  Anyon models are algebraic structures that model universal topological
properties in topological phases of matter and can be regarded as mathematical
characterization of topological order in two spacial dimensions. It is
conjectured that every anyon model, or mathematically unitary modular tensor
category, can be realized as the representation category of some chiral
conformal field theory, or mathematically vertex operator algebra/local
conformal net. This conjecture is known to be true for abelian anyon models
providing support for the conjecture. We reexamine abelian anyon models from
several different angles. First anyon models are algebraic data for both
topological quantum field theories and chiral conformal field theories. While
it is known that each abelian anyon model can be realized by a quantum abelian
Chern-Simons theory and chiral conformal field theory, the construction is not
algorithmic. Our goal is to provide such an explicit algorithm for a $K$-matrix
in Chern-Simons theory and a positive definite even one for a lattice conformal
field theory. Secondly anyon models and chiral conformal field theories
underlie the bulk-edge correspondence for topological phases of matter. But
there are interesting subtleties in this correspondence when stability of the
edge theory and topological symmetry are taken into consideration. Therefore,
our focus is on the algorithmic reconstruction of extremal chiral conformal
field theories with small central charges. Finally we conjecture that a much
stronger reconstruction holds for abelian anyon models: every abelian anyon
model can be realized as the representation category of some non-lattice
extremal vertex operator algebra generalizing the moonshine realization of the
trivial anyon model.
