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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 andboundary approaches. In the bulk we study an effective field theory which uponcoupling to a background flat $G$ gauge field furnishes a purely topologicalresponse theory. The response action evaluated on certain manifolds, withappropriate choice of background gauge field, defines a set of SPT topologicalinvariants. Further, SPTs can be gauged by summing over all isomorphism classesof flat $G$ gauge fields to obtain Dijkgraaf-Witten topological $G$ gaugetheories. These topological gauge theories can be ungauged by first introducingand then proliferating defects that spoils the gauge symmetry. This mechanismis related to anyon condensation in $2+1d$ and condensing bosonic gauge chargesin $3+1d$. In the dual boundary approach, we study $1+1d$ and $2+1d$ quantumfield theories that have $G$ 't-Hooft anomalies that can be precisely cancelledby (the response theory of) the corresponding bulk SPT. We show how toconstruct/compute topological invariants for the bulk SPTs directly from theboundary theories. Further we sum over boundary partition functions withdifferent background gauge fields to construct $G$-characters that generatetopological data for the bulk topological gauge theory. Finally, we study a$2+1d$ quantum field theory with a mixed $\mathbb{Z}_2^{T/R} \times U(1)$anomaly where $\mathbb{Z}_2^{T/R}$ is time-reversal/reflection symmetry, andthe $U(1)$ could be a 0-form or 1-form symmetry depending on the choice of timereversal/reflection action. We briefly discuss the bulk effective action andtopological response for a theory in $3+1d$ that cancels this anomaly. Thissignals the existence of SPTs in $3+1d$ protected by 0,1-form $U(1)\times\mathbb{Z}_{2}^{T,R}$.

Bosonic topological phases of matter: bulk-boundary correspondence, SPT invariants and gauging