  We establish a connection between two recently-proposed approaches to the
understanding of the geometric origin of the Fu-Kane-Mele invariant
$\mathrm{FKM} \in \mathbb{Z}_2$, arising in the context of 2-dimensional
time-reversal symmetric topological insulators. On the one hand, the
$\mathbb{Z}_2$ invariant can be formulated in terms of the Berry connection and
the Berry curvature of the Bloch bundle of occupied states over the Brillouin
torus. On the other, using techniques from the theory of bundle gerbes it is
possible to provide an expression for $\mathrm{FKM}$ containing the square root
of the Wess-Zumino amplitude for a certain $U(N)$-valued field over the
Brillouin torus.
  We link the two formulas by showing directly the equality between the above
mentioned Wess-Zumino amplitude and the Berry phase, as well as between their
square roots. An essential tool of independent interest is an equivariant
version of the adjoint Polyakov-Wiegmann formula for fields $\mathbb{T}^2 \to
U(N)$, of which we provide a proof employing only basic homotopy theory and
circumventing the language of bundle gerbes.
