We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, symmetry properties of low-energy conformal field theory (CFT) provide discrete invariants, establishing notion symmetry-enriched quantum criticality. The charges nonlocal scaling operators, or more generally defects, are imply presence localized edge modes. primarily focus on $1...

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 t...

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...