Quantum Knot Invariants and Supersymmetric Gauge Theories

Coordinators: Sergei Gukov, Mikhail Khovanov, Andrew Lobb, Piotr Sułkowski, and Cumrun Vafa

This program will focus on new relations between knot theory, supersymmetric field theories, and string theory. While there are no obvious links between any pair of these topics, all three are in fact intricately interrelated. Tremendous developments in knot theory in recent decades led to the formulation of polynomial knot invariants, such as the Jones polynomial and its generalizations. Furthermore, within the last decade it has been found that knot polynomials can be categorified, meaning that they arise as the Euler characteristics of much richer homological invariants. In recent years, it has also been realized that these results have many beautiful relations to various developments in high energy physics: supersymmetric gauge theories, topological string theory, counting of BPS states, refinement, and so on. This brings in three advantages. Firstly, knot theory provides a relatively simple (compared to the real world) playground for testing and developing new ideas in quantum field theory and string theory. Secondly, various exact computations and tools devised by physicists lead to concrete (albeit often conjectural) mathematical statements, which are hard to obtain by rigorous mathematical techniques. Thirdly, quantum invariants of knots are intimately related to other frontiers in physics, in particular to the field of topological quantum computation, which is being actively developed – aiming to revolutionize computer science and technology – in places such as Santa Barbara’s Station Q.

The program will have an interdisciplinary character, and will be devoted to topics on which various groups of physicists and mathematicians see often from different perspectives, using different lenses to understand them, and also different languages to describe them. An important goal of the program is to bring researchers from those communities together, establish a common language and explain important results to each other, summarize the status of the field, and specify important directions for future research.