Virasoro representations with central charges $\frac{1}{2}$ and 1 on the real neutral fermion Fock space $\mathit{F^{\otimes \frac{1}{2}}}$

TL;DR

This paper explores fermionic oscillator representations of the Virasoro algebra on a neutral fermion Fock space, decomposing modules with central charges 1/2 and 1, and deriving character formulas.

Contribution

It provides a new decomposition of Virasoro modules with central charges 1/2 and 1 on the neutral fermion Fock space, linking their structures and character formulas.

Findings

Decomposition of Fock space into irreducible Virasoro modules with c=1.

Decomposition of c=1/2 modules into c=1 modules.

Derivation of positive sum character formulas for c=1/2 modules.

Abstract

We study a family of fermionic oscillator representations of the Virasoro algebra via 2-point-local Virasoro fields on the Fock space $\mathit{F^{\otimes \frac{1}{2}}}$ of a neutral (real) fermion. We obtain the decomposition of $\mathit{F^{\otimes \frac{1}{2}}}$ as a direct sum of irreducible highest weight Virasoro modules with central charge $c=1$. As a corollary we obtain the decomposition of the irreducible highest weight Virasoro modules with central charge $c=\frac{1}{2}$ into irreducible highest weight Virasoro modules with central charge $c=1$. As an application we show how positive sum (fermionic) character formulas for the Virasoro modules of charge $c=\frac{1}{2}$ follow from these decompositions.