Mirror symmetry, Gröbner bases and tropical geometry

• Antragsteller: Dr. Janko Böhm
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2008 bis 2010
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 92940236

Mirror symmetry is a key link between mathematics and theoretical physics, e.g., algebraic geometry obtains new ideas in enumerative geometry from superstring theory which, in return, benefits from the study of Calabi-Yau varieties. Important insight to mirror symmetry is gained by explicit constructions computing for a given Calabi-Yau variety the corresponding mirror Calabi-Yau. In [Boehm, 2007] we develop via tropical and toric geometry a general framework for mirror symmetry leading to an algorithmic mirror construction. It unifies the known constructions such as that by Batyrev for hypersurfaces, its generalization by Batyrev and Borisov to complete intersections and that by Rødland for a particular non-complete intersection. The tropical mirror construction extends these to a considerably larger class of Calabi-Yau varieties and produces explicit new mirror examples. It comes with a natural mirror map identifying deformations and divisors. The goal of the project is to extend the understanding of the tropical mirror construction to a deeper level via the concepts of stringy E-functions, quantum cohomology, instanton numbers and T-duality, which relate naturally to the data provided by tropical mirror symmetry. Key insight will be given by computing new mirrors via the tropical mirror construction. This may lead to a further extension of the class of Calabi-Yau varieties to which it is applicable.

• DFG-Verfahren: Forschungsstipendien
• Internationaler Bezug: USA
• Gastgeber: Professor Dr. David Eisenbud