  We calculate the topological part of the electromagnetic response of Bosonic
Integer Quantum Hall (BIQH) phases in odd (spacetime) dimensions, and Bosonic
Topological Insulator (BTI) and Bosonic chiral semi-metal (BCSM) phases in even
dimensions. To do this we use the Nonlinear Sigma Model (NLSM) description of
bosonic symmetry-protected topological (SPT) phases, and the method of gauged
Wess-Zumino (WZ) actions. We find the surprising result that for BIQH states in
dimension $2m-1$ ($m=1,2,\dots$), the bulk response to an electromagnetic field
$A_{\mu}$ is characterized by a Chern-Simons term for $A_{\mu}$ with a level
quantized in integer multiples of $m!$ (factorial). We also show that BTI
states (which have an extra $\mathbb{Z}_2$ symmetry) can exhibit a
$\mathbb{Z}_2$ breaking Quantum Hall effect on their boundaries, with this
boundary Quantum Hall effect described by a Chern-Simons term at level
$\frac{m!}{2}$. We show that the factor of $m!$ can be understood by requiring
gauge invariance of the exponential of the Chern-Simons term on a general
Euclidean manifold, and we also use this argument to characterize the
electromagnetic and gravitational responses of fermionic SPT phases with $U(1)$
symmetry in all odd dimensions. We then use our gauged boundary actions for the
BIQH and BTI states to (i) construct a bosonic analogue of a chiral semi-metal
(BCSM) in even dimensions, (ii) show that the boundary of the BTI state
exhibits a bosonic analogue of the parity anomaly of Dirac fermions in odd
dimensions, and (iii) study anomaly inflow at domain walls on the boundary of
BTI states. In a series of Appendices we derive important formulas and
additional results. In particular, in Appendix A we use the connection between
equivariant cohomology and gauged WZ actions to give a mathematical
interpretation of the actions for the BIQH and BTI boundaries constructed in
this paper.
