In the geometric quantization of Lagrangian fibrations, the index
of the Spin^c Dirac operator is often observed to be equal to the number of Bohr-Sommerfeld fibers in the several examples, such as, the nonsingular Lagrangian fibrations case, projective toric varieties, Gelfand-Cetlin integrable system on the complex flag variety, the Goldman's integrable system on the moduli space of flat connections on a surface.
In this talk, for the prequantizable nonsingular Lagrangian fibrations
with a certain class of compatible complex structures let us formally"
construct a basis of the space of holomorphic sections of the prequantum line bundle such that the basis is indexed by the Bohr-Sommerfeld points and the support of each section in the basis converges to the corresponding Bohr-Sommerfeld fiber by the adiabatic limit. We will discuss the general case if time is permitted.