Bosonic topological phases of matter: bulk-boundary correspondence, SPT 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.
We also analyze a $2+1d$ quantum field theory with
a mixed $\Z_2^{T/R} \times U(1)$ anomaly
where
$\Z_2^{T/R}$ 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 $U(1)\times \Z_{2}^{T,R}$.