Symmetry fractionalization, defects, and gauging of topological phases

We examine the interplay of symmetry and topological order in 2 + 1-dimensional topological quantum phases of matter. We present a precise definition of the topological symmetry group Aut(C), which characterizes the symmetry of the emergent topological quantum numbers of a topological phase C, and we describe its relation with the microscopic symmetry of the underlying physical system. This allows us to derive a general framework to characterize and classify symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are antiunitary. We develop a theory of extrinsic defects (fluxes) associated with elements of the symmetry group, extending previous results in the literature. This provides a general classification of 2 + 1-dimensional symmetry-enriched topological (SET) phases derived from a topological phase of matter C with on-site symmetry group G. We derive a set of data and consistency conditions, the solutions of which define the algebraic theory of the defects, known as a G-crossed braided tensor category C-G(x). This allows us to systematically compute many properties of these theories, such as the number of topologically distinct types of defects associated with each group element, their fusion rules, quantum dimensions, zero modes, braiding exchange transformations, a generalized Verlinde formula for the defects, and modular transformations of the G-crossed extensions of topological phases. We also examine the promotion of the global on-site symmetry to a local gauge invariance (gauging the symmetry), wherein the extrinsic G defects are turned into deconfined quasiparticle excitations, which results in a different topological phase (C-G(x))(G) . We present systematic methods to compute the properties of (C-G(X))(G) when G is a finite group. The quantum phase transition between the topological phases (C-G(x))(G) and C can be understood to be a gauge symmetry breaking transition, thus shedding light on the universality class of a wide variety of topological quantum phase transitions. A number of instructive and/or physically relevant examples are studied in detail.