Self-dual quantum electrodynamics as boundary state of the three-dimensional bosonic topological insulator

Inspired by recent developments in constructing novel Dirac liquid boundary states of a three-dimensional (3D) topological insulator, we propose one possible two-dimensional boundary state of a 3D bosonic symmetry protected topological state with $U{\left(1\right)}_e⋊Z_2^T\times U{\left(1\right)}_s$ symmetry. This boundary theory is described by a $(2+1)$-dimensional quantum electrodynamics $\left({\mathrm{QED}}_3\right)$ with two flavors of Dirac fermions $\left(N_f=2\right)$ coupled with a noncompact U(1) gauge field, $\mathcal{L}={\sum }_{j=1}^2{\overline{\psi }}_j{\gamma }_{\mu }({\partial }_{\mu }-i{a}_{\mu }){\psi }_j-i{A}_{\mu }^s\overline{{\psi }_i}{\gamma }_{\mu }{\tau }_{ij}^z{\psi }_j+\frac{i}{2\pi }{\epsilon }_{\mu \nu \rho }{a}_{\mu }{\partial }_{\nu }{A}_{\rho }^e$, where $a_{\mu }$ is the internal noncompact U(1) gauge field, and $A_{\mu }^s$ and $A_{\mu }^e$ are two external gauge fields that couple to $U{\left(1\right)}_s$ and $U{\left(1\right)}_e$ global symmetries, respectively. We demonstrate that this theory has a “self-dual” structure, which is a fermionic analog of the self-duality of the noncompact ${\mathrm{CP}}^1$ theory with easy plane anisotropy. Under the self-duality, the boundary action takes exactly the same form except for an exchange between $A_{\mu }^s$ and $A_{\mu }^e$. The self-duality may still hold after we break one of the U(1) symmetries (which makes the system a bosonic topological insulator), with some subtleties that will be discussed.