Relation between chiral central charge and ground-state degeneracy in (2+1) -dimensional topological orders

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A bosonic topological order on $d$-dimensional closed space $\Sigma^d$ may have degenerate ground states. The space $\Sigma^d$ with different shapes (different metrics) form a moduli space ${\cal M}_{\Sigma^d}$. Thus the degenerate ground states on every point in the moduli space ${\cal M}_{\Sigma^d}$ form a complex vector bundle over ${\cal M}_{\Sigma^d}$. It was suggested that the collection of such vector bundles for $d$-dimensional closed spaces of all topologies completely characterizes the topological order. Using such a point of view, we propose a direct relation between two seemingly unrelated properties of 2+1-dimensional topological orders: (1) the chiral central charge $c$ that describes the many-body density of states for edge excitations (or more precisely the thermal Hall conductance of the edge), (2) the ground state degeneracy $D_g$ on closed genus $g$ surface. We show that $c D_g/2 \in \mathbb{Z},\ g\geq 3$ for bosonic topological orders. We explicitly checked the validity of this relation for over 140 simple topological orders. For fermionic topological orders, let $D_{g,\sigma}^{e}$ ($D_{g,\sigma}^{o}$) be the degeneracy with even (odd) number of fermions for genus-$g$ surface with spin structure $\sigma$. Then we have $2c D_{g,\sigma}^{e} \in \mathbb{Z}$ and $2c D_{g,\sigma}^{o} \in \mathbb{Z}$ for $g\geq 3$.