I will comment on the recent work on new exotic types of gapped
Topological Boundary/Interface Conditions of TQFTs (e.g. SPT and SET
states) in any dimension: https://arxiv.org/abs/1705.06728. In
contrast to known gapped boundaries/interfaces obtained via symmetry
breaking (either global symmetry breaking or Anderson-Higgs mechanism
for gauge theory), our approach is based on symmetry extension. In
this work, we show that a certain anomalous non-on-site G symmetry
along the SPT boundary becomes on-site when viewed as a larger H
symmetry, via a suitable group extension. Namely, a non-perturbative
global (gauge/gravitational) anomaly in G becomes anomaly-free in H.
This guides us to formulate exactly soluble lattice path integral and
Hamiltonian constructions of symmetric gapped boundaries applicable to
any SPT state of any finite symmetry group, including on-site unitary
and anti-unitary time-reversal symmetries. The resulting symmetric
gapped boundary can be described either by an H-symmetry extended
boundary in any spacetime dimension, or more naturally by a
topological K-gauge theory with a global symmetry G on a 3+1D bulk or
above. Apply our approach to a 1+1D boundary of 2+1D bulk, we find
that a deconfined topologically ordered boundary indeed has
spontaneous symmetry breaking with long-range order. The deconfined
symmetry-breaking phase crosses over smoothly to a confined phase
without a phase transition. (Based on JW’s 1705.06728 paper, related
1212.4863 and 1408.6514 and Refs therein.)