Smooth SYZ fibration and dual local-system torus

Let $X^{\circ }$ be a smooth open subset of an almost-Calabi–Yau $n$-fold and $B^{\circ }$ a smooth real $n$-manifold. Write $T^n=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^n$. A smooth SYZ fibration datum is a smooth proper submersion $\pi :X^{\circ }\to B^{\circ }$ whose fibres $L_b={\pi }^{-1}(b)$ are connected oriented manifolds diffeomorphic to $T^n$ and special Lagrangian, with a specified phase ${\theta }_b\in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ for each $b$. Proper means inverse images of compact subsets are compact; submersion means $d\pi$ is surjective at every point. The chosen domain $X^{\circ }$, base $B^{\circ }$, and any excluded discriminant are part of the data.

For each $b$, let $H_1(L_b;\mathbb{Z})$ denote first singular homology. The dual torus is ${\widehat{L}}_b=\mathrm{Hom}(H_1(L_b;\mathbb{Z}),U(1)),U(1)=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}\mathrm{.}$ It is the character group; equivalently it specifies one-dimensional complex representations of ${\pi }_1(L_b)$ that factor through $H_1(L_b;\mathbb{Z})$ and have image in $U(1)$. To define the family, use the homology local system $R_1{\pi }_{\ast }\mathbb{Z}$: in a smooth local trivialization of $\pi$, its sections are locally constant elements of $H_1(T^n;\mathbb{Z})$, glued by the homology maps induced by changes of trivialization. Equivalently, supply a local system $\mathcal{H}$ with a specified isomorphism $\eta :\mathcal{H}\stackrel{\sim }{\to }{R}_1{\pi }_{\ast }\mathbb{Z}$ of local systems. In fibre terms this requires ${\eta }_c{T}_{\gamma }^{\mathcal{H}}=T_{\gamma }^{H_1}{\eta }_b$ for every path $\gamma$ from $b$ to $c$, where $T_{\gamma }$ denotes parallel transport. Isomorphisms of individual fibres without this transport compatibility are insufficient. The dual family is the character-torus bundle glued by precomposition with the inverse homology transport, with fibre $\mathrm{Hom}({\mathcal{H}}_b,U(1))$. Its monodromy is this induced action along loops. Integral-affine coordinates and their monodromy may be included as further data but are not implied by this definition.

This is only a definition on the smooth locus. It asserts no fibration exists, no singular compactification, and no instanton-corrected gluing or mirror reconstruction.

Special Lagrangian with phase

Symplectic and almost-complex curve data