Index theorem, spin Chern Simons theory and fractional magnetoelectric effect in strongly correlated topological insulators

Making use of index theorem and spin Chern Simons theory, we construct an effective topological field theory of strongly correlated topological insulators coupling to a nonabelian gauge field $ SU(N) $ with an interaction constant $ g $ in the absence of the time-reversal symmetry breaking. If $ N $ and $ g $ allow us to define a t'Hooft parameter $ \lambda $ of effective coupling as $ \lambda = N g^{2} $, then our construction leads to the fractional quantum Hall effect on the surface with Hall conductance $ \sigma_{H}^{s} = \frac{1}{4\lambda} \frac{e^{2}}{h} $. For the magnetoelectric response described by a bulk axion angle $ \theta $, we propose that the fractional magnetoelectric effect can be realized in gapped time reversal invariant topological insulators of strongly correlated bosons or fermions with an effective axion angle $ \theta_{eff} = \frac{\pi}{2 \lambda} $ if they can have fractional excitations and degenerate ground states on topologically nontrivial and oriented spaces. Provided that an effective charge is given by $ e_{eff} = \frac{e}{\sqrt{2 \lambda}} $, it is shown that $ \sigma_{H}^{s} = \frac{e_{eff}^{2}}{2h} $, resulting in a surface Hall conductance of gapless fermions with $ e_{eff} $ and a pure axion angle $ \theta = \pi $.