  In arXiv:2110.06932, we argued that the chiral central charge -- a
topologically protected quantity characterizing the edge theory of a gapped
(2+1)-dimensional system -- can be extracted from the bulk by using an order
parameter called the modular commutator. In this paper, we reveal general
properties of the modular commutator and strengthen its relationship with the
chiral central charge. First, we identify connections between the modular
commutator and conditional mutual information, time reversal, and modular flow.
Second, we prove, within the framework of the entanglement bootstrap program,
that two topologically ordered media connected by a gapped domain wall must
have the same modular commutator in their respective bulk. Third, we
numerically calculate the value of the modular commutator for a bosonic lattice
Laughlin state for finite sizes and extrapolate to the infinite-volume limit.
The result of this extrapolation is consistent with the proposed formula up to
an error of about 0.7%.
