PDF《HU-EP-19/02, ZMP-HH/19-2》

arXiv:1902.

06755v2 [hep-th]

13 Jun

2019 HU-EP-19/02, ZMP-HH/19-2 Quantum Trace Formulae for the Integrals of the Hyperbolic Ruijsenaars-Schneider model Gleb Arutyunov,a Rob Klabbersb and Enrico Olivuccia a II. Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg, Luruper Chaussee 149,

22761 Hamburg, Germany Zentrum f¨ ur Mathematische Physik, Universit¨ at Hamburg, Bundesstrasse 55,

20146 Hamburg,Germany b Institut f¨ ur Physik und IRIS Adlershof, Humboldt-Universit¨ at zu Berlin, Zum Gro?en Windkanal 6, D-12489 Berlin, Germany Abstract: We conjecture the quantum analogues of the classical trace formulae for the integrals of motion of the quantum hyperbolic Ruijsenaars-Schneider model. This is done by departing from the classical construction where the corresponding model is obtained from the Heisenberg double by the Poisson reduction procedure. We also discuss some algebraic structures associated to the Lax matrix in the classical and quantum theory which arise upon introduction of the spectral parameter. Contents

1 Introduction

2 2 The classical model from reduction

4 2.1 Moment map and Lax matrix

4 2.2 Poisson structure on the reduced phase space

7 2.3 Introduction of a spectral parameter

9 3 Quantum model

13 3.1 Quantum Heisenberg double

13 3.2 Quantum R-matrices and the L-operator

15 3.3 Spectral parameter and quantum L-operator

18 4 Conclusions

23 A Derivation of the Poisson structure

23 A.1 Lax matrix and its Poisson structure

23 A.2 Dirac bracket

28 B Derivation of the spectral-dependent r-matrices

32 C

1 C

1 Introduction The Ruijsenaars-Schneider (RS) models [1, 2] continue to provide an outstanding theoretical laboratory for the study of various aspects of Liouville integrability, both at the classical and quantum level, see, for instance, [5C10]. Also, new interesting applications of these type of models were recently found in conformal ?eld theories [11]. In this work we study some aspects related to the quantum integrability of the RS model with the hyperbolic potential. Recall that the de?nition of quantum integrability relies on the existence of a quantisation map which maps a complete involutive family of classical integrals of motion into a set of commuting operators on a Hilbert space. In general, there are di?erent ways to choose a functional basis for this involutive family which is mirrored by the ring structure of the corresponding commuting operators. In particular, a classical integrable structure, most conveniently encoded into a Lax pair (L, M), produces a set of canonical integrals which are simply the eigenvalues of the Lax matrix. Their commutativity relies on the existence of the classical r-matrix [12]. Provided this matrix exists one can build up di?erent classical involutive families represented, for instance, by elementary symmetric functions of the eigenvalues of L or, alternatively, by traces TrLk for k ∈ Z. Concerning the particular class of the RS hyperbolic models, the quantisation of a family of elementary symmetric functions associated to a properly chosen L is well known and given by the Macdonald operators [2, 13]. In this paper we conjecture the quantum analogues of TrLk built up in terms of the same L-operator that is used to generate Macdonald operators through the determinant type formulae [14, 15]. In fact there appear two commuting families I± k that are given by the quantum trace formulae I± k = Tr12 Ct2 12L1 ? Rt2 21Rt2 ±12L1 . . . L1 ? Rt2 21Rt2 ±12L1 , as quantisation of the classical integrals TrLk . In particular, R and ? R are two quantum dynamical R- matrices that depend rationally on the variables Q i = eqi , where qi, i = 1,N are coordinates, and satisfy a system of equations of Yang-Baxter type. Also, R is a parametric solution of the standard quantum Yang-Baxter equation.1 Departing from I± k and introducing q = e?? h , we then ?nd that these integrals are related to the Macdonald operators Sk through the q-deformed analogues of the determinant formulae that in the classical case relate the coe?cients of characteristic polynomial of L with invariants constructed out of TrLk . The commutativity of I± k and their relation to Macdonald operators has been checked by explicit computation for su?ciently large values of N. We arrive to this expression for I± k through the following chain of arguments. It is known that the Calogero-Moser-Sutherland models and their RS generalisations can be obtained at the classical level through the hamiltonian or Poisson reduction applied to a system exhibiting free motion on one of the suitably chosen initial ?nite- or in?nite-dimensional phase spaces [16]-[23], [3C6]. For instance, the RS model with the rational potential is obtained by the hamiltonian reduction of the cotangent bundle T ? G = G ? G , where G ia Lie group and G is its Lie algebra. In [19] the corresponding reduction was developed for the Lie group G = GL(N, C) by employing a special parametrisation for the Lie algebra- valued element ? = T QT ?1 ∈ G , where Q is a diagonal matrix and T is an element of the Frobenius group F ? G. An analogous parametrisation is used for the group element g = UP?1 T ?1 ∈ G, where U is Frobenius and P is diagonal. If one writes Qi = qi and Pi = exp pi, then (pi, qi) is a system of canonical variables with the Poisson bracket {pi, qj} = δij. In the new variables the Poisson structure of the cotangent bundle is then described in terms of the triangular dynamical matrix r satisfying the classical Yang-Baxter equation (CYBE) and of another matrix ? r. The cotangent bundle is easily 1For the de?nition of other quantities, see the main text. C