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Source Spaces and Perturbations for Cluster Complexes

We define objects made of marked complex disks connected by metric line segments and construct nonsymmetric and symmetric moduli spaces of these objects. This allows choices of coherent perturbations over the corresponding versions of the Floer trajectories proposed by Cornea and Lalonde. These perturbations are intended to lead to an alternative description of the (obstructed) $A_\infty$-structures studied by Fukaya, Oh, Ohta and Ono. Given a $Pin_{\pm}$ monotone lagrangian submanifold $L \subset (M,ω)$ with minimal Maslov number $N_L \geq 2$, we define an $A_\infty$-algebra (resp. differential graded algebra) structure from the critical points of a generic Morse function on $L$. It is written as a cochain (resp. chain) complex extending the pearl complex introduced by Oh and further explicited by Biran and Cornea, equipped with its quantum product. We verify that the construction is homotopy invariant, defining a functor from a homotopy category of $Pin_{\pm}$ monotone lagrangian submanifolds $h\mathcal{L}^{mono, \pm}(M,ω)$ to the homotopy category of cochain (resp. chain) complexes $hK(Λ\text{-mod})$ where $Λ$ is a Novikov ring with coefficients in $\mathbb{Z}$.