Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions

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Quantum number fractionalization is a remarkable property
of topologically ordered states of matter, such as fractional quantum Hall
liquids, and quantum spin liquids. For a given type of topological order, there
are generally many ways to fractionalize the quantum numbers of a given
symmetry. What does it mean to have different types of fractionalization? Are
different types of fractionalization a universal property that can be used to
distinguish phases of matter? In this talk, I will answer these questions,
focusing on a simple class of topologically ordered phases, namely
two-dimensional gapped Z2 spin liquids, and I will present a symmetry
classification of these phases.  I will
also discuss efforts in progress to find microscopic models realizing different
symmetry classes.