The mirror-symmetry categorical mapping conjecture for Calabi–Yau manifolds

Let $({\mathcal Y}\rightarrow T,P)$ be a family of Calabi–Yau manifolds with a complex cusp $P\in\partial T$, let $X$ be the topological mirror of $Y$ with respect to $P$, and assume that $H^3(X,\mathbb Z)_{\rm tors}=0$. Let ${\mathcal X}\rightarrow {\mathcal M}_X$ be the marked Calabi–Yau moduli space of $X$, and let $G({\mathcal X},{\mathcal M}_X)$ denote its categorical mapping group, with cohomological subgroup $G^{\rm coh,filtr}({\mathcal X},{\mathcal M}_X)$ as defined in the source. The mirror-symmetry categorical mapping conjecture. There exists a homomorphism

$$\xi\colon \operatorname{Diff}^+(Y)\longrightarrow G({\mathcal X},{\mathcal M}_X)/(\text{translations})$$

which fits into the stated commutative diagram with the natural cohomology actions and induces an isomorphism

$$\bar\xi\colon \langle\operatorname{Diff}^{\rm coh}(Y),(-1)\rangle/(\pm1)\longrightarrow G^{\rm coh,filtr}({\mathcal X},{\mathcal M}_X)/(\pm1).$$

This proposes that diffeomorphisms of one member of a mirror pair correspond to categorical symmetries throughout the complex moduli space of the other. The claim is presented as the paper’s main conjecture and no resolution is supplied in the source.