  Topological phases of matter are described universally by topological field
theories in the same way that symmetry-breaking phases of matter are described
by Landau-Ginzburg field theories. We propose that topological insulators in
two and three dimensions are described by a version of abelian $BF$ theory. For
the two-dimensional topological insulator or quantum spin Hall state, this
description is essentially equivalent to a pair of Chern-Simons theories,
consistent with the realization of this phase as paired integer quantum Hall
effect states. The $BF$ description can be motivated from the local excitations
produced when a $\pi$ flux is threaded through this state. For the
three-dimensional topological insulator, the $BF$ description is less obvious
but quite versatile: it contains a gapless surface Dirac fermion when
time-reversal-symmetry is preserved and yields "axion electrodynamics", i.e.,
an electromagnetic $E \cdot B$ term, when time-reversal symmetry is broken and
the surfaces are gapped. Just as changing the coefficients and charges of 2D
Chern-Simons theory allows one to obtain fractional quantum Hall states
starting from integer states, $BF$ theory could also describe (at a macroscopic
level) fractional 3D topological insulators with fractional statistics of
point-like and line-like objects.
