Seminar

Skeins on branes

Knots, Strings, Symplectic Geometry and Dualities

24 September 14:00 - 15:00

Vivek Shende - University of California, Berkeley

We give a geometric interpretation of the coefficients of the HOMFLYPT polynomial of any link in the three-sphere as counts of holomorphic curves. The curves counted live in the resolved conifold where they have boundary on a shifted copy of the link conormal, as predicted by Ooguri and Vafa.. To prove this, we introduce a new method to define invariant counts of holomorphic curves with Lagrangian boundary: we show geometrically that the wall crossing associated to boundary bubbling is the framed skein relation. It then follows that counting holomorphic curves by the class of their boundary in the skein of the Lagrangian brane gives a deformation invariant curve count. This is a mathematically rigorous incarnation of the fact that boundaries of open topological strings create line defects in Chern-Simons theory, as described by Witten. The technical key to skein invariance is a new compactness result: if the Gromov limit of J-holomorphic immersions collapses a curve component, then its image has a singularity worse than a node.

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Organizers
Tobias Ekholm,
Uppsala University
Sergei Gukov
California Institute of Technology, Caltech
Vivek Shende
University of California, Berkeley

Program
Contact

Tobias Ekholm

tobias.ekholm@math.uu.se

Other
information

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