Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to substitute categories of functors for categories of smooth fibre bundles with connection. We indicate that this concept can be generalized to connections in categorified bundles, and how this generalization improves the understanding of higher dimensional parallel transport. Table of Contents

We investigate p-form electromagnetism—with the Maxwell and Kalb-Ramond fields as lowest-order cases—on discrete spacetimes, including the regular lattices commonly used in lattice gauge theory, but also more general examples. After constructing a maximally general model of discrete spacetime suitable for our purpose—a chain complex equipped with an inner product on (p + 1)-cochains—we study both the classical and quantum versions of the theory, with either R or U(1) as gauge group. We find results—such as a ‘p-form Bohm–Aharonov effect’—that depend in interesting ways on the cohomology of spacetime. We quantize the theory via the Euclidean path integral formalism, where the natural kernels in the U(1) theory are not Gaussians but theta functions. As a special case of the general theory, we show p-form electromagnetism in p + 1 dimensions has an exact solution which reduces when p = 1 to the abelian case of 2d Yang-Mills theory as studied by Migdal and Witten. Our main result describes p-form electromagnetism as a ‘chain field theory’—a theory analogous to a topological quantum field theory, but with chain complexes replacing manifolds. This makes precise a notion of time evolution in the context of discrete spacetimes of arbitrary topology.

Since Wilson’s work on lattice gauge theory in the 1970s, discrete versions of field theories have played a vital role in fundamental physics. But there is recent interest in certain higher dimensional analogues of gauge theory, such as p-form electromagnetism, including the Kalb-Ramond field in string theory, and its nonabelian generalizations. It is desirable to discretize such ‘higher gauge theories ’ in a way analogous to lattice gauge theory, but with the fundamental geometric structures in the discretization boosted in dimension. As a step toward studying discrete versions of more general higher gauge theories, we consider the case of p-form electromagnetism. We show that discrete p-form electromagnetism admits a simple algebraic description in terms of chain complexes of abelian groups. Moreover, the model allows discrete spacetimes with quite general geometry, in contrast to the regular cubical lattices usually associated with lattice gauge theory. After constructing a suitable model of discrete spacetime for p-form electromagnetism, we quantize the theory using the Euclidean path integral formalism. The main result is a description of p-form electromagnetism as a ‘chain field theory’ — a theory analogous to topological quantum field theory, but with chain complexes replacing

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module (H → D) there is a simplicial group NC(H→D), the nerve of the 1-category defined by the crossed module and its geometric realization |NC(H→D)|. Equivalence classes of principal bundles with structure group |NC(H→D) | are shown to be one-to-one with stable equivalence classes of what we call crossed module bundle gerbes. We can also associate to a crossed module a 2-category ˜ C(H→D). Then there are two equivalent ways how to view classifying spaces of NC(H→D)-bundles and hence of |NC(H→D)|-bundles and crossed module bundle gerbes. We can either apply the W-construction to NC(H→D) or take the nerve of the 2-category ˜ C(H→D). We discuss the string group and string structures from this point of view. Also a simplicial principal bundle can be equipped with a simplicial connection and a B-field. It is shown how in the case of a simplicial principal NC(H→D)-bundle these simplicial objects give the bundle gerbe connection and the bundle gerbe B-field.

Wheeler’s dream of “matter without matter ” is realized in 3d gravity, where particles show up naturally as punctures in space, or “topological defects”. My Ph.D. thesis research has been motivated by the desire to extend this idea to 4 spacetime dimensions. The starting point for this line of research lies in two key facts: 1. General relativity in 3 dimensions has a generalization known as BF theory, which retains its topological character in any dimension. One thus expects matter to show up in 4d BF theory in much the same way as in 3d gravity. 2. General relativity in 4 dimensions may be viewed as a perturbation around 4d BF theory, thanks to recent work of Freidel and Starodubtsev, based on the MacDowell–Mansouri formalism. The ultimate goal is to combine these ideas to see matter emerging in a purely geometric way in the MacDowell–Mansouri approach to gravity. In pursuit of this goal, I have studied matter in BF theory [7] as well as the geometric foundations of MacDowell–Mansouri gravity [24]. Earlier, my research focussed on lattice approaches to ‘higher gauge theory’, a generalization of gauge theory based on the mathematics of higher categories [10]. The abelian case of higher gauge theory is called p-form electromagnetism, since it involves replacing the gauge connection by a p-form. I made a

As a natural and canonical extension of Kumjian’s Fell bundles over groupoids [17], we give a definition for a double Fell bundle (a double category) over a double groupoid. We show that finite dimensional double category Fell line bundles tensored with their dual with S o-reality satisfy the finite real spectral triples axioms but not necessarily orientability. This means that these product bundles with noncommutative algebras can be regarded as noncommutative compact manifolds more general than real spectral triples as they are not necessarily orientable. By construction, they unify the noncommutative geometry axioms and hence provide an algebraic enveloping structure for finite spectral triples to give the Dirac operator D new algebraic and geometric structures that are otherwise missing in the transition from Fredholm operator to Dirac operator. The Dirac operator in physical applications as a result becomes less ad hoc. The new noncommutative space we present is a complex line bundle over a double groupoid. Its algebra is not directly analogous to the algebra of a spectral triple. As a result of its interpretation as a 2-morphism in the double category, the new structures include that the space of Ds forms part of the C ∗-algebra of the double Fell bundle inheriting a hermitian structure and as a 2-morphism in the double functor from double groupoid to double Fell bundle, has the role of 2-transport or 2-connection. This study automatically sets spectral triples in the context of higher category theory providing a possible arena for quantum gravity.