Explicit constructions of quilts with seam condition coming from symplectic reduction

Abstract

Associated to a symplectic quotient $M∕∕G$ is a Lagrangian correspondence ${\mathrm{\Lambda }}_G$ 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 exhibit the moduli space of quilted strips that would, if not for figure-eight bubbling, identify the Floer chain groups $\mathit{CF}(\mathit{\gamma },S_{\mathrm{Cl}}^1)$ and $\mathit{CF}({\mathbb{RP}}^2,T_{\mathrm{Cl}}^2)$, where γ 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 that we construct in this paper are the first concrete examples of this phenomenon.