G-Crossed Modularity of Symmetry Enriched Topological Phases

The universal properties of $$(2+1)$$ D topological phases matter enriched by a symmetry group G are described G-crossed extensions unitary modular tensor categories (UMTCs). While the fusion and braiding quasiparticles associated with order UMTC, further capture action, fractionalization, defects arising from interplay order. We describe relation between UMTC state spaces on general surfaces that may include defect branch lines boundaries carry charge. define operators in terms data represent mapping class transformations for such states torus one boundary, show these provide projective representations groups. This allows us to ensures consistent description corresponding surfaces. Our analysis also enables prove faithful extension is necessarily modular.