Bosonic topological phases of matter: Bulk-boundary correspondence, symmetry protected topological invariants, and gauging

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 Z₂T/R×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 U(1)×Z₂T,R.