# Integrability of the and Ruijsenaars-Schneider models

Abstract: We study the and Ruijsenaars-Schneider() models with interaction potential of trigonometric and rational types. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking nonrelativistic limit, we also obtain the Lax pairs for the corresponding Calogero-Moser systems.

PACS: 02.20.+b, 11.10.Lm, 03.80.+r

## I Introduction

Ruijsenaars-Schneider() and Calogero-Moser() models as integrable many-body models recently have attracted remarkable attention and have been extensively studied. They describe one-dimensional -particle system with pairwise interaction. Their importance lies in various fields ranging from lattice models in statistics physics[1, 2], the field theory to gauge theory[3, 4]. e.g. to the Seiberg-Witten theory [5] et al. Recently, the Lax pairs for the elliptic models in various root system have been given by Olshanetsky et al[6], Bordner et al[7, 8, 9, 10] and D’Hoker et al[11] respectively, while the commutative operators for model based on various type Lie algebra given by Komori [12, 13], Diejen[14, 15] and Hasegawa[1, 16]et al. An interesting result is that in Ref. [17], the authors show that for the trigonometric and models exist the same non-dynamical -matrix structure compared with the usual dynamical ones. On the other hand, similar to Hasegawa’s result that model is related to the Sklyanin algebra, the integrability of model can be depicted by Gaudin algebra[18].

As for the type model, commuting difference operators acting on the space of functions on the type weight space have been constructed by Hasegawa et al in Ref. [16]. Extending that work, the diagonalization of elliptic difference system of that type has been studied by Kikuchi in Ref. [19]. Despite of the fact that the Lax pairs for models have been proposed for general Lie algebra even for all of the finite reflection groups[10], however, the Lax integrability of model are not clear except only for -type[20, 2, 21, 22, 23, 24] and for by the authors by straightforward construction[25], i.e. the general Lax pairs for the models other than -type have not yet been obtained.

Extending the work of Ref. [25], the main purpose of the present paper is to provide the Lax pairs for the and Ruijsenaars-Schneider() models with the trigonometric and rational interaction potentials. The key technique we used is Dirac’s method on the system imposed by some constraints. We shall give the explicit forms of Lax pairs for these systems. It is turned out that the and systems can be obtain by Hamiltonian reduction of and ones. The characteristic polynomial of the Lax matrixes leads to a complete set of involutive Hamiltonians associated with the root system of and . In particular, taking their non-relativistic limit, we shall recover the systems of corresponding types.

The paper is organized as follows. The basic materials about model are reviewed in Sec. II. We also give a Lax pair associating with Hamiltonian which has a reflection symmetry with respect to the particles in the origin. The main results are showed in Secs. III and IV. In Sec. III, we present the Lax pairs of and models by reducing from that of model. The explicit forms for the Lax pairs are given in Sec. IV. The characteristic polynomials, which gives the complete sets of involutive constant motions for these systems, will also be given there. Sec. V, is devoted to derive the nonrelativistic limits of these systems which coincide with the forms given in Refs. [6] and [7]. The last section is brief summary and some discussions.

## Ii -type Ruijsenaars-Schneider model

As a relativistic-invariant generalization of the -type nonrelativistic Calogero-Moser model, the -type Ruijsenaars-Schneider systems are completely integrable whose integrability are first showed by Ruijsenaars[20, 26]. The Lax pairs for this model have been constructed in Refs. [20, 2, 21, 22, 23, 24]. Recent progress have showed that the compactification of higher dimension SUSY Yang-Mills theory and Seiberg-Witten theory can be described by this model[5]. Instanton correction of prepotential associated with system have been calculated in Ref. [27].

### ii.1 The Lax operator for A model

Let us briefly give the basics of this model. In terms of the canonical variables , enjoying in the canonical Poisson bracket

(II.1) |

we give firstly the Hamiltonian of system

(II.2) |

Notice that in Ref. [20] Ruijsenaars used another “gauge” of the momenta such that two are connected by the following canonical transformation:

(II.3) |

The Lax operator for this model has the form(for the trigonometric case)

and for the rational case

(II.4) |

where

(II.5) |

and denotes the coupling constant.

It is shown in Ref. [23] that the Lax operator satisfies the quadratic fundamental Poisson bracket

(II.6) |

where and the four matrices read as

(II.7) |

The forms of are

(II.8) |

for the trigonometric case and

(II.9) |

for the rational case. The symbol means with

Noticing that

(II.10) |

one can get the characteristic polynomials of and [28]

(II.11) | |||||

(II.12) |

where and

(II.13) | |||||

(II.14) |

Define

(II.15) |

from the fundamental Poisson bracket Eq.(II.6), we can verify that

(II.16) |

In particular, the Hamiltonian Eq.(II.2) can be rewritten as

(II.17) |

It should be remarked the set of integrals of motion Eq.(II.15) have a reflection symmetry which is the key property for the later reduction to and cases. i.e. if we set

(II.18) |

then the Hamiltonians flows are invariant with respect to this symmetry.

### ii.2 The construction of Lax pair for the model

As for the model, a generalized Lax pair has been given in Refs. [20, 2, 21, 22, 23, 24]. But there is a common character that the time-evolution of the Lax matrix is associated with the Hamiltonian . We will see in the next section that the Lax pair can’t reduce from that kind of forms directly. Instead, we give a new Lax pair which the evolution of are associated with the Hamiltonian

(II.19) |

where can be constructed with the help of matrices as follows

(II.20) |

The explicit expression of entries for is

(II.21) |

for trigonometric case and

(II.22) |

for rational case.

## Iii Hamiltonian reductions of and models from -type ones

Let us first mention some results about the integrability of Hamiltonian (II.2). In Ref. [26] Ruijsenaars demonstrated that the symplectic structure of and type systems can be proved integrable by embedding their phase space to a submanifold of and type ones respectively, while in Refs. [14, 15] and [13], Diejen and Komori, respectively, gave a series of commuting difference operators which led to their quantum integrability. However, there are not any results about their Lax representations so far, i.e. the explicit forms of the Lax matrixes , associated with a (respectively) which ensure their Lax integrability, haven’t been proposed up to now except for the special case of [25]. In this section, we concentrate our treatment to the exhibition of the explicit forms for general and systems. Therefore, some previous results, as well as new results, could now be obtained in a more straightforward manner by using the Lax pairs.

For the convenience of analysis of symmetry, let us first give vector representation of Lie algebra. Introducing an dimensional orthonormal basis of

(III.1) |

Then the sets of roots and vector weights are:

(III.2) | |||||

(III.3) |

The dynamical variables are canonical coordinates and their canonical conjugate momenta with the Poisson brackets of Eq.(II.1) . In general sense, we denote them by dimensional vectors and ,

so that the scalar products of and with the roots , , etc. can be defined. The Hamiltonian Eq.(II.2) can be rewritten as

(III.4) |

in which and is given in Eq.(II.5) for various choices of potentials. Here, the condition means that the summation is over roots such that for

So does for

### iii.1 model

The set of roots consists of two parts, long roots and short roots:

(III.5) |

in which the roots are conveniently expressed in terms of an orthonormal basis of :

(III.6) |

In the vector representation, vector weights are

(III.7) |

The Hamiltonian of model is given by

(III.8) |

From the above data, we notice that either for or Lie algebra, any root can be constructed in terms with vector weights as where By simple comparison of representation between or , one can found that if replacing with in the vector weights of algebra, we can obtain the vector weights of one. Also does for the corresponding roots. This hints us it is possible to get the model by this kind of reduction.

For model let us set restrictions on the vector weights with

(III.9) |

which correspond to the following constraints on the phase space of -type model with

(III.10) |

Following Dirac’s method[29], we can show

(III.11) |

i.e. is the first class Hamiltonian corresponding to the above constraints Eq. (III.10). Here the symbol represents that, only after calculating the result of left side of the identity, could we use the conditions of constraints. It should be pointed out that the most necessary condition ensuring the Eq. (III.11) is the symmetry property Eq. (II.18) for the Hamiltonian Eq. (II.2). So that for arbitrary dynamical variable we have

(III.12) | |||||

where

(III.13) |

and the denote the Dirac bracket. By straightforward calculation, we have the nonzero Dirac brackets of

(III.14) |

Using the above data together with the fact that is the first class Hamiltonian (see Eq. (III.11), we can directly obtain Lax representation of model by imposing constraints on Eq. (II.19)

(III.15) | |||||

(III.16) |

where

(III.17) |

so that

(III.18) |

Nevertheless, the is not the first class Hamiltonian, so the Lax pair given by many authors previously can’t reduce to case directly by this way.

### iii.2 model

The root system consists of three parts, long, middle and short roots:

(III.19) |

in which the roots are conveniently expressed in terms of an orthonormal basis of :

(III.20) |

In the vector representation, vector weights can be

(III.21) |

The Hamiltonian of model is given by

(III.22) |

By similar comparison of representations between or , one can found that if replacing with and with in the vector weights of Lie algebra, we can obtain the vector weights of one. Also does for the corresponding roots. So by the same procedure as model, it is expected to get the Lax representation of model.

For model, we set restrictions on the vector weights with

(III.23) |

which correspond to the following constraints on the phase space of -type model with

(III.24) |

Similarly, we can show

(III.25) |

i.e. is the first class Hamiltonian corresponding to the above constraints Eq. (III.24). So and can be constructed as follows

(III.26) |

while is

## Iv Lax representations of and models

### iv.1 model

The Hamiltonian of system is Eq.(III.8), so the canonical equations of motion are

(IV.1) | |||||

(IV.2) | |||||

where

(IV.3) |

The Lax matrix for model can be written in the following form for the rational case

(IV.4) |

which is a matrix whose indices are labelled by the vector weights, denoted by , can be written as

(IV.5) |

where

(IV.6) | |||||

(IV.7) | |||||

and

(IV.8) |

For the trigonometric case, we have

(IV.9) |

and

(IV.10) |

where

(IV.11) | |||||

(IV.12) | |||||

where take the value as Eq.( IV.8) with the trigonometric forms of and .

The satisfies the Lax equation

(IV.13) |

which equivalent to the equations of motion Eq.(IV.1) and Eq.(IV.2). The Hamiltonian can be rewritten as the trace of

(IV.14) |

The characteristic polynomial of the Lax matrix generates the involutive Hamiltonians

(IV.15) |

where , and Poisson commute

(IV.16) |

This can be deduced by verbose but straightforward calculation to verify that the is the first class Hamiltonian with respect to the constraints Eq.(III.10), using Eq.(II.16), (III.12) and the first formula of Eq.(III.17).

The explicit form of are