K matrix construction of symmetry-enriched phases of matter

We construct in the K matrix formalism concrete examples of symmetry-enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and nonchiral topological phases and demonstrate by our examples how the interplay between the global symmetry and the fusion algebra of the anyons of a topologically ordered system determines the existence of gapless edge modes protected by the symmetry and that a (quasi) group structure can be defined among these phases. Our examples include phases that display charge fractionalization and more exotic nonlocal anyon exchange under global symmetry that correspond to general group extensions of the global symmetry group.