CKP Hierarchy, Bosonic Tau Function and Bosonization Formulae

TL;DR

This paper advances the theory of the CKP hierarchy by establishing bosonization formulae and revealing the emergence of specific orthogonal polynomials within this framework.

Contribution

It introduces new bosonization formulae for the CKP hierarchy and identifies orthogonal polynomials that naturally arise in this context.

Findings

Bosonization formulae for CKP hierarchy are developed.

Orthogonal polynomials appear in the CKP theory, involving even and odd variables.

The paper connects CKP hierarchy with polynomial structures in Grassmann variables.

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.