Open holomorphic curves and knot invariants

02.05.2024 16:15

Knot theory concern the classification of simple closed space curves up to continuous deformations through simple closed curves. There are many different knots and to distinguish them one uses invariants, algebraic quantities that do not change under deformations. In the 1980’s new, so called, quantum invariants were discovered by Jones and son thereafter generalized. These invariants are of combinatorial nature and can be understood from a physical perspective as Wilson lines in quantum field theory, as shown by Witten. Inspired by string theory, Ooguri and Vafa stated in 1999 a conjecture that gave an enumerative geometric interpretation of knot invariants: the invariants arises as counts of certain geometric objects, 2-dimensional holomorphic curves that end on a 3-dimensional space, the conormal of a knot, in a 6-dimensional space associated to the ambient 3-dimensional space where the knot lives. The mathematical treatment of the curve count suggested by Ooguri-Vafa is not straightforward, because of so called wall-crossing phenomena the count jumps under deformation. In recent work with Shende we managed to relate the combinatorics of knot invariants to the wall crossings and thereby found the mechanism for the relation between knot invariants and enumerative geometry. The talk will describe this construction and demonstrate how it leads to a proof of the Ooguri-Vafa conjecture.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-15, Colloque de mathématiques

entrée libre