Lagrangian sections and holomorphic U(1)-connections.
It was conjectured by Strominger-Yau-Zaslow that Calabi-Yau spaces can be often fibered by special Lagrangian tori and their mirrors can be constructed by dualizing these tori. It was further suggested by Vafa that the holomorphic vector bundles and Calabi-Yau \(n\)-fold \(M\) correspond to the Lagrangian submanifolds in the mirror \(\check M\) and the stable vector bundles correspond to the special Lagrangian submanifolds in \(\check M\). In this paper it is assumed that \(M\) admits a special Lagrangian torus bifurcation, i.e., a topological fibration \(\pi: M\to B\), where \(B\) is a compact \(n\)-dimensional manifold which is locally a Lagrangian section of \(\pi\), whose fibers \(\pi^{-1}(P)\) are special Lagrangian \(n\)-tori with respect to the Kähler form \(\omega\) and a nonvanishing holomorphic \(n\)-form \(\Omega\) on \(M\), i.e., \(\omega|_{\pi^{-1}(P)}\equiv 0\) and \(\text{Im\,}\Omega|_{\pi^{-1}(P)}\equiv 0\). In this case the mirror manifold has been constructed by Mitchin as a smooth special Lagrangian torus bifraction over Bas well. The paper is devoted to the construction of a correspondence between the gauge equivalence classes of holomorphic \(U(1)\)-connections on \(M\) and Hamiltonian deformating classes of Lagrangian sections in the mirror manifold \(\check M\).
scientific article; zbMATH DE number 1818963