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We calculate the topological part of the electromagnetic response of BosonicInteger Quantum Hall (BIQH) phases in odd (spacetime) dimensions, and BosonicTopological Insulator (BTI) and Bosonic chiral semi-metal (BCSM) phases in evendimensions. To do this we use the Nonlinear Sigma Model (NLSM) description ofbosonic symmetry-protected topological (SPT) phases, and the method of gaugedWess-Zumino (WZ) actions. We find the surprising result that for BIQH states indimension $2m-1$ ($m=1,2,\dots$), the bulk response to an electromagnetic field$A_{\mu}$ is characterized by a Chern-Simons term for $A_{\mu}$ with a levelquantized in integer multiples of $m!$ (factorial). We also show that BTIstates (which have an extra $\mathbb{Z}_2$ symmetry) can exhibit a$\mathbb{Z}_2$ breaking Quantum Hall effect on their boundaries, with thisboundary 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 requiringgauge invariance of the exponential of the Chern-Simons term on a generalEuclidean manifold, and we also use this argument to characterize theelectromagnetic and gravitational responses of fermionic SPT phases with $U(1)$symmetry in all odd dimensions. We then use our gauged boundary actions for theBIQH 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 stateexhibits a bosonic analogue of the parity anomaly of Dirac fermions in odddimensions, and (iii) study anomaly inflow at domain walls on the boundary ofBTI states. In a series of Appendices we derive important formulas andadditional results. In particular, in Appendix A we use the connection betweenequivariant cohomology and gauged WZ actions to give a mathematicalinterpretation of the actions for the BIQH and BTI boundaries constructed inthis paper.

Topological Electromagnetic Responses of Bosonic Quantum Hall, Topological Insulator, and Chiral Semi-Metal phases in All Dimensions