  Quantum holonomies of closed paths on the torus $T^2$ are interpreted as
elements of the Heisenberg group $H_1$. Group composition in $H_1$ corresponds
to path concatenation and the group commutator is a deformation of the relator
of the fundamental group $\pi_1$ of $T^2$, making explicit the signed area
phases between quantum holonomies of homotopic paths. Inner automorphisms of
$H_1$ adjust these signed areas, and the discrete symplectic transformations of
$H_1$ generate the modular group of $T^2$.
