Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts

TL;DR

This paper proves a Gromov compactness theorem for pseudoholomorphic quilts with immersed Lagrangian correspondences, revealing figure eight bubbling as a key phenomenon with algebraic implications for Floer theory.

Contribution

It establishes a Gromov compactness result for strip shrinking in pseudoholomorphic quilts involving immersed correspondences, and predicts algebraic effects of figure eight bubbling.

Findings

Figure eight bubbling occurs in the limit of strip shrinking.

Geometric composition extends to a curved $A_$-bifunctor.

Floer complexes are isomorphic after a figure eight correction.

Abstract

We establish a Gromov compactness theorem for strip shrinking in pseudoholomorphic quilts when composition of Lagrangian correspondences is immersed. In particular, we show that figure eight bubbling occurs in the limit, argue that this is a codimension-$0$ effect, and predict its algebraic consequences -- geometric composition extends to a curved $A_\infty$-bifunctor, in particular the associated Floer complexes are isomorphic after a figure eight correction of the bounding cochain. An appendix with Felix Schm\"{a}schke provides examples of nontrivial figure eight bubbles.