From special Lagrangian to Hermitian–Yang–Mills via Fourier–Mukai transform

By N. Leung, Shing-Tung Yau, Eric Zaslow. 2000

237 citations recorded by the source corpus.

We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and Bcycles. In this paper, we assume that the mirror pair are dual torus fibrations with flat tori and that the A-cycle is a section. We also show that this transformation preserves the (holomorphic) Chern-Simons functional for all connections. Furthermore, on corresponding moduli spaces of supersymmetric cycles it identifies the graded tangent spaces and the holomorphic m-forms. In particular, we verify Vafa’s mirror conjecture with bundles in this special case.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c5b0a2f29506c6daa89da00198eddc54e57a4997

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Citations in the corpus

Concavity of the Lagrangian phase operator and applications (2016, methodology): “It was shown by Leung-Yau-Zaslow [25] that, in the semi-flat model of Mirror Symmetry, solutions of the dHYM equation correspond by a Fourier-Mukai transform to special Lagrangian submanifolds of the mirror.”
A class of curvature type equations. (2019, background): “A complex analogue of this equation on compact Kähler manifolds also appeared naturally from the study of Mirror Symmetry, see Leung-Yau-Zaslow [33], Collins-Jacob-Yau [9] and Collins-Yau [11].”
Mirror symmetry for toric Fano manifolds via SYZ transformations (2008, methodology): “In this subsection, we first give a brief review of semi-flat SYZ mirror transformations (for details, see Hitchin [26], Leung-Yau-Zaslow [31] and Leung [30]).”
Mirror Symmetric SU(3)-Structure Manifolds with NS Fluxes (2003, methodology): “Extending the picture developed in [5] for the exchange of branes (a”
A scan for new N = 1 vacua on twisted tori (2006, methodology): “It also appeared in the context of Calabi– Yau manifolds: for example in [68, 69] it was used to show that the D–term BPS conditions for B–branes [70], which read Im(eiθeB+F eiJ), are mapped by mirror symmetry to the stability conditions for A–branes, Im(eiθeF Ω).”