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

TL;DR

This work develops a framework for 2+1D abelian bosonic TQFTs with parity symmetry on non-orientable manifolds by embedding torus Hilbert spaces into a projective representation of $GL(2,\mathbb{Z})$. It introduces and analyzes the crosscap state, showing that the dimension of the Hilbert space on non-orientable surfaces, computed via $Z(\Sigma\times S^1)$, remains integral for time-reversal symmetric abelian theories, with $\mathbb{RP}^2$-based data controlling the global integrality. A concrete formula for the crosscap-related coefficients $M_a$ is derived, $M_a=\frac{1}{|\mathcal{A}|^{1/2}}\sum_{b\in\ker(1+\mathsf{P})} B(a,-b)\eta(b)$, and the subgroup $\mathcal{M}\subset\mathcal{A}$ of nonzero contributions is characterized along with multiplicativity constraints on the associated phase data $\eta$ and the reflection action $\mathsf{P}$. The results tie the integrality condition to a time-reversal anomaly, showing $Z(\mathbb{RP}^4)Z(\mathbb{CP}^2)=\theta_{\mathcal{M}}$, thereby linking non-orientable TQFT consistency to SPT data; these insights pave the way for extending to non-abelian or spin cases and deepen the connection between Atiyah TQFT and MTC formalisms on non-orientable 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 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.