Geometric quantization on symplectic tori

As a test case for the application of the methods of geometric quantization the symplectic torus with affine real polarizations displays two important pathologies. The first occurs at the prequantization stage. The symplectic form is not exact so we cannot expect to have a trivial Kostant line bundle. Thus we must construct a nontrivial Kostant line bundle. In this construction we see that a set of complex-valued functions on the plane with a quasiperiodicity condition is the key to working with sections of a Kostant line bundle. The second pathology arises as we pass from the prequantization stage and attempt to form the quantum space. We find that there are no nontrivial polarized sections. The quantum phase space is usually constructed from the polarized sections. A common technique for overcoming this obstruction is to use as elements in the quantum phase space, generalized sections of our Kostant fine bundle with support contained in the Bohr-Sommerfeld sets. This technique works producing a one-dimensional quantum phase space in the case of polarizations formed by lines in the plane with rational slope, the rational polarizations. In the case of irrational polarizations problems with the Bohr-Sommerfeld sets preclude the normal application of these ideas. We solve this problem by using the cohomology of the sheaf of polarized sections and find our answer is the same for both rational and irrational polarizations. Also we see our answer can be expressed in terms of distributional sections matching the standard approach in the case of rational polarizations. Finally we raise questions about the possibility of finding answers using asymptotic sections and the need to define pairings between our quantum phase spaces.