Mirror symmetry and tropical geometry
Janko Boehm
TL;DR
This paper introduces a tropical geometry-based mirror construction for Calabi-Yau varieties, unifying previous methods and applying it to complex examples like Pfaffian Calabi-Yau threefolds.
Contribution
It presents a new tropical mirror construction that generalizes existing methods and provides an explicit algorithm with practical examples.
Findings
Reproduces Batyrev's mirror for hypersurfaces
Recovers Rodland's mirror for degree 14 Calabi-Yau
Provides an explicit tropical mirror construction algorithm
Abstract
Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rodland for the degree 14 Calabi-Yau threefold in PP^6 defined by the Pfaffians of a general linear 7x7 skew-symmetric matrix. We provide the necessary background knowledge entering into the tropical mirror construction such as toric geometry, Groebner bases, tropical geometry, Hilbert schemes and deformations. The tropical approach yields an algorithm which we illustrate in a series of explicit examples.
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Taxonomy
TopicsNonlinear Waves and Solitons · Algebraic structures and combinatorial models · Algebraic Geometry and Number Theory