## Mid-Term Report Summary - FIELDS-KNOTS (Quantum fields and knot homologies)

The project "Quantum Fields and Knot Homologies" is 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, and at the same time one of the most mysterious areas in mathematics. On one hand, its main focus are knots, such as those that can be tied on a piece of a rope and known to us from our everyday life, and on the other hand, it is intimately related to the most abstract questions of contemporary mathematics. Some problems in knot theory – for example characterization and classification of all knots, as well as those with more practical dimension – are so complicated that they cannot be solved by means of the mathematical apparatus currently known to us. It turns out, however, that methods from quantum physics, related to quantum field theory and string theory, surprisingly often make it possible to find astounding solutions to these problems, even though their pure mathematical proof is for the time being beyond our capacities. Analysis of such relations is the main purpose of this project. In more technical terms, there are four main, interrelated areas of this project, devoted respectively to: knot homologies and superpolynomials, super-A-polynomial, three-dimensional supersymmetric N=2 theories, topological recursion and quantization. In the first half of the project we have worked on all of those topics and made several important discoveries in each area.

The research team in the project is led by the Principal Investigator Dr. Piotr Sułkowski. This team includes a group – involving postdoctoral researchers and PhD students – 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).

In the first half of the project, our research results have been described in 34 papers. As it would not be possible to review all our results in this limited space, let us summarize just one our discovery, presented in the paper “Knots, BPS states, and algebraic curves” by Stavros Garoufalidis, Piotr Kucharski, and Piotr Sułkowski (published in a prestigious Commun. Math. Phys. journal). In this paper we analyzed certain physical objects, known as BPS states, which arise in string theory, and in an appropriate setting they also characterize knots. One important problem in physics is to count such BPS states, and if some knot can be associated to a given set of BPS states, their multiplicities provide an interesting knot invariant. We managed, for the first time, to provide general and explicit formulas for various classes of such BPS multiplicities, associated to various knots. Our formulas have an interesting structure of ratios of two quantities. However, it is a prediction from physics that the resulting numbers count BPS states, so they must be integer. Therefore numerators in our formulas must be divisible by denominators. It turns out that such divisibility properties are very non-obvious, and they provide very interesting predictions in number theory, which is yet another important branch of mathematics. In conclusion, our results in a fascinating way connect ideas from string theory, knot theory, and lead to deep statements in number theory. These connections also illustrate, that our project and methodology are truly interdisciplinary.

These and other results of the project have been obtained in fruitful collaborations involving the core of the research team at the University of Warsaw, as well as collaborators abroad. It should also be stressed, that establishing our new group at the University of Warsaw has raised a lot attention, and it appears that this group has already become an important place not only in (Eastern) Europe, but also in the world, where research in these areas of mathematical and high energy physics is conducted. Among other outcomes, one important manifestation of the impact of our research have been the organization of 6 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), American Institute for Mathematics (Palo Alto and San Jose, USA), Simons Center for Geometry and Physics (Stony Brook, New York), as well as at the host institution – University of Warsaw. These conferences have been attended by renowned experts in the areas of the project (both physicists and mathematicians), consolidated our community, and established directions for further studies.

To sum up, the first half of 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. We are convinced that the second half of the project will be successful to the same extent.

The research team in the project is led by the Principal Investigator Dr. Piotr Sułkowski. This team includes a group – involving postdoctoral researchers and PhD students – 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).

In the first half of the project, our research results have been described in 34 papers. As it would not be possible to review all our results in this limited space, let us summarize just one our discovery, presented in the paper “Knots, BPS states, and algebraic curves” by Stavros Garoufalidis, Piotr Kucharski, and Piotr Sułkowski (published in a prestigious Commun. Math. Phys. journal). In this paper we analyzed certain physical objects, known as BPS states, which arise in string theory, and in an appropriate setting they also characterize knots. One important problem in physics is to count such BPS states, and if some knot can be associated to a given set of BPS states, their multiplicities provide an interesting knot invariant. We managed, for the first time, to provide general and explicit formulas for various classes of such BPS multiplicities, associated to various knots. Our formulas have an interesting structure of ratios of two quantities. However, it is a prediction from physics that the resulting numbers count BPS states, so they must be integer. Therefore numerators in our formulas must be divisible by denominators. It turns out that such divisibility properties are very non-obvious, and they provide very interesting predictions in number theory, which is yet another important branch of mathematics. In conclusion, our results in a fascinating way connect ideas from string theory, knot theory, and lead to deep statements in number theory. These connections also illustrate, that our project and methodology are truly interdisciplinary.

These and other results of the project have been obtained in fruitful collaborations involving the core of the research team at the University of Warsaw, as well as collaborators abroad. It should also be stressed, that establishing our new group at the University of Warsaw has raised a lot attention, and it appears that this group has already become an important place not only in (Eastern) Europe, but also in the world, where research in these areas of mathematical and high energy physics is conducted. Among other outcomes, one important manifestation of the impact of our research have been the organization of 6 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), American Institute for Mathematics (Palo Alto and San Jose, USA), Simons Center for Geometry and Physics (Stony Brook, New York), as well as at the host institution – University of Warsaw. These conferences have been attended by renowned experts in the areas of the project (both physicists and mathematicians), consolidated our community, and established directions for further studies.

To sum up, the first half of 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. We are convinced that the second half of the project will be successful to the same extent.