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

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