Quantization of Chern-Simons topological invariants for H-type and L-type quantum systems

In 2+1-dimensions (2+1D), a gapped quantum phase with no symmetry (i.e. a topological order) can have a thermal Hall conductance $\kappa_{xy}=c \frac{\pi^2 k_B^2}{3h}T$, where the dimensionless $c$ is called chiral central charge. If there is a $U_1$ symmetry, a gapped quantum phase can also have a Hall conductance $\sigma_{xy}=\nu \frac{e^2}{h}$, where the dimensionless $\nu$ is called filling fraction. In this paper, we derive some quantization conditions of $c$ and $\nu$, via a cobordism approach to define Chern--Simons topological invariants which are associated with $c$ and $\nu$. In particular, we obtain quantization conditions that depend on the ground state degeneracies on Riemannian surfaces, and quantization conditions that depend on the type of spacetime manifolds where the topological partition function is non-zero.