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 \rtimes 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 $a_\mu$ is the internal noncompact U(1) gauge field, $A^s_\mu$ and $A^e_\mu$ are two external gauge fields that couple to $U(1)_s$ 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 $A^s_\mu$ and $A^e_\mu$. 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.