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.

References

Consequently, it is unclear whether or not the Fukaya category generally carries a strictly cyclic structure over fields which do not contain $\mathbb{R}$.

— Infinity inner products and open Gromov--Witten invariants  (2406.08693 - Haney, 2024) in Section 1 (Introduction)

The corresponding question is whether its point-like theory can be defined over Z[1/2].

— Open WDVV equations for non-commutative families of bounding cochains  (2610.13209 - Kosloff, 9 Oct 2026) in Section 1, subsection “Arithmetic of OGW” (Section 1.4)

Under the hypotheses of the character construction, does the existence of a real point-like bounding cochain ensure that the boundary character extends to a character of the full quantum algebra with real coefficients, after adjoining any necessary Novikov roots?

— Open WDVV equations for non-commutative families of bounding cochains  (2610.13209 - Kosloff, 9 Oct 2026) in Section 1, subsection “Open Gromov–Witten theory with an anti-symplectic involution” (Section 1.5)