Boson-fermion correspondence of type D-A and multi-local Virasoro representations on the Fock space $\mathit {F^{\otimes \frac{1}{2}}}$F⊗12

Abstract

We construct the bosonization of the Fock space \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 of a single neutral fermion by using a 2-point local Heisenberg field. We decompose \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 as a direct sum of irreducible highest weight modules for the Heisenberg algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_{\mathbb {Z}}$\end{document}HZ, and thus we show that under the Heisenberg \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_{\mathbb {Z}}$\end{document}HZ action the Fock space \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 of the single neutral fermion is isomorphic to the Fock space \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes 1}}$\end{document}F⊗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 \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12. We construct a family of 2-point-local Virasoro fields with central charge \documentclass[12pt]{minimal}\begin{document}$-2+12\lambda -12\lambda ^2, \ \lambda \in \mathbb {C}$\end{document}−2+12λ−12λ2,λ∈C, on \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12. We construct a W1 + ∞ representation on \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 and show that under this W1 + ∞ action \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes \frac{1}{2}}}$\end{document}F⊗12 is again isomorphic to \documentclass[12pt]{minimal}\begin{document}$\mathit {F^{\otimes 1}}$\end{document}F⊗1.