The origin of holomorphic states in Landau levels from non-commutative geometry, and a new formula for their overlaps on the torus

Holomorphic functions that characterize states in a two-dimensional Landau level been central to key developments such as the Laughlin state. Their origin has historically been attributed to a special property of "Schr\"odinger wavefunctions" of states in the "lowest Landau level". It is shown here that they instead arise in any Landau level as a generic mathematical property of the Heisenberg description of the non-commutative geometry of guiding centers. When quasiperiodic boundary conditions are applied to compactify the system on a torus, a new formula for the overlap between holomorphic states, in the form of a discrete sum rather than an integral, is obtained. The new formula is unexpected from the previous "lowest-Landau level Schr\"odinger wavefunction" interpretation.