We analyze
2+1d and
3+1d bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group
G by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat
G gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat
G gauge fields to obtain Dijkgraaf-Witten topological
G gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in
2+1d and condensing bosonic gauge charges in
3+1d. In the dual boundary approach, we study
1+1d and
2+1d quantum field theories that have
G 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct
G characters that generate topological data for the bulk topological gauge theory. Finally, we study a
2+1d quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where
Z2T/R is time-reversal/reflection symmetry, and the
U(1) could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in
3+1d that cancels this anomaly. This signals the existence of SPTs in
3+1d protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.
We construct an exactly sovable commuting projector Hamiltonian for (2+1)D bosonic topological insulator which is one of symmetry-protected topological (SPT) phases protected by U(1) and time-reversal
Z2T symmetry, where the symmetry group is U(1)
⋊Z2T. The model construction is based on the decorated domain-wall interpretation of the
E∞-page of a spectral sequence of a cobordism group that classifies the SPT phases in question. We demonstrate nontriviality of the model by showing an emergence of a Kramers doublet when the system is put on a semi-infinite cylinder
(−∞,0]×S1 with an inserted
π-flux. The surface anomaly manifests itself as a non-onsite representation of the U(1)
⋊Z2T symmetry. Anomaly matching on a boundary is discussed within a simple boundary theory.