On dimensions of (2+1)D abelian bosonic topological systems on non-orientable manifolds

We give a framework to describe abelian bosonic topological systems with parity symmetry on a torus in terms of the projective representation of $GL(2,\mathbb{Z})$. However, this information alone does not guarantee that we can assign Hilbert spaces to non-orientable surfaces in a way compatible with the gluing axiom of topological quantum field theory. Here, we show that we may assign Hilbert spaces with integer dimensions to non-orientable surfaces in the case of abelian bosonic topological systems with time-reversal symmetry, which can be seen as a necessary condition for the existence of topological quantum field theories.