Explicit constructions of quilts with seam condition coming from symplectic reduction

TL;DR

This paper constructs explicit quilted surfaces with seam conditions derived from symplectic reduction, demonstrating a novel figure eight bubbling phenomenon that obstructs Floer chain group isomorphisms.

Contribution

It provides the first concrete examples of figure eight bubbles in quilted Floer theory, linking symplectic reduction to seam conditions and Floer chain group obstructions.

Findings

Constructed quilts with seam conditions from symplectic reduction.

Produced explicit figure eight bubbles obstructing Floer isomorphisms.

Identified the first concrete examples of figure eight bubbling phenomena.

Abstract

Associated to a symplectic quotient $M/\!/G$ is a Lagrangian correspondence $\Lambda_G$ from $M/\!/G$ to $M$. In this note, we construct in two examples quilts with seam condition on such a correspondence, in the case of $S^1$ acting on $\mathbb{CP}^2$ with symplectic quotient $\mathbb{CP}^2/\!/ S^1 = \mathbb{CP}^1$. First, we study the quilted strips that would, if not for figure eight bubbling, identify the Floer chain groups $CF(\gamma,S_{\text{Cl}}^1)$ and $CF(\mathbb{RP}^2,T_{\text{Cl}}^2)$, where $\gamma$ is the connected double-cover of $\mathbb{RP}^1$. Second, we answer a question due to Akveld-Cannas da Silva-Wehrheim by explicitly producing a figure eight bubble which obstructs an isomorphism between two Floer chain groups. The figure eight bubbles we construct in this paper are the first concrete examples of this phenomenon.