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

Abstract

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 HermitianYang-Mills equations. That is, the transformation relates supersymmetric A-and B-cycles. 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 super symmetric 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. e-print archive: http://xxx.lanl.gov/math.DG/0005118

Non-Kähler SYZ Mirror Symmetry

“…The SYZ transform we use in this paper follows from [LYZ00,Leu05]. In this section we briefly review the setting and definitions which will be used in the rest of the paper.…”

“…The semi-flat A-branes on X are mirror to semi-flat B-branes onX under Fourier-Mukai transform. The readers are referred to [LYZ00] for details.…”

“…The survey by Fu [Fu10] gave an excellent introduction to this topic. In this paper we focus on the semi-flat setting [LYZ00,Leu05], namely we consider Lagrangian fibrations away from singular fibers. We will see that interesting correspondences between the Type IIA and IIB systems and their RR fluxes already appear in such a simple setting.…”

Mirror Symmetric SU(3)-Structure Manifolds with NS Fluxes

“…This is also true for the other components. 5 As already emphasized, one has to bear in mind that the almost complex structure is in general not integrable, so that dz i is not to be understood as the differential of a hypothetical coordinate z i .…”

“…In the Calabi-Yau case, this exchange is implicit in many applications of mirror symmetry. For example, the even periods and the D-brane charge can be written using e B+iJ , and its exchange with Ω was used in mapping [5] stringy-corrected DUY equations [6] to the special lagrangian condition; e B+iJ was also used in formulating the concept of Π-stability [7].…”