The geometry of Lagrangian fibrations has been studied by a number of authors, and turns out to be quite rigid; as one example, the Arnold-Liouville theorem implies that the base $B$ of a Lagrangian fibration $M\to B$ can be equipped with an integral affine structure. In the presence of a prequantization $L \to M$, we can further equip B with what we call an "integral-integral affine structure," which gives us a notion of "integral points" in B.

In this talk we will review some facts about geometric quantization and Lagrangian fibrations and discuss their interaction, and prove an "independence of polarization" result for Lagrangian torus fibrations. Along the way, we will encounter a simple fact in "integral-integral affine geometry" that on the surface has nothing to do with quantization and whose proof is surprisingly tricky.

This is joint work with Yael Karshon and Takahiko Yoshida.