Coupling a QFT to a TQFT and Duality

Dijkgraaf-Witten (DW) theories are of recent interest to the condensed matter community, in part because they represent topological phases of matter, but also because they characterize the response theory of certain symmetry protected topological (SPT) phases. However, as yet there has not been a comprehensive treatment of the spectra of these models in the field theoretic setting -- the goal of this work is to fill the gap in the literature, at least for a selection of DW mo...

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...

The goal of this paper is to introduce some of the major ideas behind extended topological quantum field theories with an emphasis on explicit examples and calculations. The statement of the Cobordism Hypothesis is explained and immediately used to classify framed and oriented extended two dimensional topological quantum field theories. The passage from framed theories to oriented theories is equivalent to giving homotopy fixed points of an $SO(n)$ action on the space of fiel...

The physics of quantum gravity is discussed within the framework of topological quantum field theory. Some of the principles are illustrated with examples taken from theories in which space-time is three dimensional.

This thesis discusses various aspects of duality in quantum field theory and string theory. In the first part we consider duality in topological quantum field theories, concentrating on the Donaldson and Seiberg-Witten theories as (dual) approaches to the study of four-manifolds. Lower-dimensional variants of these theories are also discussed. In the second part of the thesis we discuss duality in 2D sigma models by studying the interplay between renormalization group flows -...

This paper develops a detailed lattice-continuum correspondence for all common examples of Abelian gauge theories, with and without matter. These rules for extracting a continuum theory out of a lattice one represent an elementary way to rigorously define continuum gauge theories. The focus is on (2+1)D but the techniques developed here work in all dimensions. The first half of this paper is devoted to pure Maxwell theory. It is precisely shown how continuum Maxwell theory ...

A brief introduction to Topological Quantum Field Theory as well as a description of recent progress made in the field is presented. I concentrate mainly on the connection between Chern-Simons gauge theory and Vassiliev invariants, and Donaldson theory and its generalizations and Seiberg-Witten invariants. Emphasis is made on the usefulness of these relations to obtain explicit expressions for topological invariants, and on the universal structure underlying both systems.

These third-year lecture notes are designed for a 1-semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second-year subjects: multivariable calculus, introduction to quantum mechanics and basic electromagnetism. Keywords: quantum mechanics/field theory, path integral, Hodge decomposition, Chern-Simons and Yang-Mills gauge theories, conformal field theory

These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June. I review some recent work on duality in four dimensional Maxwell theory on arbitrary four manifolds, as well as a new set of topological invariants known as the Seiberg-Witten invariants. Much of the necessary background material is given, including a crash course in topological field theory, cohomology of manifolds, topological gauge theory and the rudiments of four manifold...

We introduce a framework for internal topological symmetries in quantum field theory, including "noninvertible symmetries" and "categorical symmetries". This leads to a calculus of topological defects which takes full advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called "gauging" and "c...