  We consider the quantum theory of a two-form gauge field on a space-time
which is a direct product of time and a spatial manifold, taken to be a compact
five-manifold with no torsion in its cohomology. We show that the Hilbert space
of this non-chiral theory is a certain subspace of a tensor product of two
spaces, that are naturally interpreted as the Hilbert spaces of a chiral and
anti-chiral two-form theory respectively. We also study the observable
operators in the non-chiral theory that correspond to the electric and magnetic
field strengths, the Hamiltonian, and the exponentiated holonomy of the
gauge-field around a spatial two-cycle. All these operators can be decomposed
into contributions pertaining to the chiral and anti-chiral sectors of the
theory.
