Partial collapsing degeneration of Floer trajectories and adiabatic gluing

TL;DR

This paper develops a gluing theorem for Floer trajectories undergoing partial collapsing degeneration, providing new proofs of isomorphism properties in Lagrangian Floer theory and related complexes.

Contribution

It introduces a novel gluing theorem for partial collapsing Floer trajectories, applicable to Lagrangian boundary problems, and offers new proofs of key isomorphism results in Floer theory.

Findings

Proves a new gluing theorem for partial collapsing Floer trajectories.

Provides a direct proof of chain isomorphism between Morse-Bott Floer complex and pearly complex.

Offers an alternative proof of the PSS map isomorphism without target rescaling.

Abstract

We study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic $\varepsilon$-family and its reversal adiabatic gluing, as the prototype of the partial collapsing degeneration of $2$-dimensional (perturbed) $J$-holomorphic maps to $1$-dimensional gradient segments. We consider the case when the Floer equations are $S^1$-invariant on parts of their domains whose adiabatic limits have positive lengths as $\varepsilon \to 0$, which we call thimble-flow-thimble configurations. The main gluing theorem we prove also applies to the case with Lagrangian boundaries such as in the problem of recovering holomorphic disks out of pearly configurations. In particular, our gluing theorem gives rise to a new direct proof of the chain isomorphism property between the Morse-Bott version of Lagrangian intersection Floer complex of $L$ by Fukaya-Oh-Ohta-Ono and the pearly complex of $L$ by Lalonde and Biran-Cornea (for monotone Lagrangian submanifolds). It also provides another proof of the present authors' earlier proof of the isomorphism property of the PSS map without involving the target rescaling and the scale-dependent gluing. (This is a rewritten version of our previous arXiv posting, arXiv:1103.3525.)