Boson-fermion correspondence of type D-A and multi-local Virasoro representations on the Fock space $F^{\otimes \frac{1}{2}}$

Abstract

We construct the bosonization of the Fock space $F^{\otimes \frac{1}{2}}$ of a single neutral fermion by using a 2-point local Heisenberg field. We decompose the Fock space $F^{\otimes \frac{1}{2}}$ as a direct sum of irreducible highest weight modules for the Heisenberg algebra ${\mathcal{H}}_{\mathbb{Z}}$, and thus we show that under the Heisenberg ${\mathcal{H}}_{\mathbb{Z}}$ action the Fock space $F^{\otimes \frac{1}{2}}$ of the single neutral fermion is isomorphic to the Fock space $F^{\otimes 1}$ of a pair of charged free fermions, thereby constructing the boson-fermion correspondence of type D-A. As a corollary we obtain the Jacobi identity equating the graded dimension formulas utilizing both the Heisenberg and the Virasoro gradings on $F^{\otimes \frac{1}{2}}$. We construct a family of 2-point-local Virasoro fields with central charge $-2+12\lambda -12{\lambda }^2,\lambda \in \mathbb{C}$, on the Fock space $F^{\otimes \frac{1}{2}}$. We construct a $W_{1+\infty }$ representation on $F^{\otimes \frac{1}{2}}$ and show that under the $W_{1+\infty }$ action $F^{\otimes \frac{1}{2}}$ is again isomorphic to $F^{\otimes 1}$.

23 pages. References added, some typos corrected

References in corpus (1)

• Multilocal fermionization