Explicit constructions of quilts with seam condition coming from symplectic reduction

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.

9 pages, 3 figures

References in corpus (4)

• Quilted Floer Cohomology
• Quilted Floer trajectories with constant components
• A Chiang-type Lagrangian in CP2
• Lagrangian Floer homology of the Clifford torus and real projective space in odd dimensions