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The non-Abelian geometric phases of the robust degenerate ground states wereproposed as physically measurable defining properties of topological order in1990. In this paper we discuss in detail such a quantitative characterizationof topological order, using generic Abelian fractional quantum Hall states asexamples. We find a way to determine even the pure U(1) phase of thenon-Abelian geometric phases. We show that the non-Abelian geometric phases notonly contain information about the quasi-particle statistics, they also containinformation about the chiral central charge Delta c mod 24 of the edge states,as well as the Hall viscosity. Thus, the non-Abelian geometric phases (both theAbelian part and the non-Abelian part) provide a quite complete way tocharacterize 2D topological order.

Modular transformation and bosonic/fermionic topological orders in Abelian fractional quantum Hall states