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

Understand

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.