Self-dual quantum electrodynamics as boundary state of the three-dimensional bosonic topological insulator

Inspired by recent developments in constructing novel Dirac liquid boundary states of a three-dimensional (3D) topological insulator, we propose one possible two-dimensional boundary state of a 3D bosonic symmetry protected topological state with U(1)(e) x Z(2)(T) x U(1)(s) symmetry. This boundary theory is described by a (2 + 1)-dimensional quantum electrodynamics (QED(3)) with two flavors of Dirac fermions (N-f = 2) coupled with a noncompact U(1) gauge field, L = Sigma(2)(j=1) (psi) over bar (j)gamma(mu)(partial derivative(mu) - ia(mu))psi(j) - iA(mu)(s)(psi) over bar (i)gamma(mu)tau(2)(ij)psi(j) + i/2 pi epsilon(mu v rho)a(mu)partial derivative(v)A(rho)(e), where a mu is the internal noncompact U(1) gauge field, and A(mu)(s) and A(mu)(e) 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 analog 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 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.