Torus fibrations and localization of index I

TL;DR

This paper introduces a local Riemann-Roch number for open symplectic manifolds with integrable systems, relating it to Bohr-Sommerfeld fibers and singular fibers, using a Witten deformation approach.

Contribution

It defines a new local Riemann-Roch number for open symplectic manifolds with integrable systems and describes its relation to singular Lagrangian fibrations.

Findings

Riemann-Roch number expressed as sum over nonsingular Bohr-Sommerfeld fibers and singular fiber contributions.

Application of Witten's deformation to a Hilbert bundle in the proof.

Framework for understanding index localization in symplectic geometry.

Abstract

We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the number of nonsingular Bohr-Sommerfeld fibers and a contribution of the singular fibers. A key step of the proof is formally explained as a version of Witten's deformation applied to a Hilbert bundle.