Lagrangian sections and holomorphic U(1)-connections.
type: Item
instance of: scholarly article
title: Lagrangian sections and holomorphic U(1)-connections.
zbMATH Open document ID: 1064.32020
published in: Pacific Journal of Mathematics
publication date: 2002-10-22T00:00:00Z
review text: 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 \(\check M\).
reviewed by: S. Ivashkovich
Mathematics Subject Classification ID: 32Q25
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 14J32
zbMATH DE Number: 1818963
zbMATH Keywords: Calabi-Yau manifold
zbMATH Keywords: Lagrangian submanifold
zbMATH Keywords: mirror symmetry
author: Jingyi Chen
MaRDI profile type: Publication
full work available at URL: pjm.2002.203.139
OpenAlex ID: W2081909402
DOI: 10.2140/PJM.2002.203.139
description: scientific article; zbMATH DE number 1818963