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In 2+1-dimensions (2+1D), a gapped quantum phase with no symmetry (i.e. atopological 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 centralcharge. If there is a $U_1$ symmetry, a gapped quantum phase can also have aHall conductance $\sigma_{xy}=\nu \frac{e^2}{h}$, where the dimensionless $\nu$is called filling fraction. In this paper, we derive some quantizationconditions of $c$ and $\nu$, via a cobordism approach to define Chern--Simonstopological invariants which are associated with $c$ and $\nu$. In particular,we obtain quantization conditions that depend on the ground state degeneracieson Riemannian surfaces, and quantization conditions that depend on the type ofspacetime manifolds where the topological partition function is non-zero.

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