Fig. 1: LLL projection and flux insertion in the quantum Hall liquid.

The LLL projection of a quantum Hall liquid on a torus maps the two-dimensional state into a one-dimensional ring of particles in orbital space. Inserting a flux quanta through one of the two cycles of the torus (depicted in red and blue) corresponds to a unitary operation that acts between the different ground states in the torus. In the one-dimensional representation, these unitary operations correspond to translation of the guiding centers by one orbital (T), or multiplication by a phase (U), depicted by blue and red arrows, respectively.