  Effective field theories that describes the dynamics of a conserved U(1)
current in terms of "hydrodynamic" degrees of freedom of topological phases in
condensed matter are discussed in general dimension $D=d+1$ using the
functional bosonization technique. For non-interacting topological insulators
(superconductors) with a conserved U(1) charge and characterized by an integer
topological invariant [more specifically, they are topological insulators in
the complex symmetry classes (class A and AIII) and in the "primary series" of
topological insulators in the eight real symmetry classes], we derive the
BF-type topological field theories supplemented with the Chern-Simons (when $D$
is odd) or the $\theta$-term (when $D$ is even). For topological insulators
characterized by a $\mathbb{Z}_2$ topological invariant (the first and second
descendants of the primary series), their topological field theories are
obtained by dimensional reduction. Building on this effective field theory
description for non-interacting topological phases, we also discuss, following
the spirit of the parton construction of the fractional quantum Hall effect by
Block and Wen, the putative "fractional" topological insulators and their
possible effective field theories, and use them to determine the physical
properties of these non-trivial quantum phases.
