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Quantum fields and knot homologies

Final Report Summary - FIELDS-KNOTS (Quantum fields and knot homologies)

The project "Quantum Fields and Knot Homologies" was devoted to the analysis of relations between physics and mathematics, and more precisely: between quantum field theory and string theory, and mathematical knot theory and random matrix theory. Knot theory is a fascinating branch of mathematics. Its main focus are knots, such as those that we tie on a piece of a rope; however, it is also intimately related to the most abstract questions of contemporary mathematics. Some problems in knot theory are so complicated that they cannot be solved by means of the mathematical apparatus currently known. It turns out, however, that methods from quantum field theory and string theory surprisingly often make it possible to find astounding solutions to these problems. Analysis of such relations was the main purpose of this project. More precisely, there were four main, interrelated areas of this project, devoted to: knot homologies and superpolynomials, super-A-polynomial, three-dimensional supersymmetric N=2 theories, topological recursion and quantization. In the course of the project we have made various important discoveries in all these areas, and (in our opinion) have gone significantly beyond the state of the art.

The research team in the project was led by the Principal Investigator Prof. Piotr Sułkowski. His team included a group at the Faculty of Physics, University of Warsaw (the Host Institution), as well as renowned scientists from Caltech, Institute for Advanced Study (Princeton), University of California Davis, Institut de Physique Theorique in Saclay (France), and Instituto Superior Tecnico in Lisbon (Portugal). It is worth stressing, that establishing the new group at the University of Warsaw has raised a lot attention; it appears that this group has already become an important, internationally recognized place, where research in mathematical and high energy physics is conducted.

The results of the project are presented in 87 papers, one PhD thesis (in addition, three more PhD theses resulting from the project should be defended soon), and two MSc. theses. As it is impossible to review all our results here, let us summarize just one discovery, referred to as the “knots-quivers correspondence”. This discovery was originally presented in two papers by Piotr Kucharski, Markus Reineke, Marko Stosic and Piotr Sułkowski: “BPS states, knots and quivers” (Phys. Rev. D (2017)) and “Knots-quivers correspondence” (Adv. Theor. Math. Phys. (2019)), and its consequences are analyzed in several other papers. The correspondence in question unifies various areas of the project mentioned above, and provides a new, very interesting viewpoint on various problems in knot theory and high energy physics. From physics perspective, this correspondence states that interactions of an important class of BPS states, assigned to lagrangian branes in toric Calabi-Yau manifolds, are effectively described by a supersymmetric quantum mechanics, whose structure is encoded in an appropriate quiver. Such a quiver describes interactions between pairs of a finite number of fundamental BPS states, and based on this data it enables to determine multi-body interactions (the structure of bound states of BPS states). We determined effective theories and associated quivers explicitly for many such systems, including various infinite families. From mathematical point of view, in the context of knot theory, our results show that to a given knot one can assign a quiver, which encodes information about various invariants of this particular knot. The knots-quivers correspondence has many other deep consequences in mathematics and physics; it has been noticed and appreciated internationally, it raised interest of many renowned physicists and mathematicians, and has been presented and discussed in many prestigious conferences and workshops.

Among other outcomes, one important manifestation of the impact of our research has been organization of 13 conferences and workshops by the Principal Investigator and collaborators, devoted to the topics of the project. These meetings took place in most prestigious locations, such as BIRS (Banff, Canada), Aspen Center for Physics (Aspen, USA), American Institute for Mathematics (Palo Alto and San Jose, USA), Simons Center for Geometry and Physics (Stony Brook, New York), Kavli Institute for Theoretical Physics (University of California Santa Barbara), and the University of Warsaw. These conferences were attended by renowned experts in the areas of the project (both physicists and mathematicians), they consolidated our community, and established directions for further studies.

To sum up, the project has been very successful, regarding both major research discoveries, as well as establishing a new group, and having impact on the whole community.