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

We study $3d$ (or $(3+1)d$) Quantum Electrodynamics (QED) realized on the boundary of $4d$ (or $(4+1)d$) bosonic symmetry protected topological (BSPT) states, using a systematic nonlinear sigma model (NLSM) field theory description of BSPT states. We demonstrate that many of these QED states have an exact electric-magnetic duality due to the symmetry of the BSPT states in the $4d$ bulk. The gauge charge and Dirac monopole both carry projective representations of the bulk symmetry, and the emergent gapless photons of the QED phase also transform nontrivially under the bulk symmetry. Some of these QED boundary states can be further driven into a $3d$ $\mathbb{Z}_2$ topological order, and the statistics and symmetry transformation of its point particle and vison loop excitations guarantee that this topological order cannot be driven into a trivial confined or Higgs phase. With a finite fourth dimension, the entire system becomes a $3d$ lattice, the self-dual QED and the $\mathbb{Z}_2$ topological order can coexist on two opposite boundaries respectively, which together constitute an exotic $3d$ self-dual "topological photon phase".