Source Spaces and Perturbations for Cluster Complexes

We define objects made of marked complex disks connected by metric line segments and construct two sequences of moduli spaces of these objects, referred as the ⊗ version (nonsymmetric) and the • version (symmetric). This allows choices of coherent perturbations over the corresponding versions of the Floer trajectories proposed by Cornea and Lalonde ([CL]). These perturbations are intended to lead to an alternative geometric description of the (obstructed) A_∞ and L_∞ structures studied by Fukaya, Oh, Ohta and Ono ([FOOO2],[ FOOO]) and Cho ([Cho]).

Given a Pin_± monotone lagrangian submanifold L ⊂ (M, ω) with minimal Maslov number ≥ 2, we define an A_∞-algebra structure from the critical points of a generic Morse function on L. We express this structure as a cochain complex extending the pearl complex introduced by Oh ([ Oh]) and further explicited by Biran and Cornea ([BC]), equipped with its quantum product. This could also be seen as an alternative geometric description of a Fukaya category of (M, ω) with L as its only object, a hamiltonian relative version appearing in [ Sei]. Using spaces of quilted clusters, we verify, using more general quilted cluster spaces, that this defines a functor from a homotopy category of Pin_± monotone lagrangian submanifolds [special characters omitted](M, ω) to the homotopy category of cochain complexes hK(Λ-mod) where Λ is an appropriate Novikov ring.