 Ising Virasoro modules from a Majorana fermion

A chiral Majorana fermion with modes $\{b_r,b_s\}={\delta }_{r+s,0}$ and $b_r^{†}=b_{-r}$ has $T=-:\psi \partial \psi :/2$ and central charge $1/2$. The even and odd fermion-number parts of its Neveu–Schwarz sector give unitary modules with lowest weights $0$ and $1/2$. In its Ramond sector, a choice of ground-state zero-mode eigenvalue gives the module of weight $1/16$; the other choice gives an isomorphic Virasoro algebra module. Their cyclic highest-weight submodules are irreducible: a positive-energy invariant proper submodule would have a lowest vector orthogonal to every descendant of the original highest vector, which is impossible in a positive inner-product cyclic module.