Bulk-boundary correspondence for intrinsically-gapless SPTs from group cohomology

Recent studies on gapless SPT(gSPT) have revealed interesting relations between their topological edge modes and emergent anomalies. In this work we take a further step towards systematical construction of gSPT from cohomology data and establish a direct connection between emergent anomaly, group extension, and topological edge states for any internal symmetry group in any dimension. In 2 and higher dimensions the emergent anomaly can also be satisfied by anomalous symmetry enriched topological order, which we refer to as quotient-symmetry enriched topological order(QSET) that is sharply distinguished from non-anomalous SETs by an edge phase transition.