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

Abstract

A bosonic topological order on d-dimensional closed space ςd may have degenerate ground states. The space ςd with different shapes (different metrics) form a moduli space Mςd. Thus the degenerate ground states on every point in the moduli space Mςd form a complex vector bundle over Mςd. It has been 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: (i) 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) and (ii) the ground-state degeneracy Dg on the closed genus-g surface. We show that cDg/2Z, g≥3, for bosonic topological orders. We explicitly check the validity of this relation for over 140 simple topological orders. For fermionic topological orders, we let Dg,σe (Dg,σo) be the degeneracy with an even (odd) number of fermions on the genus-g surface with spin structure σ. Then we have 2cDg,σeZ and 2cDg,σoZ for g≥3.