unconstrained system describing the dynamics of the physical dynamical degrees of freedom. We see that the reduced Hamiltonian on the principal orbit strata is exactly the Hamiltonian of the EulerCalogeroMoser system of type , i.e., is of the inversesquare interacting 3particle system with internal degrees of freedom and related to the root system of the simple Lie algebra [10, 11] embedded in the fourth order external potential (36).
ii.3 Singular stratum
Introduction of the additional constraints
(37) 
or
(38) 
defines the invariant two and one dimensional strata. One can repeat the above consideration for these singular strata and derive, correspondingly, the following unconstrained Hamiltonians:
ii.3.1 Twodimensional strata
(39) 
where the constant denotes a value of the square of the particle internal spin.
ii.3.2 Onedimensional strata
(40) 
Iii EulerCalogeroMoser system as a free motion on space of symmetric matrices
In order to discuss the relation between the YangMills mechanics and the EulerCalogeroMoser system, it is useful to represent the later in the form of a nondegenerate matrix model. Let us consider the Hamiltonian system with the phase space spanned by the symmetric matrices and with the noncanonical symplectic form
(41) 
The Hamiltonian of the system defined as
(42) 
describes a free motion in the matrix configuration space.
The following statement is fulfilled:
The Hamiltonian (42) rewritten in special coordinates
coincides with the EulerCalogeroMoser Hamiltonian
(43) 
with nonvanishing Poisson brackets for the canonical variables ^{1}^{1}1 This system is the spin generalization of the CalogeroMoser model. Particles are described by their coordinates and momenta together with internal degrees of freedom of angular momentum type . The analogous model has been introduced in [1] where the internal degrees of freedom satisfy the following Poisson brackets relations
(45) 
To find the adapted set of coordinates in which the Hamiltonian (42) coincides with the EulerCalogeroMoser Hamiltonian (43), let us introduce new variables
(46) 
where the orthogonal matrix is parameterized by the elements, e.g., the Euler angles and denotes a diagonal matrix. This point transformation induces the canonical one which we can obtain using the generating function
(47) 
Using the representation
(48) 
where the matrices form an orthogonal basis in the space of the symmetric matrices under the scalar product
(49) 
one can find that and components are represented via the right invariant vectors fields
(50) 
From this, it is clear that the Hamiltonian (42) coincides with the EulerCalogeroMoser Hamiltonian (43).
The integration of the Hamilton equations of motion
(51) 
derived with the help of Hamiltonian (42), gives the solution of the EulerCalogeroMoser Hamiltonian system as follows: for the coordinates we need to compute the eigenvalues of the matrix , while the orthogonal matrix , which diagonalizes , determines the time evolution of internal variables.
Iv YangMills mechanics through the discrete reduction of EulerCalogeroMoser system
In this section, we shall demonstrate how the YangMills mechanics arises from the higher dimensional matrix model after projection onto a certain invariant submanifold determined by the discrete symmetries. Let us consider the classical Hamiltonian system of particles on a line with internal degrees of freedom embedded in external field with the potential and described by the Hamiltonian
(52) 
The particles are described by their coordinates and momenta together with the internal degrees of freedom of angular momentum type . The nonvanishing Poisson brackets are
(53) 
The external potential is constructed in terms of the superpotential
(54) 
with given as ^{2}^{2}2 Writing the superpotential in an invariant form as
(55) 
Below it is useful to treat the internal degrees of freedom entering into the Hamiltonian (52) in the Cartesian form
(56) 
where the internal variables and combine
the canonical pairs
with the canonical symplectic form.
The Hamiltonian (52) has the following discrete
symmetries [26]:

Parity
(57) 
Permutation symmetry
(58)
where is the element of the permutation group . The manifold of phase space defined as
(59)  
(60) 
is invariant under the action of the symmetry group where
(61) 
and is specified as
In order to project onto the manifold described by constraints (59)(60) , we use the Dirac method to deal with the second class constraints. Let us introduce the Dirac brackets between the arbitrary functions and of all variables as
(62) 
where denote all second class constraints ,
(63)  
(64) 
with the canonical algebra
(65)  
(66) 
Thus, the fundamental Dirac brackets are
(67) 
After the introduction of these new brackets, one can treat all constraints in the strong sense. Letting the constraint functions vanish, the system with Hamiltonian (52) reduces to the following one
(68) 
where
(69) 
Expression (68) for coincides with the Hamiltonian of the YangMills mechanics after taking into account that after projection onto the constraint shell (CS) (63)(64) , the potential (54) reduces to the potential of YangMills mechanics
(70) 
V Lax pair representation for YangMills mechanics in zero coupling limit
The conventional perturbative scheme of nonAbelian gauge theories starts with the zero approximation of the free theory. However, the limit of the zero coupling constant is not quite trivial. If the coupling constant in the initial YangMills action vanish, the nonAbelian gauge symmetry reduces to the symmetry. In this section, we shall discuss this free theory limit for the case of the unconstrained YangMills mechanics. The solution of the corresponding zero coupling limit of the YangMills mechanics in the form of a Lax representation will be given. The relation between (52) and (68) allows one to construct the Lax pair for the free part of the Hamiltonian (68) () using the known Lax pair for the EulerCalogeroMoser system (52) without an external potential term ().
According to the work of S.Wojciechowski [2] , the Lax pair for the system with Hamiltonian
(71) 
is
(72)  
(73) 
and the equations of motion in Lax form are
(74)  
(75) 
where the matrix .
The introduction of Dirac brackets allows one to use the Lax pair of higher dimensional EulerCalogeroMoser model (namely ) for the construction of Lax pairs () of free YangMills mechanics by performing the projection onto the constraint shell (63)(64)
(76) 
Thus, the explicit form of the Lax pair matrices for the free YangMills mechanics is given by the following matrices
(77) 
and
(78) 
The equations of motion for the YangMills mechanics in the zero constant coupling limit read in a Lax form as
(79)  
(80) 
where the matrix is
(81) 
Vi Acknowledgments
We are grateful to V.I. Inozemtsev, M.D. Mateev and HP. Pavel for discussions. We would like to thank E. Langmann for useful comments on the relations between 2dimensional YangMills theory on a cylinder and CalogeroMoser systems. B.G. Dimitrov is also acknowledged for reading of the manuscript. The work of A.M.K. was supported in part by the Russian Foundation for Basic Research under grant No. 960100101.
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