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Inspired by the recent developments of constructing novel Dirac liquidboundary 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 isdescribed by a $(2+1)d$ quantum electrodynamics (QED$_3$) with two flavors ofDirac 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 $U(1)_e$ globalsymmetries respectively. We demonstrate that this theory has a "self-dual"structure, which is a fermionic analogue of the self-duality of the noncompactCP$^1$ theory with easy plane anisotropy. Under the self-duality, the boundaryaction 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 somesubtleties that will be discussed.

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