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Skeins on Branes
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We study 1-parameter families of holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds. We show that the expected codimension one phenomena can be organized to match the HOMFLYPT skein relations from quantum topology. It follows that counting holomorphic curves by the class of their boundaries in the skein module of the Lagrangian gives a deformation invariant result. This is a mathematically rigorous incarnation of Witten's assertion that boundaries of open topological strings create line defects in Chern-Simons theory. Using this theory, we rigorously establish the following prediction of Ooguri and Vafa: the coefficients of the HOMFLYPT polynomial of a link in the three-sphere count the holomorphic curves in the resolved conifold, with boundary on (a push-off of) the link conormal.
Forward citations
Cited by 2 Pith papers
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Compatibility of quantum trace and UV-IR maps
Quantum trace and quantum UV-IR maps fit into a natural commutative square; the surface case proves the Neitzke-Yan conjecture and the 3-manifold case recovers the 3d quantum trace map from the UV-IR map.
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Calabi-Yau Deformation Quantization
A Calabi-Yau version of Kontsevich's formality morphism is recorded, yielding canonical closed deformation quantizations for unimodular holomorphic Poisson Calabi-Yau manifolds.
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