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

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.