# The second bosonization of the CKP hierarchy

## TL;DR

This paper develops a second bosonization framework for the CKP hierarchy's Hirota bilinear equation, revealing a new untwisted Heisenberg action and connecting it to super vertex algebra structures.

## Contribution

It introduces a novel untwisted Heisenberg action on the Fock space and links the highest weight vectors to the symplectic fermions vertex algebra.

## Key findings

- Decomposition of Fock space into irreducible Heisenberg modules.
- Identification of a super vertex algebra structure in the highest weight space.
- Explicit expression of the generating field via boson vertex operators.

## Abstract

In this paper we discuss the second bosonization of the Hirota bilinear equation for the CKP hierarchy introduced by Date, Jimbo, Kashiwara and Miwa. We show that there is a second, untwisted, Heisenberg action on the Fock space, in addition to the twisted Heisenberg action suggested by Date, Jimbo, Kashiwara and Miwa and studied by van de Leur, Orlov and Shiota. We derive the decomposition of the Fock space into irreducible Heisenberg modules under this action. We show that the space spanned by the highest weight vectors of the irreducible Heisenberg modules has a structure of a super vertex algebra, specifically the symplectic fermions 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.