Equivariant completion: topological quantum field theory and beyond

For the last 25 years the functorial approach to topological quantum field theory has been contributing deep insights in topology, algebra, representation theory, and geometry. More recently, partly motivated by the proof of the cobordism hypothesis, certain refinements of TQFT have enjoyed increased attention. The natural language here is that of higher categories; in particular, two-dimensional defect TQFT is described in terms of bicategories with adjoints. Inspired by orbifolds in quantum field theory and the foundational work of Fuchs-Runkel-Schweigert on rational conformal field theory, in 2012 Runkel and I developed the theory of equivariant completion. From a bicategory with adjoints it produces another, bigger, such bicategory in terms of Frobenius monoids and their bimodules. This new bicategory enjoys several good properties, but most importantly we developed elegant methods to construct interesting equivalences in this setting; the only ingredient necessary is a 1- morphism with invertible quantum dimension. One class of such applications are ADE orbifolds: we proved new relations (surprising to many experts) between matrix factorisation categories associated to simple singularities (and accordingly between singularity categories and derived categories of Dynkin quivers). The objective of the present proposal is twofold: reap the rewards of equivariant completion in a variety of other applications, and develop the general theory further. In this way equivariant completion appears both as a goal in itself, and as a method to obtain new results in various important fields of pure mathematics and mathematical physics. The potential of ADE orbifolds alone is already surprisingly rich in range and depth. For example, we will use them to relate Khovanov-Rozansky link invariants of A- and D-type (and possibly construct new E-type invariants); ADE orbifolds should be extended to the level of Calabi-Yau completions, to become a relation between derived categories of Ginzburg algebras; they have an important role to play in relating stability conditions from quadratic differentials on Riemann surfaces in the context of 2d/4d correspondence. In addition to these and several further applications described in the present proposal, we shall also pursue extensions of the abstract theory of equivariant completion which in turn will increase the success of more applications. Two particularly important directions here are a formulation in terms of a universal property (which will also help uncover further general structures and establish coherence results), as well as developing a three-dimensional variant of equivariant completion as an operation on tricategories (which by its physical origin `must` be possible).

The project was placed at the border region between theoretical physics and mathematics. Accordingly, its main goals were twofold: (1) to better understand fundamental aspects of quantum physics, by rigorously describing a large class of models, so-called topological quantum field theories; (2) to systematically and conceptually differentiate between low-dimensional geometric object (that also feature as simplified spacetime models) in a purely algebraic language. One specific main result is the construction and application of a theory of "generalised orbifolds": it produces new topological quantum field theories out of algebraic data, hence giving a new means to map the space of all topological quantum field theories. State sum models and the process of gauging the action of symmetry groups turn out to be special cases of the generalised orbifold construction, revealing the unifying nature of the latter. These methods were in particular applied to a class of three-dimensional models (of Reshetikhin-Turaev type) which are intimately linked to the theory of 3-manifolds, and to theoretical models for topological quantum computers.