# Mirror Symmetry of Fourier—Mukai Transformation for Elliptic Calabi—Yau Manifolds

# Mirror Symmetry of Fourier—Mukai Transformation for Elliptic Calabi—Yau Manifolds

Mirror symmetry conjecture says that for any Calabi–Yau (CY) manifold *M* near the large complex/symplectic structure limit, there is another CY manifold *X*, called the mirror manifold, such that the B-model superstring theory on *M* is equivalent to the A-model superstring theory on *X*, and vice versa. Mathematically speaking, it roughly says that the complex geometry of *M* is equivalent to the symplectic geometry of *X*, and vice versa. It is conjectured that this duality can be realized as a Fourier-type transformation along fibers of special Lagrangian fibrations on *M* and *X*, called the SYZ mirror transformation F^{SY Z}. This chapter addresses the following two questions: (i) What is the SYZ transform of the elliptic fibration structure on *M*? (ii) What is the SYZ transform of the FM transform F^{FM} _{cx}? Section 15.2 reviews the SYZ mirror transformation and show that the mirror manifold to an elliptically fibered CY manifold has a twin Lagrangian fibration structure. Section 15.3 reviews the FM transform in complex geometry in general and also for elliptic manifolds. Section 15.4 first defines the symplectic FM transform between Lagrangian cycles on *X* and *Y*, and then defines twin Lagrangian fibrations, giving several examples of them, and studies their basic properties. Section 15.5 shows that the SYZ transformation of the complex FM transform between *M* and *W* is the symplectic FM transform between *X* and *Y*, which is actually the identity transformation.

*Keywords:*
mirror symmetry, Fourier–Mukai transformation, Calabi-Yau, A-model superstring theory

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