Coupling a QFT to a TQFT and Duality

Following my plenary lecture on ICMP2000 I review my results concerning two closely related topics: topological quantum field theories and the problem of quantization of gauge theories. I start with old results (first examples of topological quantum field theories were constructed in my papers in late seventies) and I come to some new results, that were not published yet.

In this note, we discuss electric-magnetic duality between a pair of 4d topological field theories (TQFTs) by considering their compactifications to 2 dimensions. These TQFTs control the long-distance behavior of loop and surface operators in 4d gauge theories with gapped phases. These were recently used in work by S. Gukov and A. Kapustin in detecting phases not distinguishable by the Wilson-'t Hooft criterion and by A. Kapustin and the author to construct discrete theta-ang...

We present a summary of the progress made in the last few years on topological quantum field theory in four dimensions. In particular, we describe the role played by duality in the developments which led to the Seiberg-Witten invariants and their relation to the Donaldson invariants. In addition, we analyze the fruitful framework that this connection has originated. This analysis involves the study of topological quantum field theories which contain twisted N=2 supersymmetric...

It is argued that quantum gravity has an interpretation as a topological field theory provided a certain constraint from the path intergral measure is respected. The constraint forces us to couple gauge and matter fields to gravity for space - time dimensions different from 3. We then discuss possible models which may be relevant to our universe.

We study topological field theory describing gapped phases of gauge theories where the gauge symmetry is partially Higgsed and partially confined. The TQFT can be formulated both in the continuum and on the lattice and generalizes Dijkgraaf-Witten theory by replacing a finite group by a finite 2-group. The basic field in this TQFT is a 2-connection on a principal 2-bundle. We classify topological actions for such theories as well as loop and surface observables. When the topo...

The relationship between the sources of physical fields and the fields themselves is investigated with regard to the coupling of topological information between them. A class of field theories that we call topological field theories is defined such that both the field and its source represent de Rham cocycles in varying dimensions over complementary subspaces and the coupling of one to the other is by way of an isomorphism of the those cohomology spaces, which we refer to as ...

In these lectures we present a general introduction to topological quantum field theories. These theories are discussed in the framework of the Mathai-Quillen formalism and in the context of twisted N=2 supersymmetric theories. We discuss in detail the recent developments in Donaldson-Witten theory obtained from the application of results based on duality for N=2 supersymmetric Yang-Mills theories. This involves a description of the computation of Donaldson invariants in term...

In many gauge theories, the existence of particles in every representation of the gauge group (also known as completeness of the spectrum) is equivalent to the absence of one-form global symmetries. However, this relation does not hold, for example, in the gauge theory of non-abelian finite groups. We refine this statement by considering topological operators that are not necessarily associated with any global symmetry. For discrete gauge theory in three spacetime dimensions,...

We summarize basic features of quantum field theories with discrete symmetry $\mathbb{Q}/\mathbb{Z}$ (possibly higher form, global or gauged). The classification of representations and anomalies is quite rich and involves the ring of profinite integers. As a main example we consider in detail 3d topological Dijkgraaf-Witten $\mathbb{Q}/\mathbb{Z}$ gauge theories. We also briefly discuss relevance for some previously considered physical systems. In particular we comment on a r...

The composite particle duality extends the notions of both flux attachment and statistical transmutation in spacetime dimensions beyond 2+1$\text{D}$. It constitutes an exact correspondence that can be understood either as a theoretical framework or as a dynamical physical mechanism. The immediate implication of the duality is that an interacting quantum system in arbitrary dimensions can experience a modification of its statistical properties if coupled to a certain gauge fi...