Representations of $a_{\infty }$ and $d_{\infty }$ with central charge 1 on the single neutral fermion Fock space $F^{\otimes \frac{1}{2}}$

Abstract

We construct a new representation of the infinite rank Lie algebra $a_{\infty }$ with central charge $c=1$ on the Fock space $F^{\otimes \frac{1}{2}}$ of a single neutral fermion. We show that $F^{\otimes \frac{1}{2}}$ is a direct sum of irreducible integrable highest weight modules for $a_{\infty }$ with central charge $c=1$. We prove that as $a_{\infty }$ modules $F^{\otimes \frac{1}{2}}$ is isomorphic to the Fock space $F^{\otimes 1}$ of the charged free fermions. As a corollary we obtain the decompositions of certain irreducible highest weight modules for $d_{\infty }$ with central charge $c=\frac{1}{2}$ into irreducible highest weight modules for $d_{\infty }$ with central charge $c=1$.

Contribution to the XXIst International Conference on Integrable Systems and Quantum symmetries (ISQS-21), Prague, Czech Republic, 2013

References in corpus (2)

• Vertex Algebras and Algebraic Curves
• Torus n-Point Functions for $\mathbb{R}$-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds

Cited by in corpus (1)

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