The Spectral Flow of a Restriction to a Subspace and the Maslov Indices in a Symplectic Reduction

TL;DR

This paper establishes a general formula connecting spectral flow and Maslov indices in symplectic reduction, enhancing understanding of spectral invariants in infinite-dimensional symplectic geometry.

Contribution

It provides a comprehensive formula linking spectral flow of quadratic forms and their restrictions, and relates Maslov indices before and after symplectic reduction.

Findings

Derived a general spectral flow restriction formula

Connected Maslov indices of original and reduced paths

Applied results to symplectic reduction scenarios

Abstract

We prove in full generality a formula that relates the spectral flow of a continuous path of quadratic forms of Fredholm type with the spectral flow of the restrictions of the forms to a fixed closed finite codimensional subspace. We then apply this to obtain a formula relating the Maslov index of a continuous path in a Fredholm Lagrangian Grassmannian with the Maslov index of its symplectic reduction by a closed finite codimensional coisotropic subspace.