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Constructions with Bundle Gerbes Constructions with Bundle Gerbes Stuart Johnson Thesis submitted for the degree of Doctor of Philosophy in the School of Pure Mathematics University of Adelaide arXiv:math/0312175v1 [math.DG] 9 Dec 2003 13 December 2002 Abstract This thesis develops the theory of bundle gerbes and examines a number of useful constructions in this theory. These allow us to gain a greater insight into the struc- ture of bundle gerbes and related objects. Furthermore they naturally lead to some interesting applications in physics. i ii Statement of Originality This thesis contains no material which has been accepted for the award of any other degree or diploma at any other university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being made available for loan and photocopying. Stuart Johnson Adelaide, 13 December, 2002 iii iv Acknowledgement Firstly I would like to thank my supervisor Michael Murray. He has been extremely helpful, insightful and encouraging at all times and it has been a great pleasure to work with him. I would also like to thank Alan Carey for valuable assistance in the early stages of my work on this thesis and for his continuing support. Thanks are also due to Danny Stevenson for many interesting and helpful conversations and to Ann Ross for always being of assistance with administrative matters. I would also like to acknowledge the support of a University of Adelaide postgraduate scholarship. Finally I should acknowledge everyone, past and present, at Kathleen Lumley Col- lege as well as all of the other wonderful people that I have met in Adelaide and who have made my time here so enjoyable. v vi Contents 1 Introduction 1 2 Bundle Gerbes and Deligne Cohomology 5 2.1 AReviewofSheafCohomology ...................... 5 2.2 U(1)-Functions ............................... 7 2.3 U(1)-Bundles ................................ 9 2.4 Bundle0-Gerbes .............................. 10 2.5 BundleGerbes ............................... 15 3 Examples of Bundle Gerbes 21 3.1 TautologicalBundleGerbes . 21 3.2 TrivialBundleGerbes ........................... 24 3.3 LiftingBundleGerbes ........................... 27 3.4 TorsionBundleGerbes ........................... 28 3.5 CupProductBundleGerbes. 30 4 Other Geometric Realisations of Deligne Cohomology 37 4.1 TheBundleGerbeHierarchy. 37 4.2 Gerbes.................................... 38 4.3 Bundle2-Gerbes .............................. 39 4.4 Z-Bundle0-Gerbes ............................. 51 4.5 BpS1-Bundles................................ 53 4.6 Comparing the Various Realisations . 55 5 Holonomy and Transgression 57 5.1 Holonomy of U(1)-Bundles......................... 57 5.2 HolonomyofBundleGerbes . .. .. 62 5.3 HolonomyofBundle2-Gerbes . 65 5.4 AGeneralHolonomyFormula . 67 5.5 Transgression for Closed Manifolds . 71 6 Parallel Transport and Transgression with Boundary 73 6.1 ParallelTransportforBundles. 73 6.2 LoopTransgressionofBundleGerbes . 77 6.3 AGeneralFormulaforParallelTransport. 81 6.4 LoopTransgressionofBundle2-Gerbes . 86 vii 7 Further Results on Holonomy and Transgression 89 7.1 SomeGeneralResults............................ 89 7.2 The Holonomy of the Tautological Bundle Gerbe . 92 7.3 HolonomyReconstruction . .. .. 93 7.4 ReconstructionviaTransgression . 97 7.5 ReconstructionofBundle2-Gerbes . 100 7.6 GeometricTransgression . 103 7.7 GaugeTransformations. 107 8 Applications 113 8.1 TheWess-Zumino-WittenAction . 113 8.2 TheChern-SimonsAction . 115 8.3 D-BranesandAnomalyCancellation . 121 8.4 Axiomatic Topological Quantum Field Theory . 127 Bibliography 132 viii Chapter 1 Introduction Bundle gerbes were introduced by Murray [36] as geometric realisations of classes in H3(M, Z) on a manifold M. By geometric realisation we mean an equivalence class of geometric objects which is isomorphic to H3(M, Z). An example of this in lower degree is the relationship between isomorphism classes of principal U(1)-bundles, or equivalently line bundles, over M and the Chern class in H2(M, Z). The interest in H3(M, Z) was motivated by the appearance of such integral cohomology classes in a number of situations including central extensions of structure groups of principal bun- dles and Wess-Zumino-Witten (WZW) theory. There were already a number of other realisations of H3(M; Z), of particular interest are the gerbes of Giraud as described in Brylinski’s book [5]. These are, from a rather simplistic view, defined as sheaves of groupoids. The idea of bundle gerbe theory was to define realisations which did not involve sheaves. Instead it is possible to build a realisation out of principal U(1)- bundles. π Essentially a bundle gerbe over a manifold M consists of a submersion Y M and → a U(1)-bundle P Y [2] over the fibre product → Y [2] = Y Y = (y,y′) Y 2 π(y)= π(y′) ×π { ∈ | } The fibres of P are required to carry a certain associative product structure which is called the bundle gerbe product. In [36] bundle gerbe connections and curving were defined. Together these form a higher analogue of connections on U(1)-bundles. They also correspond to connective structures and curvings on gerbes [5]. A bundle gerbe connection is a connection, A, on the bundle P which is compatible with the bundle gerbe product. Denote the curvature [2] of this connection by F . Let π1 and π2 be the projections of each component Y Y . ∗ ∗→ A curving is a 2-form, η, on Y which satisfies δ(η) = F where δ(η) = π2 η π1 η. A bundle gerbe with connection and curving defines a class in the Deligne cohomology− 3 group H (M, Z(3)D) which may be thought of as the hypercohomology of a complex of sheaves, H2(M, U(1) Ω1(M) Ω2(M)). M → → The bundle gerbe construction has also been used to consider higher degree Cechˇ and Deligne classes [13]. In particular Stevenson has developed a theory of bundle 2- gerbes ([44],[43]) which have an associated class in H4(M, Z), and when given a higher 4 analogue of connection and curving give rise to a class in H (M, Z(4)D). There is a cup product in Deligne cohomology which was described by Esnault and Viehweg [17] and which has been given a geometric interpretation by Brylinski and McLaughlin ([5],[8]). 1 A generalisation of the concept of holonomy to bundle gerbes was first considered in [36]. Just as the holonomy of a U(1)-bundle with connection associates an element of U(1) to every loop in the base, the holonomy of a bundle gerbe with connection and curving associates an element of U(1) to every closed surface in the base. In one particular case bundle gerbe holonomy has been used to describe the WZW action [12]. These considerations lead to the transgression formulae derived by Gawedski [23] for dealing with such actions in general settings. These formulae generalise bundle holonomy and parallel transport to higher degree Deligne classes. Completely general transgression formulae have been given by Gomi and Terashima ([25],[26]). The relevance of Cechˇ and Deligne cohomology classes to applications in physics has been well established. For example Dijkgraaf and Witten [15] have used differential characters to find a general Chern-Simons Theory, Gomi [24] has considered the relation between gerbes and Chern-Simons theory and Freed and Witten [21] have considered the role of Deligne classes in anomaly cancellation in D-brane theory. As well as the WZW case we have already alluded to there are a number of other examples discussed in [12]. The basic aim of this thesis is to provide further development of the theory of bundle gerbes. The goal has been to develop this theory in a way which keeps in mind the need for a balance between an abstract approach which readily accommodates generalisation and an approach which more easily allows application of the theory and which could be of interest to a wider audience. For the first factor the most important feature is the bundle gerbe hierarchy principle which is a guiding principle for relating bundle gerbe type constructions corresponding to Deligne cohomology in various degrees. For the second factor we show how various constructions may be described in geometric terms, often allowing manipulation of diagrammatic representations of bundle gerbes to take the place of complicated calculations. With these factors as a guide we have described a number of constructions involving bundle gerbes. Some of these which have already been developed elsewhere are given in different forms or with a different emphasis to demonstrate the hierarchy principle or to relate more easily to our applications. We also describe some constructions which are new to bundle gerbe theory. Finally we show how these constructions are useful in applications of bundle gerbe theory to physics. We begin with a review of the basic features of Deligne cohomology and introduce the bundle gerbe family of geometric realisations via bundle 0-gerbes as an alternative to U(1)-bundles. This would appear to be a complicated approach however it simplifies the transition from bundles to bundle gerbes and allows us to develop some features of bundle gerbe theory in a setting which is still relatively familiar. We then define bundle gerbes and explain their role as representatives of degree 3 Deligne cohomology. In Chapter 3 we consider some important examples of bundle gerbes. Tautological bundle gerbes [36] are introduced by first defining a bundle 0-gerbe, helping to gain a feel for the bundle gerbe hierarchy. Trivial bundle gerbes are discussed in some detail since they play an important role in many constructions. In particular we give a detailed account of the distinction between trivial bundle gerbes which by definition have trivial Cechˇ class and D-trivial bundle gerbes which have trivial Deligne class. We then consider torsion bundle gerbes which are defined as bundle gerbes with a torsion Cechˇ class.
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