The second bosonization of the CKP hierarchy

Abstract

In this paper we discuss the second bosonization of the Hirota bilinear equation for the CKP hierarchy introduced in the work of Date et al. [J. Phys. Soc. Jpn. 50(11), 3813–3818 (1981)]. We show that there is a second, untwisted, Heisenberg action on the Fock space, in addition to the twisted Heisenberg action suggested by Date et al. [J. Phys. Soc. Jpn. 50(11), 3813–3818 (1981)] and studied in the work of van de Leur et al. [SIGMA 8, 28 (2012)]. We derive the decomposition of the Fock space into irreducible Heisenberg modules under this action. We show that the vector space spanned by the highest weight vectors of the irreducible Heisenberg modules has a structure of a super vertex algebra, specifically the symplectic fermion vertex algebra. We complete the second bosonization of the CKP Hirota equation by expressing the generating field via exponentiated boson vertex operators acting on a polynomial algebra with two infinite sets of variables.