Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 036, 28 pages      arXiv:1102.0087      https://doi.org/10.3842/SIGMA.2012.036
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

CKP Hierarchy, Bosonic Tau Function and Bosonization Formulae

Johan W. van de Leur a, Alexander Yu. Orlov b and Takahiro Shiota c
a) Mathematical Institute, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
b) Nonlinear Wave Processes Laboratory, Oceanology Institute, 36 Nakhimovskii Prospect, Moscow 117851, Russia
c) Mathematics Department, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

Received January 17, 2012, in final form June 07, 2012; Published online June 22, 2012

Abstract
We develop the theory of CKP hierarchy introduced in the papers of Kyoto school [Date E., Jimbo M., Kashiwara M., Miwa T., J. Phys. Soc. Japan 50 (1981), 3806-3812] (see also [Kac V.G., van de Leur J.W., Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 369-406]). We present appropriate bosonization formulae. We show that in the context of the CKP theory certain orthogonal polynomials appear. These polynomials are polynomial both in even and odd (in Grassmannian sense) variables.

Key words: integrable system; Pfaffian; Hafnian; symmetric functions; Schur type functions.

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