Integral-integral affine geometry, geometric quantization, and Riemann-Roch

TL;DR

This paper proves that for certain symplectic manifolds with Lagrangian torus fibrations, the Riemann-Roch number equals the count of Bohr-Sommerfeld fibers, illustrating the independence of polarization in geometric quantization.

Contribution

It provides a simple proof linking Riemann-Roch numbers to Bohr-Sommerfeld fibers in integral-integral affine settings, highlighting the independence of polarization phenomenon.

Findings

Riemann-Roch number equals the number of Bohr-Sommerfeld fibers.

Total volume of integral-integral affine manifolds equals the count of integer points.

The proof simplifies understanding of geometric quantization in this context.

Abstract

We give a simple proof that, for a pre-quantized compact symplectic manifold with a Lagrangian torus fibration, its Riemann-Roch number coincides with its number of Bohr-Sommerfeld fibres. This can be viewed as an instance of the "independence of polarization" phenomenon of geometric quantization. The base space for such a fibration acquires a so-called integral-integral affine structure. The proof uses the following simple fact, whose proof is trickier than we expected: on a compact integral-integral affine manifold, the total volume is equal to the number of integer points.