Topological phases protected by crystalline symmetries and internal
symmetries are shown to enjoy fascinating one-to-one correspondence in
classification. Here we investigate the physics content behind the abstract
correspondence in three or higher-dimensional systems. We show correspondence
between anomalous boundary states, which provides a new way to explore the
quantum anomaly of symmetry from its crystalline equivalent counterpart. We
show such correspondence directly in two scenarios, including the anomalous
symmetry-enriched topological orders (SET) and critical surface states. (1)
First of all, for the surface SET correspondence, we demonstrate it by
considering examples involving time-reversal symmetry and mirror symmetry. We
show that one 2D topological order can carry the time reversal anomaly as long
as it can carry the mirror anomaly and vice versa, by directly establishing the
mapping of the time reversal anomaly indicators and mirror anomaly indicators.
Besides, we also consider other cases involving continuous symmetry, which
leads us to introduce some new anomaly indicators for symmetry from its
counterpart. (2) Furthermore, we also build up direct correspondence for (near)
critical boundaries. Again taking topological phases protected by time reversal
and mirror symmetry as examples, the direct correspondence of their (near)
critical boundaries can be built up by coupled chain construction that was
first proposed by Senthil and Fisher. The examples of critical boundary
correspondence we consider in this paper can be understood in a unified
framework that is related to \textit{hierarchy structure} of topological O(n)
nonlinear sigma model, that generalizes the Haldane's derivation of O(3)
sigma model from spin one-half system.Comment: 17 pages, 5 figure