Kuranishi structures with compatible forgetful maps and submersive boundary evaluations

Establish the existence of Kuranishi structures on the moduli spaces of pseudoholomorphic disks with boundary on a Lagrangian submanifold that are simultaneously (i) compatible with forgetful maps of marked points and (ii) submersive for all boundary evaluation maps. Such structures are needed to implement cyclic symmetry at the chain level for Lagrangian Floer-theoretic constructions using models like smooth singular chains.

Background

In constructing cyclic pairings for Lagrangian Floer theory, one seeks Kuranishi structures on moduli spaces of pseudoholomorphic disks that are compatible with forgetful maps of marked points and ensure submersivity of boundary evaluation maps at all boundary marked points. This combination of properties enables cyclic symmetry needed to define strictly cyclic A_infty structures in chain models beyond de Rham forms.

Existing approaches (e.g., Fukaya’s framework) achieve submersivity only for a single boundary evaluation map, and compatibility with all forgetful maps together with simultaneous submersivity is not currently available. This limitation obstructs extending cyclic structures to settings where integration pairings are not available, such as over general fields.