4D TQFT and categorified Hall algebras

Quantum field theory is the tool used to describe and study fundamental particles. It is one of the most successful and precise scientific theories with a wide range of applications: from particle accelerators to superconductors and quantum computers. Quantum fields depend on the underlying geometry of space and time. In some cases the fields are independent of the measurment of length and duration and only depend on the underlying shape of space and time. In such cases the quantum field theory is called topological. Topological Quantum Field Theories (TQFTs) are now in the frontier of research in both physics and mathematics.

The mathematical construction of TQFTs is based on a collection of axioms, much like, for example, Euclidean geometry. This makes TQFTs a mathematical as well as physical theory. TQFTs were invented by physicists in the late 80's in an effort to make a mathematicallly rigorous construction of Quantum Field Theories. They were quickly understood to be of wide mathematical interest, being related to, among other things, Knot Theory and the theory of four-manifolds in Algebraic Topology, the theory of moduli spaces in algebraic geometry and Quantum Groups. For almost three decades TQFT has been one of the central research areas in mathematics and theoretical physics. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory. However substantial parts of the theory, in particular the generalization to the higher-dimensional case, still remain to be constructed. This is the ultimate goal of the proposed project.

The way we propose to achieve it is to create a coherent framework unifying several recent results in the approach to TQFT. One of the main tools currently at our disposal is Higher Category Theory which is a novel and very active area of pure mathematics. The fundamental idea behind the categorical approach is that formulating the problems on the most abstract level often leads to new insights and creation of radically new approaches to their solutions. For this reason the categorical language is becoming more and more widespread in Pure Mathematics and even in some areas of more applied disciplines, such as Chemistry and Computer Science. Higher Category Theory seeks to develop this language further in order to apply it to additional areas of mathematics, generally speaking of a more geometric flavor, including TQFT.

The second important object in our approach are Quantum Groups. Quantum groups were originally defined at around the same time as TQFTs. They were originally conceived as algebraic objects, that is their definition involved formulas and equations. Very soon they were found to have a number of surprising and deep connections to different areas of mathematics, in particular to TQFT. In several early influential works on the subject, Quantum Groups were shown to give rise to invariants of three-dimensional manifolds, i.e. objects of a purely geometric nature. In this way it became apparent that Quantum Groups contain a rich stock of geometric information.

To construct invariants of four-dimensional manifolds we must take this definition of Quantum Groups one level deeper, defining a similar object with an additional layer of information. Such a process is called categorification. We intend to do so using the wealth of research on Quantum Groups accumulated since their introduction and the new ideas coming from Higher Category Theory. This should then allow us to develop a systematic approach to constructing four-dimensional TQFTs as well as constructing new and concrete examples of these theories. The ultimate result would be to produce an algorithm assigning various numbers to a four-dimensional object, expressing it's deep intrinisic properties.

TQFT has contributed much to the understanding of various branches of mathematics and theoretical physics. In the course of this work we will bring together a variety of sophisticated techniques from geometric representation theory, higher algebra, categorification, algebraic geometry, and more. This promises to provide new insights both in our object of study and in these various fields.

The project will provide new methods for constructions of both TQFTs and categorifications. Although the focus is Hall algebraic constructions, the methods developed should apply in many other settings such as categorical structures arising from Hilbert spaces and quiver varieties. Thus we expect that the the results of the project will be useful in wide range of areas in pure mathematics.

This project will contrubute insights and tools required to turn the work of the physicists Witten and Kapustin on the relationship between 4d TQFTs and the geometric Langlands programme into a mathematical approach to proving the geometric Langlands conjecture which is one of the main conjectures of geometric representation theory.

4d TQFTs arise in physics as topological twists of supersymmetric Yang-Mills theories. These 4d QFTs are defined using path integrals and still don't have a rigorous definition. Giving a rigorous definition of these 4d TQFTs is a difficult and important open problem and would lead to a better understanding of the theories and their possible applications.

The PI and PDRA's will give talks and publish articles about the results of the project. They will organize a workshop in the end of the second year of the project to disseminate information and to orient the project for its final year. They will also will organize a seminar on TQFT directed at both mathematicians and physicists with the goal of bridging the gap between the two points of view on the subject.

To achieve a wider dissemination of knowledge about the field and the results of the research the PI and PDRA's will organize public lectures about TQFT and its connection to physics and our world in collaboration with Oxford colleges and mathematical societies.