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: $\mathcal{L} = \sum_{j = 1}^2 \barψ_j γ_μ(\partial_μ- i a_μ) ψ_j - i A^{s}_μ\bar{ψ_i} γ_μτ^z_{ij} ψ_j + \frac{i}{2π} ε_{μνρ} a_μ\partial_νA^{e}_ρ$, where $a_μ$ is the internal noncompact U(1) gauge field, $A^s_μ$ and $A^e_μ$ 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 $A^s_μ$ and $A^e_μ$. 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.

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