  We classify distinct types of quantum number fractionalization occurring in
two-dimensional topologically ordered phases, focusing in particular on phases
with Z2 topological order, that is, on gapped Z2 spin liquids. We find that the
fractionalization class of each anyon is an equivalence class of projective
representations of the symmetry group, corresponding to elements of the
cohomology group H^2(G,Z2). This result leads us to a symmetry classification
of gapped Z2 spin liquids, such that two phases in different symmetry classes
cannot be connected without breaking symmetry or crossing a phase transition.
Symmetry classes are defined by specifying a fractionalization class for each
type of anyon. The fusion rules of anyons play a crucial role in determining
the symmetry classes. For translation and internal symmetry, braiding
statistics plays no role, but can affect the classification when point group
symmetries are present. For square lattice space group, time reversal and SO(3)
spin rotation symmetry, we find 2,098,176 ~ 2^21 distinct symmetry classes. We
give an explicit construction of symmetry classes for square lattice space
group symmetry in the toric code model. Via simple examples, we illustrate how
information about fractionalization classes can in principle be obtained from
the spectrum and quantum numbers of excited states. Moreover, the symmetry
class can be partially determined from the quantum numbers of the four
degenerate ground states on the torus. We also extend our results to arbitrary
Abelian topological orders, and compare our classification with the related
projective symmetry group classification of parton mean-field theories. Our
results provide a framework for understanding and probing the sharp
distinctions among symmetric Z2 spin liquids, and are a first step toward a
full classification of symmetric topologically ordered phases.
