Abstract
The Gopakumar-Vafa conjecture predicts that the Gromov-Witten invariants of a Calabi-Yau 3-fold can be canonically expressed in terms of integer invariants called BPS numbers. Using the methods of symplectic Gromov-Witten theory, we prove that the Gopakumar-Vafa conjecture holds for any symplectic Calabi-Yau 6-manifold, and hence for Calabi-Yau 3-folds. The results extend to all symplectic 6-manifolds and to the genus zero GW invariants of semipositive manifolds.
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author = {Bryan, Jim and Pandharipande, Rahul},
title = {B{PS} states of curves in {C}alabi-{Y}au 3-folds},
journal = {Geom. Topol.},
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author = {Bryan, Jim and Pandharipande, Rahul},
title = {The local {G}romov-{W}itten theory of curves},
note = {With an appendix by Bryan, C. Faber, A. Okounkov and Pandharipande},
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doi = {10.1090/S0894-0347-06-00545-5},
url = {http://dx.doi.org/10.1090/S0894-0347-06-00545-5},
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url = {http://dx.doi.org/10.1007/s002229900028},
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} -
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