  Topological holography is a holographic principle that describes the
generalized global symmetry of a local quantum system in terms of a topological
order in one higher dimension. This framework separates the topological data
from the local dynamics of a theory and provides a unified description of the
symmetry and duality in gapped and gapless phases of matter. In this work, we
develop the topological holographic picture for (1+1)d quantum phases,
including both gapped phases as well as a wide range of quantum critical
points, including phase transitions between symmetry protected topological
(SPT) phases, symmetry enriched quantum critical points, deconfined quantum
critical points and intrinsically gapless SPT phases. Topological holography
puts a strong constraint on the emergent symmetry and the anomaly for these
critical theories. We show how the partition functions of these critical points
can be obtained from dualizing (orbifolding) more familiar critical theories.
The topological responses of the defect operators are also discussed in this
framework. We further develop a topological holographic picture for conformal
boundary states of (1+1)d rational conformal field theories. This framework
provides a simple physical picture to understand conformal boundary states and
also uncovers the nature of the gapped phases corresponding to the boundary
states.
