CKP Hierarchy, Bosonic Tau Function and Bosonization Formulae

Abstract

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.

The two bosonizations of the CKP hierarchy: Overview and character identities

“…In the untwisted case, the vector space spanned by the highest weight vectors has a structure equivalent to the symplectic fermion vertex algebra, as we showed in [Ang17]. In the twisted case, although in [vOS12] the authors don't use the language of vertex algebras, their calculations show that the vector space spanned by the highest weight vectors has a structure of a twisted fermion vertex algebra (in the sense of [ACJ14]), see Theorem 2.13 and Corollary 2.14. In Section II we also recount two of the gradings we will use in Section III: the charge grading, by the 0-mode h Z 0 of the untwisted Heisenberg field, (2.15), and the degree grading, by the 0-mode L 0 of one of the Virasoro fields, (2.18).…”

“…The degree grading is also used in [vOS12], although without its connection to the Virasoro field. Next, we will need some notations for the corresponding indexing sets in the decompositions.…”

“…In this conference proceedings, we start by discussing two important, albeit surprising, properties of the algebraic Hirota equation (1.1) and its Fock space: that there are no finite sum solutions (Lemma 2.3), but that there are series solutions in the series completion of the Fock space (Proposition 2.5 and its corollary). Specifically, as the name CKP suggests, Proposition 2.5 shows that the Hirota equation (1.1) commutes with the action of the c ∞ Lie algebra (this was remarked upon and used in [vOS12], but without proof). Next, we review the two bosonizations of the CKP hierarchy in parallel, by addressing each of the three stages above.…”

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We discuss the Hirota bilinear equation for the CKP hierarchy introduced in [DJKM81b], and its algebraic properties. We review in parallel the two bosonizations of the CKP hierarchy: one arising from a twisted Heisenberg algebra ([vOS12]), and the second from an untwisted Heisenberg algebra ([Ang17]). In particular, we recount the decompositions into irreducible Heisenberg modules and the (twisted) fermionic structures of the spaces spanned by the highest weight vectors under the two Heisenberg actions.

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The “ghost” symmetry in the CKP hierarchy

“…where dλ ≡ ∞ dλ 2πi = Res λ=∞ . By now, the CKP hierarchy has attracted many researches [3][4][5][6][7][8][9][10][11][12][13]. In contrast to the KP and the BKP cases, there seems not a single tau function to describe the CKP hierarchy in the form of Hirota bilinear equations (that is, Hirota's equations are no longer of the type P (D)τ · τ = 0) [2].…”