  We ask which topological phases can and cannot be realized by exactly soluble
string-net models. We answer this question for the simplest class of
topological phases, namely those with abelian braiding statistics.
Specifically, we find that an abelian topological phase can be realized by a
string-net model if and only if (i) it has a vanishing thermal Hall conductance
and (ii) it has at least one Lagrangian subgroup --- a subset of quasiparticles
with particular topological properties. Equivalently, we find that an abelian
topological phase is realizable if and only if it supports a gapped edge. We
conjecture that the latter criterion generalizes to the non-abelian case. We
establish these results by systematically constructing all possible abelian
string-net models and analyzing the quasiparticle braiding statistics in each
model. We show that the low energy effective field theories for these models
are multicomponent U(1) Chern-Simons theories, and we derive the K-matrix
description of each model. An additional feature of this work is that the
models we construct are more general than the original string-net models, due
to several new ingredients. First, we introduce two new objects $\gamma,
\alpha$ into the construction which are related to $\mathbb{Z}_2$ and
$\mathbb{Z}_3$ Frobenius-Schur indicators. Second, we do not assume parity
invariance. As a result, we can realize topological phases that were not
accessible to the original construction, including phases that break
time-reversal and parity symmetry.
