  The string-net approach by Levin and Wen and the local unitary transformation
approach by Chen, Gu and Wen provided ways to systematically label non-chiral
topological orders in 2D. In those approaches, different topologically ordered
many-body wave functions were characterized by different fixed-point tensors.
Though extremely powerful, the resulting fixed-point tensors were mathematical
abstractions and thus lacked a physical interpretation. As a result it was hard
to judge if two different fixed-point tensors described the same quantum phase
or not. We want to improve that approach by giving a more physical description
of the topological orders. We find that the non-Abelian Berry's phases, $T$-
and $S$-matrices, of the topological protected degenerate ground states on a
torus give rise to a more physical description of topological orders. Using the
Verlinde conjecture, we can even choose the canonical basis for the $T$- and
$S$-matrices. It is conjectured that the $T$ and $S$-matrices form a complete
and one-to-one characterization of non-chiral topological orders and can
replace the fixed-point tensor description to give us a more physical label for
topological orders. As a result, all the topological properties can be obtained
from the $T$- and $S$-matrices, such as number of quasiparticle types (from the
dimension of $T$ or $S$), the quasiparticle statistics (from the diagonal
elements of $T$), the quantum dimensions of quasiparticles (from the first row
of $S$), \etc.
