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

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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 Z₂ᵗ × U(1)ₛ symmetry. This boundary theory is described by a (2+1)d quantum electrodynamics (QED₃) with two flavors of Dirac fermions (Nf = 2) coupled with a noncompact U(1) gauge field: L = ∑ⱼ ₌ ₁² ψⱼ γμ(∂μ- i aμ) ψⱼ - i Aˢμψᵢ γμτᶻij ψⱼ + i/2π εμνᵨ aμ∂νAᵉρ, where aμ is the internal noncompact U(1) gauge field, Aˢμ and Aᵉμ 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 CP¹ theory with easy plane anisotropy. Under the self-duality, the boundary action takes exactly the same form except for an exchange between Aˢμ and Aᵉμ. 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.