Self-dual Quantum Electrodynamics as Boundary State of the three dimensional Bosonic Topological Insulator

Inspired by the recent developments of constructing novel Dirac liquid boundary states of the $3d$ topological insulator, we propose one possible $2d$ boundary state of the $3d$ bosonic symmetry protected topological state with $U(1{)}_e⋊Z_2^T\times U(1{)}_s$ symmetry. This boundary theory is described by a $(2+1)d$ quantum electrodynamics (QED${}_3$) with two flavors of Dirac fermions ($N_f=2$) coupled with a noncompact U(1) gauge field: $ $\mathcal{L}$ = \sum_{j = 1}^2 \bar{\psi}j \gamma\mu (\partial_\mu - i a_\mu) \psi_j - i A^{s}\mu \bar{\psi_i} \gamma\mu \tau^z_{ij} \psi_j + \frac{i}{2\pi} \epsilon_{\mu\nu\rho} a_\mu \partial_\nu A^{e}\rho $,where$a\mu$istheinternalnoncompactU(1)gaugefield,$A^s_\mu$and$A^e_\mu$aretwoexternalgaugefieldsthatcoupleto$U(1)_s$and$U(1)_e$ global symmetries respectively. We demonstrate that this theory has a “self-dual” structure, which is a fermionic analogue of the self-duality of the noncompact 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.