From Special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai Transform

Details From Special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai Transform From Special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai Transform

2000-05-11

arxiv.org/abs/math/0005118v1

Adv.Theor.Math.Phys. 4 (2002) 1319-1341

Mathematics

Differential Geometry

20 pages

Scientific paper

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 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 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.

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