Quantum Holonomies and the Heisenberg Group

TL;DR

This paper explores the relationship between quantum holonomies on a torus and the Heisenberg group, revealing how group operations relate to path concatenation, signed area phases, and modular transformations.

Contribution

It establishes a novel interpretation of quantum holonomies as elements of the Heisenberg group, linking geometric phases to algebraic structures and symmetries of the torus.

Findings

Quantum holonomies correspond to elements of the Heisenberg group.

Group composition models path concatenation and encodes signed area phases.

Modular transformations are generated by symplectic automorphisms of 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 $T^2$.