Quantum holonomies and the Heisenberg group.

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 π 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 . [ABSTRACT FROM AUTHOR]