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We study $3d$ (or $(3+1)d$) Quantum Electrodynamics (QED) realized on theboundary of $4d$ (or $(4+1)d$) bosonic symmetry protected topological (BSPT)states, using a systematic nonlinear sigma model (NLSM) field theorydescription of BSPT states. We demonstrate that many of these QED states havean exact electric-magnetic duality due to the symmetry of the BSPT states inthe $4d$ bulk. The gauge charge and Dirac monopole both carry projectiverepresentations of the bulk symmetry, and the emergent gapless photons of theQED phase also transform nontrivially under the bulk symmetry. Some of theseQED boundary states can be further driven into a $3d$ $\mathbb{Z}_2$topological order, and the statistics and symmetry transformation of its pointparticle and vison loop excitations guarantee that this topological ordercannot be driven into a trivial confined or Higgs phase. With a finite fourthdimension, the entire system becomes a $3d$ lattice, the self-dual QED and the$\mathbb{Z}_2$ topological order can coexist on two opposite boundariesrespectively, which together constitute an exotic $3d$ self-dual "topologicalphoton phase".

Self-dual Quantum Electrodynamics on the boundary of 4d Bosonic Symmetry Protected Topological States