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Correspondence of monoidal structures on Fukaya categories with the derived tensor on Perf(X) under HMS

Determine whether the symmetric monoidal structures constructed on generalized Donaldson–Fukaya categories for (almost) toric fibrations correspond, under homological mirror symmetry, to the standard derived tensor product ⊗^{L}_{O_X} on Perf(X).

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Background

A symmetric monoidal structure on generalized Donaldson–Fukaya categories has been constructed for smooth torus fibrations and announced for almost toric fibrations with focus–focus singularities.

The authors note that it remains unsettled whether these monoidal structures match the usual derived tensor product on the mirror Perf(X) under homological mirror symmetry.

References

It is recently announced in that the construction of this monoidal structure can be extended to singular cases, more precisely, to almost toric fibrations only with focus-focus singularities in the sense of *{\href{https://arxiv.org/pdf/math/0210033}{Definition 4.2, Definition 4.5.}, while it seems to be still unknown if those monoidal structures correspond to $\tens_X$ on $\perf X$ under homological mirror symmetry.

— Polarizations on a triangulated category (2502.15621 - Ito, 21 Feb 2025) in Section 5.1, Example (Monoidal structures on Fukaya categories)