Journal
ADVANCES IN MATHEMATICS
Volume 224, Issue 1, Pages 169-245Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.aim.2009.11.007
Keywords
Mirror symmetry; Tropical geometry
Categories
Funding
- NSF [0805328]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0805328] Funding Source: National Science Foundation
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This paper explores the relationship between mirror symmetry for P-2, a, the level of big quantum cohomology, and tropical geometry. The mirror of P-2 is typically taken to be ((C-x)(2), W), where W is a Landau-Ginzburg potential of the form x + y + 1/xy. The complex moduli space of the mirror is the universal unfolding of W, and oscillatory integrals produce a Frobenius manifold structure oil this universal unfolding. We show that W can be deformed by counting Maslov index two tropical disks, and the natural parameters appearing in this deformation are then the flat coordinates on the moduli space. Furthermore, the oscillatory integrals are shown to read off directly tropical curve counts from the potential. Thus we show in fact that mirror symmetry for P-2 is equivalent in a strong sense to tropical curve counting formulas, including tropical formulas for gravitational descendent invariants. (C) 2009 Elsevier Inc. All rights reserved.
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