Recently, general methods of bosonization beyond 1+1 dimensions have been developed. In this article, we review these bosonizations and extend them to the case with boundary conditions. In particular, we study the case when the bulk theory is a $G$-symmetry protected topological phase and the boundary theory has a $G$ 't Hooft anomaly. We discuss how, when the anomaly is not realizable in a bosonic system, the $G$ symmetry algebra becomes modified in the bosonization of the anomalous theory. This gives us a useful tool for understanding anomalies of fermionic systems, since there is no way to compute a boundary gauge variation of the anomaly polynomial, as one does for anomalies of bosonic systems. We take the chiral anomalies in 1+1D and the parity/time reversal anomalies in 2+1D as case studies. We also provide a derivation of new constraints in SPT phases with domain defects decorated by $p+ip$ superconductors and Kitaev strings.

44 pages, 1 figure, many updates in section 2 giving an overview of bosonization and SPT phases, including 1+1D calculations of bosonizations of Kitaev phases