Local Mirror Symmetry at Higher Genus
نویسندگان
چکیده
We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function from higher genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P case. The Kodaira-Spencer theory of gravity, tailored to this local geometry, can be solved to compute this partition function. Then, using the results of Gopakumar and Vafa [1] and the local mirror map [2], the partition function can be rewritten in terms of expansion coefficients, which are found to be integers. We verify, through localization calculations in the A-model, many of these Gromov-Witten predictions. The integrality is a mystery, mathematically speaking. The asymptotic growth (with degree) of the invariants is analyzed. Some suggestions are made towards an enumerative interpretation, following the BPS-state description of Gopakumar and Vafa. ∗ email: [email protected] ∗∗ email: [email protected]
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