Bosonic and ghost contractions

Take real-transpose-symmetric $F_{\mathcal{B}},F_Q$, and set

where star is the Hermitian kernel adjoint.

Let ${\mathcal{N}}_{+}\subset \mathcal{N}$ mean that the first covector is future-directed. In signature $(-,+,+,+)$, this means its metric-raised vector is future-directed.