Physics and mathematics of knot homologies: June 1 – 5, 2015

Organized by Eugene Gorsky, Sergei Gukov, Mikhail Khovanov, Vivek Shende, and Piotr Sulkowski

Dates: June 1 – 5, 2015

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The aim of this workshop is to exchange current ideas in physics and mathematics related to knot homologies. Intimate relations between knots and physics, found around 25 years ago in the seminal work by Witten, grew into a fascinating web of correspondences involving topological and supersymmetric quantum field and string theories. In mathematics, (perhaps) the most important modern idea in knot theory is the program of categorification and formulation of knot homologies; this program is still being completed, and one of outstanding questions related to knot homologies is their geometric meaning.

In the last decade several knot homology theories has been developed. Some of them, such as Heegaard-Floer homology, are based on the ideas from symplectic geometry and carry a lot of deep information about the geometry of a given knot. Several other theories, developed by Khovanov and Rozansky, are more combinatorial and are defined in terms of a knot diagram. It is expected that Khovanov-Rozansky homology carry some geometric information too: for example, a slice genus bound in terms of Khovanov homology has been found. Nonetheless, while several geometric interpretations for the Khovanov-Rozansky homologies has been developed, these approaches share the same problem: the size of the chain complex grows very fast with the complexity of a knot and by far outgrow the size of its homology. As a result, no cleargeometric understanding of the homology generators is known.

Recently progress in this direction has been made for algebraic knots, where an explicit conjectural description of Khovanov-Rozansky homology in terms of the homology of Hilbert schemes of points on a singular curves has been found. This conjecture was partially verified on the level of Euler characteristics. For torus knots, further works revealed the intimate relation between these Hilbert schemes and representation theory of double affine Hecke and rational Cherednik algebras.

In parallel with all these developments, many new ideas related to knot homologies have been established in physics. Firstly, the so-called 3d-3d correspondence has been formulated, which relates Chern-Simons theory on a knot complement to a 3-dimensional supersymmetric N=2 gauge theory. Secondly, it has been found that partition functions and indices of these 3d N=2 theories are built from universal holomorphic blocks. Thirdly, a new class of algebraic curves associated to knots, so-called super-A-polynomials, akin to Seiberg-Witten curves, has been discovered. Furthermore, existence of an intricate structure of differentials in knot homologies has been postulated based on properties of BPS states, and a refinement of Chern-Simons theory, responsible for the homological grading, has been formulated. All these developments are very interesting physically, and relate to the mathematical conjectures mentioned above.

The aim of this workshop is to provide an up to date summary of all the above approaches, to formulate a list of (currently) most important problems, and to propose paths which might lead to their solution. We believe that the workshop will be a source of mutual inspirations for physicists and mathematicians, and hope that the combined effort of both our communities will lead to solutions of those problems in the foreseeable future.

This workshop is a part of the Spring 2015 program, Knot homologies, BPS states, and SUSY gauge theories, which is organized by Sergei Gukov, Mikhail Khovanov, and Piotr Sulkowski. This program takes place from March 16 – June 12, 2015.