In the first stages of the project, we have developed a rich machinery for dealing with these topological field theories in the non-modular setting, i.e. the completely general setting of rigid braided tensor categories. These constructions build on cutting-edge techniques of several teams in algebraic topology and homotopy theory -- Lurie, Ayala-Francis-Tannaka, Haugseng, Scheimbauer, Freed-Teleman, and many others -- to define the theory. They furthermore apply a powerful set of tools from category theory and geometric representation theory to transform the abstract formalism given to us by the homotopy theorists into very concrete sets of equations, which in turn capture a number of important quantum symmetries studied in the quantum groups literature for decades. This immediately provides a conduit between the two a priori quite distant fields, which we may exploit to prove many new results. Our current state of the art is as follows: we have constructed and thoroughly analyzed the resulting algebraic structures coming from surfaces -- that is, spaces in dimension 2. We have now turned our attention to a rigorous construction of the long-predicted invariants in dimension three. These have been foreshadowed in works of Walker, Freed-Teleman, Lurie, Crane-Yetter-Kauffman, and many others, but only in recent years, and only with the tools we have been building can these constructions be brought to their full strength.
In the final stages of the product we have obtained a number of important applications of our earlier foundational work. We have proved a long-standing and well-known conjecture of the physicist Edward Witten, stating that certain very natural Hilbert spaces of quantum states on a 3-dimensional space time, comprised of so-called Wilson loop expectations, is actually a finite-dimensional vector space. In related work we proved that the boundary operators for this theory -- coming from the value of the theory on surfaces -- have a strong finiteness and invertibility property relative to their classical degeneration. This established a refined version of the Unicity Conjecture of Bonahon and Wong. Finally, we established a fundamental conjecture of Freed, Teleman and Walker, which describes the general algebraic structure one needs to capture four-dimensional symmetries -- anomalies, in physical parlance -- acting on 3-dimensional topological field theories. In establishing their conjecture, we in fact found new and important examples extending the original formulation. Finally, we have taken the first step towards understanding the algebraic structures involved in spaces with defects, by computing categories of surface operators in a 4-dimensional theory in the framework of cluster algebras and canonical quantization.
