We present a construction of an integrable model as a projective type limit of Calogero-Sutherland models of N fermionic particles, when N tends to infinity. Explicit formulas for limits of Dunkl operators and of commuting Hamiltonians by means of vertex operators are given.
REFERENCES
1.
Alexandrov
, A.
and Zabrodin
, A.
, “Free fermions and tau-functions
,” J. Geom. Phys.
67
, 37
–80
(2013
).2.
Andric
, I.
, Jevicki
, A.
, and Levine
, H.
, “On the large-N limit in symplectic matrix models
,” Nucl. Phys. B
215
, 307
(1983
).3.
Awata
, H.
, Matsuo
, Y.
, Odake
, S.
, and Shiraishi
, J.
, “Collective field theory, Calogero-Sutherland model and generalized matrix models
,” Phys. Lett. B
347
(1
), 49
–55
(1995
).4.
Awata
, H.
, Matsuo
, Y.
, and Yamamoto
, T.
, “Collective field description of spin Calogero-Sutherland models
,” J. Phys. A: Math. Gen.
29
, 3089
–3098
(1996
).5.
Bernard
, D.
, Gaudin
, M.
, Haldane
, F. D. M.
, and Pasquier
, V.
, “Yang-Baxter equation in spin chains with long range interactions
,” J. Phys. A: Math. Gen.
26
, 5219
(1993
).6.
Dunkl
, C. F.
, “Differential-difference operators associated to reflection groups
,” Trans. Am. Math. Soc.
311
(1
), 167
–183
(1989
).7.
Heckman
, G. J.
, “An elementary approach to the hypergeometric shift operators of Opdam
,” Invent. Math.
103
(1
), 341
–350
(1991
).8.
Kato
, Y.
and Kuramoto
, Y.
, “Exact solution of the Sutherland model with arbitrary internal symmetry
,” Phys. Rev. Lett.
74
, 1222
(1995
).9.
Khoroshkin
, S. M.
and Matushko
, M. G.
, “Matrix coefficients of vertex operators and fermionic limit of spin Calogero–Sutherland system
” (unpublished).10.
Khoroshkin
, S. M.
, Matushko
, M. G.
, and Sklyanin
, E. K.
, “On spin Calogero–Moser system at infinity
,” J. Phys. A: Math. Theor.
50
(11
), 115203
(2017
).11.
Macdonald
, I. G.
, Symmetric Functions and Hall Polynomials
(Oxford University Press
, 1998
).12.
Nazarov
, M. L.
and Sklyanin
, E. K.
, “Sekiguchi-Debiard operators at infinity
,” Commun. Math. Phys.
324
(3
), 831
–849
(2013
).13.
Nazarov
, M. L.
and Sklyanin
, E. K.
, “Integrable hierarchy of the quantum Benjamin-Ono equation
,” Symmetry, Integrability Geom.: Methods Appl.
9
, 078
(2013
).14.
Polychronakos
, A. P.
, “Exchange operator formalism for integrable systems of particles
,” Phys. Rev. Lett.
69
(5
), 703
(1992
).15.
Sergeev
, A. N.
and Veselov
, A. P.
, “Calogero-Moser operators in infinite dimension
,” eprint arXiv:0910.1984 (2009
).16.
Sergeev
, A. N.
and Veselov
, A. P.
, “Dunkl operators at infinity and Calogero-Moser systems
,” Int. Math. Res. Not.
2015
(21
), 10959
–10986
.17.
Stanley
, R. P.
, Enumerative Combinatorics
(Cambridge University Press
, Cambridge
, 1997
), Vol. 2.18.
Uglov
, D.
, “Yangian actions on higher level irreducible integrable modules of affine .
,” preprint arXiv:math/9802048 (1998
).© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.