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

Abstract

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. We show that the two bosonizations give rise to four different diagonalizable grading operators on the CKP Fock space, not all of them commuting among each other. We compute the various graded dimensions related to these four grading operators. We prove a sum-vs-product identity relating the bosonic vs fermionic descriptions under the untwisted Heisenberg action, utilizing the charge and the degree grading operators. As a corollary, the resulting identities relate the CKP hierarchy, the Dyson crank of a partition and the Hammond-Lewis birank of a distinct integer bipartition.

Tau functions of the charged free bosons

“…where 2) and : : is the normal ordered product [16]. For convenience we collect commutation relations of the fields h(z)(h ∈ h) and Y (e γ , z) as follows:…”