Gauge Theory Approach Towards an Explicit Solution of the (Classical) Elliptic Calogero-Moser System

DOI

10.2991/jnmp.2005.12.s1.35How to use a DOI?

Abstract

We discuss the relation of the trigonometric Calogero-Moser (CM) system to YanMills gauge theories and its generalization to the elliptic case. This yields a liearization of the time evolution of the elliptic CM system and suggests two promising strategies for finding a fully explicit solution of this model. We also present a large class of integrable spin-particle systems generalizing the elliptic CM system. Prologue. Integrable systems associated with the name of Calogero [1, 2, 3, 4] seem to have a strong attraction to me: on two different occasions I started to work on seemingly unrelated problems, made progress, but at some point realized that the natural application of our results was, unexpectedly for me, in the context of Calogero-type systems. One such occasion was a project which was initially aiming at a better understanding of certain apects of the fractional quantum Hall effect [5] but eventually led to a method for solving the quantum elliptic Calogero-Sutherland system [6]. The present paper is based on the other such occasion which started as a project on two dimensional quantum chromodynamics [7, 8, 9] but eventually led to an alternative solution method for classical trigonometric Calogero-Moser-type systems [10, 11] (this somewhat curious story is told in more detail in the Epilogue of this paper). I will shortly review the solution method thus obtained for the trigonometric Calogero-Moser system, explaining in particular its particle physics motivation. I then show that this method has a natural generalization to the elliptic case, and this leads to an explicit linearization of the time evolution of the (classical) elliptic Calogero-Moser (eCM) model. I finally discuss possible strategies to turn this result into a fully explicit solution of the eCM model. I should mention various closely related papers [12, 13, 14, 15, 16, 17, 18, 19] and in particular [20, 21] whose relation to our work will be also mentioned. However, I have tried to keep my discussion self-contained. Copyright c 2005 by E Langmann 424 E Langmann This paper is based on notes which I wrote in 1999. They have remained unpublished up to now since I have felt I was just one step away from a fully explicit solution of the eCM system. However, I did not succeed with the final step since 1999, and I hope it could be helpful to make available what I have: somebody else might be interested and able to complete this story. Francesco Calogero is an outstanding example for me, not only as a scientist but also as a person. It is a pleasure for me to dedicate this paper to him.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Download article (PDF)

TY  - JOUR
AU  - Edwin Langmann
PY  - 2005
DA  - 2005/01/01
TI  - Gauge Theory Approach Towards an Explicit Solution of the (Classical) Elliptic Calogero-Moser System
JO  - Journal of Nonlinear Mathematical Physics
SP  - 423
EP  - 439
VL  - 12
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2005.12.s1.35
DO  - 10.2991/jnmp.2005.12.s1.35
ID  - Langmann2005
ER  -