On the Dimensions of (2+1)D Abelian Bosonic Topological Systems on Nonorientable Manifolds

Abstract 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 nonorientable surfaces in a way compatible with the gluing axiom of topological quantum field theory. Here, we show that we can assign Hilbert spaces with integer dimensions to nonorientable surfaces in the case of Abelian bosonic topological systems with time-reversal symmetry.