Local Mirror Symmetry: Calculations and Interpretations
T.-M. Chiang, A. Klemm, S.-T. Yau, E. Zaslow
TL;DR
This paper explores local mirror symmetry through mathematical calculations and interpretations, aligning A-model localization results with B-model Picard-Fuchs solutions, and examining invariants of singular surfaces.
Contribution
It provides new mathematical insights into local mirror symmetry, including explicit calculations and interpretations of invariants for singular and smooth surfaces.
Findings
A-model calculations match B-model Picard-Fuchs solutions
Local invariants of singular surfaces agree with smooth cases
Interpretation of Gromov-Witten-type numbers in enumerative geometry
Abstract
We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the Gromov-Witten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of singular surfaces agree with the smooth cases when they occur as complete intersections.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · Geometric and Algebraic Topology