nLab Calogero model

Overview

Calogero models comprise several integrable systems in classical mechanics and their relatives; they are describing interacting systems of $n$ particles on a circle or on a line. Basic variants include Calogero-Moser system, Calogero-Sutherland system and “relativistic” Ruijsenaars models?. Calogero-Moser system was a historical source of the study of Dunkl operators and Cherednik’s double Hecke algebras. The Calogero-Moser system can also be related to the rational $n$-soliton solutions of rational KP equation; the elucidation of the relation between the soliton solutions and solutions of the Calogero-Moser system is rather deep and is called Calogero-Moser correspondence. Its modern formulation involves noncommutative algebraic geometry.

Some special functions come out of analysis of Calogero models, like Jack polynomials.

Literature

• Pavel Etingof, Lectures on Calogero-Moser systems, pdf
• D. Kazhdan, B. Kostant, S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure Appl. Math. 31 (1978), 481-508, MR478225, doi
• H. Airault, H. P. McKean, J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), no. 1, 95–148; MR0649926, doi
• E. Mukhin, V. Tarasov, A. Varchenko, KZ characteristic variety as the zero set of classical Calogero-Moser Hamiltonians, arxiv/1201.3990
• Sergio Andraus, Makoto Katori, Seiji Miyashita, Calogero-Moser systems as a diffusion-scaling transform of Dunkl processes on the line, arxiv/1211.6561
• scholarpedia Calogero-Moser system
• Yuri Berest, Oleg Chalykh, Deformed Calogero–Moser operators and ideals of rational Cherednik algebras, Commun. Math. Phys. 400, 133–178 (2023) doi arXiv:2002.08691

Review:

• Martin Hallnäs, Calogero-Moser-Sutherland systems, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.12932]

On Calogero-Moser correspondence:

• I. M. Kričever, Rational solutions of the Kadomcev-Petviašvili equation and the integrable systems of $N$ particles on a line, Funkcional. Anal. i Priložen. 12 (1978), no. 1, 76–78, MR488139
• Takahiro Shiota, Calogero-Moser hierarchy and KP hierarchy, J. Math. Phys. 35 (1994), no. 11, 5844–5849, MR95i:58095
• George Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian, With an appendix by I. G. Macdonald. Invent. Math. 133 (1998), no. 1, 1–41, MR99f:58107, doi
• Yuri Berest, George Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry (with an Appendix by M. Van den Bergh), Internat. Math. Res. Notices 26 (2002), 1347–1396.
• Yuri Berest, George Wilson, Mad subalgebras of rings of differential operators on curves, Advances in Math. 212 no. 1 (2007), 163–190.
• Victor Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett. 8 (2001), 377-400.
• Yuri Berest, Oleg Chalykh, $A_\infty$-modules and Calogero-Moser spaces, J. Reine Angew. Math. 607 (2007), 69–112, MR2009f:16019, doi