  Time-reversal invariant three-dimensional topological insulators can be
defined fundamentally by a topological field theory with a quantized axion
angle theta of zero or pi. It was recently shown that fractional quantized
values of theta are consistent with time-reversal invariance if deconfined,
gapped, fractionally charged bulk excitations appear in the low-energy spectrum
due to strong correlation effects, leading to the concept of a fractional
topological insulator. These fractionally charged excitations are coupled to
emergent gauge fields which ensure that the microscopic degrees of freedom, the
original electrons, are gauge-invariant objects. A first step towards the
construction of microscopic models of fractional topological insulators is to
understand the nature of these emergent gauge theories and their corresponding
phases. In this work, we show that low-energy effective gauge theories of both
Abelian or non-Abelian type are consistent with a fractional quantized axion
angle if they admit a Coulomb phase or a Higgs phase with gauge group broken
down to a discrete subgroup. The Coulomb phases support gapless but
electrically neutral bulk excitations while the Higgs phases are fully gapped.
The Higgs and non-Abelian Coulomb phases exhibit multiple ground states on
boundaryless spatial 3-manifolds with nontrivial first homology, while the
Abelian Coulomb phase has a unique ground state. The ground state degeneracy
receives an additional contribution on manifolds with boundary due to the
induced boundary Chern-Simons term.
