Mirror symmetry, Gröbner bases and tropical geometry
Final Report Abstract
Mirror symmetry is a key link between mathematics and theoretical physics, for example, 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. We consider a degeneration of Calabi-Yau varieties to a union of planes. Gröbner bases and tropical geometry allow us to associate combinatorial data to such a degeneration. Using this data we can formulate a general framework for mirror symmetry leading to an algorithmic mirror construction. It directly specializes to the famous construction by Batyrev for hypersurfaces and its generalization by Batyrev and Borisov to complete intersections. The tropical mirror construction extends this construction into the realm of the considerably larger class of non-complete intersection Calabi-Yau varieties. For example, we obtain known and new mirror pairs of Pfaffian and determinental varieties. Tropical mirror symmetry comes with a natural mirror map identifying deformations and divisors. Construction of new examples of mirror pairs is a key way to a deeper understanding of mirror symmetry. This is an important application of the tropical mirror symmetry framework. We develop, implement and document the key algorithms in the package SRdeformations, which is available as part of the open source computer algebra system Macaulay2. Interesting examples of degenerations of Calabi-Yau varieties come from the boundary complexes ∆(d, m) of cyclic polytopes of dimension d with m vertices. We prove a fundamental recursion for the Stanley-Reisner ring of ∆(d, m +1) in terms of the Stanley-Reisner rings of ∆(d, m) and ∆(d −2, m −1). It is actually a recursion for the resolutions of the Stanley-Reisner rings. As an application we obtain results in combinatorics. The implementation of tropical mirror symmetry gives a tool to address various problems in mirror symmetry, hence leading to new projects. Those include, for example, hypergeometric systems for non-complete intersections, subcanonical mirror symmetry, or mirror symmetry of Enriques surfaces by passing involutions to the mirror side via the deformation-divisor mirror map.
Publications
- On the structure of Stanley-Reisner rings associated to cyclic polytopes, Osaka J. Math., 20pp
Janko Böhm, Stavros Papadakis
- SRdeformations – Deformations of Stanley-Reisner rings and tropical mirror symmetry (2009)
Janko Böhm
