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

We construct the bosonization of the Fock space $\mathit{F^{\otimes \frac{1}{2}}}$ of a single neutral fermion by using a 2-point local Heisenberg field. We decompose the Fock space $F^{\otimes \frac{\mathit{1}}{\mathit{2}}}$ 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 $\mathit{F^{\otimes \frac{1}{2}}}$ofthesingleneutralfermionisisomorphictotheFockspace$$F^{\otimes \mathit{1}}$$ 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 $\mathit{F^{\otimes \frac{1}{2}}}$. 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 $F^{\otimes \frac{\mathit{1}}{\mathit{2}}}$. We construct a $W_{1+\mathrm{\infty }}$ representation on $F^{\otimes \frac{\mathit{1}}{\mathit{2}}}$ and show that under the $W_{1+\mathrm{\infty }}$ action $F^{\otimes \frac{\mathit{1}}{\mathit{2}}}$ is again isomorphic to $F^{\otimes \mathit{1}}$.