Strictly cyclic structures on the Fukaya category over non-real fields

Determine whether the Fukaya category admits a strictly cyclic structure (i.e., a cyclic pairing compatible with its A_infty operations) over ground fields that do not contain the real numbers. Resolving this would clarify whether open Gromov–Witten potentials and related constructions can be defined intrinsically over arbitrary characteristic-zero fields without relying on integration over \mathbb{R}.

Background

Cyclic symmetry in Lagrangian Floer theory underpins the construction of open Gromov–Witten potentials invariant under gauge and geometric changes. In the de Rham model, integration yields a cyclic pairing over \mathbb{R}, but the lack of Kuranishi structures with fully submersive boundary evaluations across all marked points prevents transferring this cyclic symmetry to other chain-level models.

As a result, it is unclear whether strictly cyclic structures exist for the Fukaya category over fields not containing \mathbb{R}. Establishing or refuting their existence would determine whether standard OGW potentials can be defined over more general base fields.