Konstruktionen und Beispiele für die SYZ-Vermutung der Mirror-Symmetrie

• Applicant: Dr. Marco Kühnel
• Subject Area: Mathematics
• Term: from 2003 to 2005
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5392380

Mirror Symmetry is a notion for the phenomenon discovered in physics that a universe consisting of strings instead of particles can be encoded equally by two different three-dimensional complex manifolds. Such pairs of manifolds are called mirror pairs. Supersymmetry forces the manifolds to obey some restrictions for the curvature. Manifolds satisfying these restrictions are called Calabi-Yau manifolds. The existence of a mirror partner for an arbitrary Calabi-Yau manifold is not yet clear. Originating also in physics, the conjecture of Strominger, Yau and Zaslow (SYZ conjecture) states that the mirror partners are geometrically connected by dualizing special families of tori inside the manifolds. The research project aims to find such families of tori and to formulate and prove the correct procedure of dualizing. This has been done only for so called toric Calabi-Yau manifolds. To develop the theory in a more general context it is important to have non-toric examples. For this purpose we will have a close look at some class of manifolds already discussed in the applicant´s thesis. Having started with this, the target will be a formulation of the SYZ-conjecture as general as possible. This would also be a way to settle the existence problem of mirror partners.

• DFG Programme: Research Fellowships