We generalize gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are replaced by linearized analogs. Starting from physical considerations, we derive an axiomatic definition of Hopf algebra gauge theory, including locality conditions under which the theory for a general ribbon graph can be assembled from local data in the neighborhood of each vertex. For a vertex neighborhood with [Formula: see text] incoming edge ends, the algebra of non-commutative ‘functions’ of connections is dual to a two-sided twist deformation of the [Formula: see text]-fold tensor power of the gauge Hopf algebra. We show these algebras assemble to give an algebra of functions and gauge-invariant subalgebra of ‘observables’ that coincide with those obtained in the combinatorial quantization of Chern–Simons theory, thus providing an axiomatic derivation of the latter. We then discuss holonomy in a Hopf algebra gauge theory and show that for semisimple Hopf algebras this gives, for each path in the embedded graph, a map from connections into the gauge Hopf algebra, depending functorially on the path. Curvatures — holonomies around the faces canonically associated to the ribbon graph — then correspond to central elements of the algebra of observables, and define a set of commuting projectors onto the subalgebra of observables on flat connections. The algebras of observables for all connections or for flat connections are topological invariants, depending only on the topology, respectively, of the punctured or closed surface canonically obtained by gluing annuli or discs along edges of the ribbon graph.

The ‘observer space’ of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose taking observer space as fundamental in general relativity, and prove integrability conditions under which spacetime can be reconstructed as a quotient of observer space. Additional field equations on observer space then descend to Einstein’s equations on the reconstructed spacetime. We also consider the case where no such reconstruction is possible, and spacetime becomes an observer-dependent, relative concept. Finally, we discuss applications of observer space, including a geometric link between covariant and canonical approaches to gravity.

We show that general relativity can be viewed as a higher gauge theory involving a categorical group, or 2-group, called the teleparallel 2-group. On any semi-Riemannian manifold M, we first construct a principal 2-bundle with the Poincaré 2-group as its structure 2-group. Any flat metric-preserving connection on M gives a flat 2-connection on this 2-bundle, and the key ingredient of this 2-connection is the torsion. Conversely, every flat strict 2-connection on this 2-bundle arises in this way if M is simply connected and has vanishing 2nd deRham cohomology. Extending from the Poincaré 2-group to the teleparallel 2-group, a 2-connection includes an additional piece: a coframe field. Taking advantage of the teleparallel reformulation of general relativity, which uses a coframe field, a flat connection and its torsion, this lets us rewrite general relativity as a theory with a 2-connection for the teleparallel 2-group as its only field.

We reinterpret special relativity, or more precisely its de Sitter deformation, in terms of 3d conformal geometry, as opposed to (3+1)d spacetime geometry. An inertial observer, usually described by a geodesic in spacetime, becomes instead a choice of ways to reverse the conformal compactification of a Euclidean vector space up to scale. The observer's "current time," usually given by a point along the geodesic, corresponds to the choice of scale in the decompactification. We also show how arbitrary conformal 3-geometries give rise to "observer space geometries," as defined in recent work, from which spacetime can be reconstructed under certain integrability conditions. We conjecture a relationship between this kind of "holographic relativity" and the "shape dynamics" proposal of Barbour and collaborators, in which conformal space takes the place of spacetime in general relativity. We also briefly survey related pictures of observer space, including the AdS analog and a representation related to twistor theory.

We study the dynamics of gauge theory and general relativity using fields of local observers, thus maintaining local Lorentz symmetry despite a space/time splitting of fields. We start with Yang-Mills theory, where observer fields are defined as normalized future-timelike vector fields. We then define observers without a fixed geometry, and find these play two related roles in general relativity: splitting fields into spatial and temporal parts, and 'breaking' gauge symmetry, effectively reducing the spacetime SO(n, 1) connection to an observer-dependent spatial SO(n) connection. In both gauge theory and gravity, the observer field reduces the action to canonical form, without using gauge fixing. In the 4d gravity case, the result is a manifestly Lorentz covariant counterpart of the Ashtekar-Barbero formulation. We also explain how this leads geometrically to a picture of general relativity in terms of 'observer space' rather than spacetime – a setting where both spacetime symmetry and the dynamical description are simultaneously available.

The `observer space' of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose taking observer space as fundamental in general relativity, and prove integrability conditions under which spacetime can be reconstructed as a quotient of observer space. Additional field equations on observer space then descend to Einstein's equations on the reconstructed spacetime. We also consider the case where no such reconstruction is possible, and spacetime becomes an observer-dependent, relative concept. Finally, we discuss applications of observer space, including a geometric link between covariant and canonical approaches to gravity.

In Ashtekar's Hamiltonian formulation of general relativity, and in loop quantum gravity, Lorentz covariance is a subtle issue that has been strongly debated. Maintaining manifest Lorentz covariance seems to require introducing either complex-valued fields, presenting a significant obstacle to quantization, or additional (usually second class) constraints whose solution renders the resulting phase space variables harder to interpret in a spacetime picture. After reviewing the sources of difficulty, we present a Lorentz covariant, real formulation in which second class constraints never arise. Rather than a foliation of spacetime, we use a gauge field y, interpreted as a field of observers, to break the SO(3, 1) symmetry down to a subgroup SO(3)(y). This symmetry breaking plays a role analogous to that in MacDowell-Mansouri gravity, which is based on Cartan geometry, leading us to a picture of gravity as "Cartan geometrodynamics." We study both Lorentz gauge transformations and transformations of the observer field to show that the apparent breaking of SO(3, 1) to SO(3) is not in conflict with Lorentz covariance.

In gravity, breaking symmetry from a group G to a group H plays the role of describing geometry in relation to the geometry of the homogeneous space G/H. The deep reason for this is Cartan's 'method of equivalence,' giving, in particular, an exact correspondence between metrics and Cartan connections. I argue that broken symmetry is thus implicit in any gravity theory, for purely geometric reasons. As an application, I explain how this kind of thinking gives a new approach to Hamiltonian gravity in which an observer field spontaneously breaks Lorentz symmetry and gives a Cartan connection on space.

A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners - features not seen in ordinary group representation theory. We study irreducible and indecomposable representations and intertwiners. We also study "irretractable" representations - another feature not seen in ordinary group representation theory. Finally, we argue that measurable categories equipped with some extra structure deserve to be considered "separable 2-Hilbert spaces", and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.

Hamiltonian gravity, relying on arbitrary choices of ‘space,’ can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between ‘spatial’ and ‘temporal’ variables. The key is viewing dynamical fields from the perspective of a field of observers—a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the ‘space of observers’ is fundamental, and spacetime geometry itself may be observer-dependent.

The geometric content of the MacDowell–Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell–Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous ‘model spacetime’, including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti-de Sitter or other models. A ‘Cartan connection’ gives a prescription for parallel transport from one ‘tangent model spacetime’ to another, along any path, giving a natural interpretation of the MacDowell–Mansouri connection as ‘rolling’ the model spacetime along physical spacetime. I explain Cartan geometry, and ‘Cartan gauge theory’, in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell–Mansouri gravity, as well as its more recent reformulation in terms of BF theory, in the context of Cartan connections.

Just as 3d state sum models, including 3d quantum gravity, can be built using categories of group representations, "2-categories of 2-group representations" may provide interesting state sum models for 4d quantum topology, if not quantum gravity. Here we focus on the "Euclidean 2-group", built from the rotation group SO(4) and its action on the group of translations of 4d Euclidean space. We explain its infinite-dimensional unitary representations, and construct a model based on the resulting representation 2-category. This model, with clear geometric content and explicit "metric data" on triangulation edges, shows up naturally in an attempt to write the amplitudes of ordinary quantum field theory in a background independent way.

We study topologically massive (2+1)-dimensional gravity with a negative cosmological constant. The masses of the linearized curvature excitations about AdS3 backgrounds are not only shifted from their flat background values but also, more surprisingly, split according to chirality. For all finite values of the topological mass, we find a single bulk degree of freedom with positive energy, and exhibit a complete set of normalizable, finite-energy wave packet solutions. This model can also be written as a sum of two higher-derivative Chern–Simons theories, weighted by the central charges of the boundary conformal field theory. At two particular 'critical' values of the couplings, one of these central charges vanishes, and linearized topologically massive gravity becomes equivalent to topologically massive electromagnetism; however, the physics of the bulk wave packets remains unaltered here.

Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3, 1) and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'.

We report a measurement of the angular distributions of Drell-Yan dimuons produced using an 800 GeV/c proton beam on a deuterium target. The muon angular distributions in the dilepton rest frame have been measured over the kinematic range 4.5<m{mu mu}<15 GeV/c{2}, 0<p{T}<4 GeV/c, and 0<x{F}<0.8. No significant cos2phi dependence is found in these proton-induced Drell-Yan data, in contrast with the situation for pion-induced Drell-Yan data. The data are compared with expectations from models which attribute the cos2phi distribution to a QCD vacuum effect or to the presence of the transverse-momentum-dependent Boer-Mulders structure function h{1}{perpendicular}. Constraints on the magnitude of the sea-quark h{1}{perpendicular} structure functions are obtained.

After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that loop-like defects in 4d BF theory obey exotic statistics governed by the ‘loop braid group’. This group has a set of generators that switch two loops just as one would normally switch point particles, but also a set of generators that switch two loops by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can give a presentation of the whole loop braid group, which turns out to be isomorphic to the ‘braid permutation group’ of Fenn, Rimányi and Rourke. In the context 4d BF theory this group naturally acts on the moduli space of flat Gbundles on the complement of a collection of unlinked unknotted circles in R. When G is unimodular, this gives a unitary representation of the loop braid group. We also discuss ‘quandle field theory’, in which the gauge group G is replaced by a quandle.

We report a high statistics measurement of Upsilon production with an 800 GeV/c proton beam on hydrogen and deuterium targets. The dominance of the gluon-gluon fusion process for Upsilon production at this energy implies that the cross section ratio, sigma(p+d-->Upsilon)/2sigma(p+p-->Upsilon), is sensitive to the gluon content in the neutron relative to that in the proton. Over the kinematic region 0<x(F)<0.6, this ratio is found to be consistent with unity, in striking contrast to the behavior of the Drell-Yan cross section ratio sigma(p+d)(DY)/2sigma(p+p)(DY). This result shows that the gluon distributions in the proton and neutron are very similar. The Upsilon production cross sections are also compared with the p+d and p+Cu cross sections from earlier measurements.