  The strong time-reversal symmetric (TRS) topological insulator (TI) in three
space dimensions features gapless surface states in the form of massless Dirac
fermions. We study these surface states with the method of bosonization, and
find that the resulting bosonic theory has a topological contribution due to
the parity anomaly of the surface Dirac fermions. We argue that the presence of
a quantum anomaly is, in fact, the main reason for the existence of a surface
state, by the principle that anomalies of a surface and bulk must cancel.
Inspecting other classes of topological insulators, we argue that this
principle holds in general. Moving beyond purely topological considerations, we
incorporate the dynamics of the surface electron states into the bosonic
theory. Additionally, we discuss the thermodynamics of the bosonic theory and
propose a representation of the surface Dirac fermions in terms of the bosonic
fields.
