Topological Gauge Theory, Cartan Geometry, and Gravity by Derek Keith Wise B.S. Physics (Abilene Christian University, Texas) 1998 B.A. Mathematics (Abilene Christian University, Texas) 1998 M.S. Mathematics (University of California, Riverside) 2004 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, RIVERSIDE Committee in charge: Dr. John C. Baez, Chairperson Dr. Michel L. Lapidus Dr. Stefano Vidussi June 2007 Topological Gauge Theory, Cartan Geometry, and Gravity Copyright 2007 by Derek Keith Wise The dissertation of Derek Keith Wise is approved: Chair Date Date Date University of California, Riverside June 2007 iv Acknowledgments First, I thank John Baez for his exceptional guidance, mentorship, and patience. I've learned much from John, and in many ways. It is hard to imagine having a better advisor, or getting along better with one's advisor. I also wish to thank Michel Lapidus and Stefano Vidussi for serving on my thesis committee. During this work, I was involved in two collaborations of particular relevance to this work. John Baez and Alissa Crans collaborated with me on the exotic statistics project [12] which forms an important part of this thesis. John and I were also involved in a project with Aristide Baratin, Laurent Freidel, and Jeffrey Morton [10]. I am sincerely grateful to each of these collaborators. I am indebted to Xiao{Song Lin, first because his work on the loop braid group was the impetus for our work on exotic statistics. I also owe a him a great debt of gratitude for all that I learned from him as a student, and for his inspiring example. These are unfortunately debts that can no longer be repaid. I miss you, Xiao{Song. In the course my graduate studies, I also benefitted greatly from numerous dis- cussions with many others, especially Jeff Morton, Jim Dolan, Josh Willis, Sam Nelson, Alissa Crans, Miguel Carrion Alvarez,´ Toby Bartels, Aristide Baratin, Prasad Senesi, Blake Winter, Mike Stay, Alex Hoffnung, and Danny Stevenson. v i Abstract Topological Gauge Theory, Cartan Geometry, and Gravity by Derek Keith Wise Doctor of Philosophy in Mathematics University of California, Riverside Dr. John C. Baez, Chair We investigate the geometry of general relativity, and of related topological gauge theories, using Cartan geometry. Cartan geometry|an ‘infinitesimal’ version of Klein's Erlanger Programm|allows us to view physical spacetime as tangentially approximated by a homogeneous `model spacetime', such as de Sitter or anti de Sitter. This idea leads to a common geometric foundation for 3d Chern{Simons gravity, as studied by Witten, and 4d MacDowell{Mansouri gravity. We describe certain topological gauge theories, including BF theory|a natural extension of 3d gravity to arbitrary dimensions|as `Cartan gauge theories' in which the gauge field is replace by a `Cartan connection' modeled on some Klein geometry G=H. Cartan-type BF theory has solutions that say spacetime is locally isometric to the G=H itself; in this case Cartan geometry reduces to the theory of `geometric structures'. This leads to generalizations of 3d gravity based on other 3d Klein geometries, including those in Thurston's classification of 3d Riemannian model geometries. For BF theory in n-dimensional spacetime, we also describe codimension-2 `branes' as topological defects. These branes|particles in 3d spacetime, strings in 4d, and so on| are shown to be classified by conjugacy classes in the gauge group G of the theory. They also exhibit `exotic statistics' which are neither Bose{Einstein nor Fermi{Dirac, but are governed by representations of generalizations of the braid group known as `motion groups'. These representations come from a natural action of the motion group on the moduli space of flat G-bundles on space. We study this in particular detail in the case of strings in 4d BF ii theory, where Lin has called the motion group the `loop braid group', LBn. This makes 4d BF theory with strings into a `loop braided quantum field theory'. We also use ideas from `higher gauge theory' to study particles as topological defects in 4d BF theory, and find they are classified by adjoint orbits in the Lie algebra of the gauge group. Including both particles and strings in 4d BF theory leads to interesting effects, such as exotic particle/string statistics and a duality between Bohm{Aharonov effects for particles and strings. iii Contents 1 Introduction 1 I Geometric Structures and Topological Gauge Theory 18 2 Homogeneous spacetimes and Klein geometry 19 2.1 Klein geometry . 19 2.2 Metric Klein geometry . 22 2.3 Survey of homogeneous spacetimes . 23 2.3.1 3d spacetimes . 24 2.3.2 Contractions and Wick rotations . 28 2.3.3 4d spacetimes . 38 3 Geometric structures and flat bundles 40 3.1 Geometric manifolds . 41 3.2 The flat bundle associated to a geometric structure . 42 3.3 Moduli space of geometric structures . 47 3.4 Model geometries and exotic spacetimes . 48 3.4.1 3d model geometries . 51 3.4.2 Exotic Lorentzian spacetimes . 57 4 Topological gauge theories 59 4.1 From general relativity to BF theory . 59 4.2 BF theory and geometric structures . 64 4.3 Generalized 3d gravities . 68 4.4 Chern-Simons theory and 3d gravity . 72 4.5 3d Galilean general relativity and the Newtonian limit of 3d gravity . 75 4.6 Transforms between Chern{Simons theories . 77 II Particles and Strings 78 5 Statistics and Motion Groups 79 5.1 The Hamiltonian picture of BF theory . 79 iv 5.2 Canonical quantization and motion group statistics . 81 6 Point particles in 3d BF theory 85 6.1 Quandle field theory . 88 6.2 ISO(2; 1) Chern{Simons gravity and spin . 94 6.3 SO(3; 1) Chern{Simons gravity . 95 6.4 SO(2; 2) Chern{Simons gravity . 99 6.5 Particle types and contractions . 101 7 Strings in 4d BF Theory 103 7.1 The loop braid group . 103 7.2 Loop braid statistics and representations . 112 8 Higher gauge theory and particles in 4d BF theory 117 8.1 The idea of a p-connection . 117 8.2 From groups to 2-groups . 119 8.2.1 2-groups as 2-categories . 119 8.2.2 Constructing 2-groups . 120 8.3 2-connections and 2-groups for 4d BF theory . 122 8.4 Particles . 125 8.5 Particle/string statistics and Bohm{Aharonov duality . 127 8.6 Adjoint orbits . 130 III Cartan Geometry and Gravity 133 9 Cartan geometry 134 9.1 Ehresmann connections . 134 9.2 Definition of Cartan geometry . 136 9.3 Geometric interpretation: rolling Klein geometries . 137 9.4 Reductive Cartan geometry . 140 10 Cartan-type gauge theory 147 10.1 A sequence of bundles . 147 10.2 Parallel transport in Cartan geometry . 149 10.3 Holonomy and development . 151 10.4 Cartan-type BF Theory . 152 11 From Palatini to MacDowell{Mansouri 155 11.1 The Palatini formulation of general relativity . 155 11.2 Equivalence of formulations . 158 11.3 The coframe field . 161 11.4 MacDowell{Mansouri gravity . 163 11.5 Lagrangians for gravity and topological gauge theories . 166 11.6 Generalized MacDowell{Mansouri theory . 166 v Appendices A Presentations of the loop braid group 169 B The Lie algebras so(p; q) and iso(p; q) 177 C Hodge duality in Lie algebras 178 C.1 Hodge duality for inner product spaces. 178 C.2 so(4), so(3; 1), and so(2; 2) . 178 D Identities for forms and tensors 180 D.1 Permutation symbols . 180 D.2 Lie algebra-valued differential forms . 180 Bibliography 183 1 Chapter 1 Introduction One long standing theme in theoretical work on quantum gravity has been to exploit relationships between general relativity and gauge theory. The reason is clear: we know how to quantize the gauge theories of particle physics. But ordinary gauge theories are very different from gravity in an essential way. A typical gauge theory, such as Yang{Mills theory, uses the geometry of spacetime, as encoded in the metric, in its definition. Gravity, on the other hand, is a kind of `gauge theory' that determines the spacetime geometry itself. Topological gauge theories represent a sort of compromise. On one hand, such theories are formulated in essentially the same language as, say, Yang{Mills theory, and one can try quantizing them using similar methods. On the other hand, they are more similar to gravity in that they do not require any fixed background structure. While being simpler than general relativity, they thus share with it the many of the conceptual issues related to quantizing a generally covariant theory. What makes a gauge theory `topological' is a somewhat subjective matter. One possibility is that it should be describable using the functorial definition of topological field theory, or some slight generalization. But this is too strong a requirement to include some of the most interesting examples. A more practical requirement is that all solutions of a `topological gauge theory' should be locally the same up to gauge transformations. Such theories are more interesting when the topology of spacetime is more interesting, since solutions that look completely trivial on a local scale may yet have interesting global prop- erties. Intuitively, it is natural that topological gauge theories of this sort should be related to topological invariants. This is indeed true: the interplay between pure topology and gauge theories has been enormously fruitful, particularly in work on 3-manifold invariants 2 and knot theory, but more recently also for 4-manifolds. But one should not be misled into thinking the difference between topological gauge theories and general relativity is too much like the difference between topology and geometry.