  The quantum spin Hall effect shares many similarities (and some important
differences) with the quantum Hall effect for the electric charge. As with the
quantum (electric charge) Hall effect, there exists a correspondence between
bulk and boundary physics that allows to characterize the quantum spin Hall
effect in diverse and complementary ways. In this paper, we derive from the
network model that encodes the quantum spin Hall effect, the so-called Z_2
network model, a Dirac Hamiltonian in two dimensions. In the clean limit of
this Dirac Hamiltonian, we show that the bulk Kane-Mele Z_2 invariant is
nothing but the SU(2) Wilson loop constructed from the SU(2) Berry connection
of the occupied Dirac-Bloch single-particle states. In the presence of
disorder, the non-linear sigma model (NLSM) that is derived from this Dirac
Hamiltonian describes a metal-insulator transition in the standard
two-dimensional symplectic universality class. In particular, we show that the
fermion doubling prevents the presence of a topological term in the NLSM that
would change the universality class of the ordinary two-dimensional symplectic
metal-insulator transition. This analytical result is fully consistent with our
previous numerical studies of the bulk critical exponents at the
metal-insulator transition encoded by the Z_2 network model. Finally, we
improve the quality and extend the numerical study of boundary multifractality
in the Z_2 topological insulator. We show that the hypothesis of
two-dimensional conformal invariance at the metal-insulator transition is
verified within the accuracy of our numerical results.
