  Intrinsically gapless symmetry protected topological phases (igSPT) are
gapless systems with SPT edge states with properties that could not arise in a
gapped system with the same symmetry and dimensionality. igSPT states arise
from gapless systems in which an anomaly in the low-energy (IR) symmetry group
emerges from an extended anomaly-free microscopic (UV) symmetry We construct a
general framework for constructing lattice models for igSPT phases with
emergent anomalies classified by group cohomology, and establish a direct
connection between the emergent anomaly, group-extension, and topological edge
states by gauging the extending symmetry. In many examples, the edge-state
protection has a physically transparent mechanism: the extending UV symmetry
operations pump lower dimensional SPTs onto the igSPT edge, tuning the edge to
a (multi)critical point between different SPTs protected by the IR symmetry. In
two- and three- dimensional systems, an additional possibility is that the
emergent anomaly can be satisfied by an anomalous symmetry-enriched topological
order, which we call a quotient-symmetry enriched topological order (QSET) that
is sharply distinguished from the non-anomalous UV SETs by an edge phase
transition. We construct exactly solvable lattice models with QSET order.
