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

TL;DR

This paper explores the algebraic structures of the CKP hierarchy through two different bosonizations, revealing new identities and connections to partition statistics, and analyzing their implications on the hierarchy's properties.

Contribution

It provides a comprehensive comparison of the two bosonizations of the CKP hierarchy and derives new identities linking bosonic and fermionic descriptions.

Findings

Four different grading operators on the CKP Fock space identified.

A sum-vs-product identity relating bosonic and fermionic descriptions proved.

Connections established between CKP hierarchy and partition statistics like Dyson crank.

Abstract

We discuss the Hirota bilinear equation for the CKP hierarchy introduced by Date, Jimbo, Kashiwara and Miwa in J. Phys. Soc. Japan. 50(11), 3813-3818 (1981), and its algebraic properties. We review in parallel the two bosonizations of the CKP hierarchy: one arising from a twisted Heisenberg algebra (van de Leur, Orlov and Shiota, SIGMA 8, 28 (2012)), and the second from an untwisted Heisenberg algebra (Anguelova, J. Math. Phys. 58(7), 071707 (2017)). 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.