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 $.
Categories: cond-mat.str-el hep-th math-ph math.MP