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Skeins on branes
Tobias Ekholm1, 2 orcid ; Vivek Shende3, 4 orcid
1Department of mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
2Institut Mittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden
3Center for Quantum Mathematics, Syddansk Univ., Campusvej 55 5230 Odense, Denmark
4Department of mathematics, UC Berkeley, 970 Evans Hall, Berkeley CA 94720, USA
Publications Mathématiques de l'IHÉS, Online first, pp. 1-62
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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 [42].

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.

Received:
Revised:
Accepted:
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DOI: 10.5802/pmihes.30
Classification: 53D12
Keywords: Holomorphic curve, Lagrangian boundary condition, HOMFLYPT

Tobias Ekholm  1 , 2 ; Vivek Shende  3 , 4

1 Department of mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
2 Institut Mittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden
3 Center for Quantum Mathematics, Syddansk Univ., Campusvej 55 5230 Odense, Denmark
4 Department of mathematics, UC Berkeley, 970 Evans Hall, Berkeley CA 94720, USA 
Tobias Ekholm; Vivek Shende. Skeins on branes. Publications Mathématiques de l'IHÉS, Online first, pp. 1-62
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