A remark on monoidal structure and homological mirror symmetry

TL;DR

This paper explores how the monoidal structure of the Fukaya category of a symplectic manifold can determine its homological mirror functor, linking algebraic and geometric aspects of mirror symmetry.

Contribution

It demonstrates that the monoidal structure of the Fukaya category uniquely determines the homological mirror functor, filling a gap in the understanding of mirror symmetry.

Findings

The monoidal structure recovers the mirror $Y$ via the Balmer spectrum.

The monoidal structure determines the homological mirror functor.

Clarifies the relationship between monoidal structures and mirror symmetry.

Abstract

For a symplectic geometry $X$, suppose the (derived) Fukaya category $\mathrm{Fuk}(X)$ of $X$ is equipped with a monoidal structure. Then its Balmer spectrum recovers a mirror $Y$ of $X$ if there exists homological mirror symmetry $\mathrm{Fuk}(X)\cong D^b\mathrm{coh}(Y)$ and the monoidal structure is the mirror of the standard one of $D^b\mathrm{coh}(Y)$. In this short note, we fill one gap of this story in the literature: we show that the monoidal structure determines the homological mirror functor $\mathrm{Fuk}(X)\to D^b\mathrm{coh}(Y)$.