Occurrence of co-isotropic branes in the split-closure of the affine-Lagrangian Fukaya category

Determine whether co-isotropic branes occur, possibly non-obviously, in the split-closure of the Fukaya category generated by affine Lagrangians on a symplectic torus.

Background

The discussion distinguishes the Fukaya category generated by affine Lagrangians from a potentially larger Fukaya category containing co-isotropic branes. Space-filling branes provide concrete examples of co-isotropic objects expected to correspond to coherent sheaves that are not obviously in the image of the affine-Lagrangian construction.

Resolving whether such objects already belong to the split-closure of the affine-Lagrangian category would clarify whether the larger category is genuinely necessary for homological mirror symmetry in this setting.

References

As far as we understand it remains possible that they appear in a non-obvious way in the split-closure of Fukaya's category of affine Lagrangians.

— The derived category of a generalized complex torus  (2610.21960 - Antweiler et al., 15 Oct 2026) in Section 1, subsection “Noncommutative mirror symmetry”

Higher compositions will be considered in future work.

— The derived category of a generalized complex torus  (2610.21960 - Antweiler et al., 15 Oct 2026) in Section 1, subsection “Lagrangians and strict deformation quantization”