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

A Lie 2-algebra is a ‘categorified ’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.

I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that certain 2-categories of principal 2-bundles are equivalent to certain 2-categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2-bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2-category of 2-bundles over a given 2-space under a given 2-group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2-transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2-space is the 2-space corresponding to a given space and the 2-group is the automorphism 2-group of a given group, then this 2-category is equivalent to the 2-category of gerbes over that space under that group (being described by the same cohomological data).

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → P U(H) to higher categorical central extensions, like the String-extension BU(1) → String(G) → G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures ” whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class.

Abstract. What remains of a geometrical notion like that of a principal bundle when the base space is not a manifold but a coarse graining of it, like the poset formed by a base for the topology ordered under inclusion? Motivated by the search for a geometrical framework for developing gauge theories in algebraic quantum field theory, we give, in the present paper, a first answer to this question. The notions of transition function, connection form and curvature form find a nice description in terms of cohomology, in general non-Abelian, of a poset with values in a group G. Interpreting a 1–cocycle as a principal bundle, a connection turns out to be a 1–cochain associated in a suitable way with this 1–cocycle; the curvature of a connection turns out to be its 2–coboundary. We show the existence of nonflat connections, and relate flat connections to homomorphisms of the fundamental group of the poset into G. We discuss holonomy and prove an analogue of the Ambrose-Singer theorem. 1.

For locally homotopy trivial fibrations, one can define transition functions gαβ: Uα ∩ Uβ → H = H(F) where H is the monoid of homotopy equivalences of F to itself but, instead of the cocycle condition, one obtains only that gαβgβγ is homotopic to gαγ as a map of Uα ∩ Uβ ∩ Uγ into H. Moreover, on multiple intersections, higher homotopies arise and are relevant to classifying the fibration. The full theory was worked out by the first author in his 1965 Notre Dame thesis [17]. Here we present it using language that has been developed in the interim. We also show how this points a direction ‘on beyond gerbes’.

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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

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity. 1

Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2-functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the Dijkgraaf-Witten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a higher-categorical version of Vect, denoted 2Vect, a bicategory of 2-vector spaces. Along the way, we prove several results showing how to construct 2-vector spaces of Vect-valued presheaves on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of “pullback and pushforward ” of presheaves gives both the morphisms and 2-morphisms in 2Vect for the extended TQFT, and that these