Quantum Holonomies and the Heisenberg Group

Abstract

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 $T^2$.

8 pages, 3 figures

References in corpus (3)

• Erlangen Programme at Large: An Overview
• Quantum geometry from 2+1 AdS quantum gravity on the torus
• A quantum Goldman bracket in 2+1 quantum gravity