Free chiral Majorana fermion conformal field theory ($h_{\psi }=1/2,c=1/2$)

A real chiral Majorana fermion with contraction $\psi (z)\psi (w)\sim 1/(z-w)$ has stress tensor $T=-:\psi \partial \psi :/2$. Fermionic Wick contractions give $T(z)\psi (w)\sim \psi (w)/(2(z-w{)}^2)+\partial \psi (w)/(z-w)$. Double contractions in $TT$ give $1/(4(z-w{)}^4)$, so the central charge is $1/2$. Fermion bilinears of different species are weight-one currents; equal-species undifferentiated bilinears vanish by antisymmetry.

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.