Abstract
Quantum holonomies of closed paths on the torus T² are interpreted as elements of the Heisenberg group H₁. Group composition in H₁ corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group π₁ of T², making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of H₁ adjust these signed areas, and the discrete symplectic transformations of H₁ generate the modular group of T².