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Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation

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Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation Severin Bunk Submitted for the degree of Doctor of Philosophy Department of Mathematics arXiv:1709.06174v1 [math-ph] 18 Sep 2017 School of Mathematics and Computer Sciences September 2017 The copyright in this thesis is owned by the author. Any quotation from the thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. Abstract We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We start by briefly recalling the construction of the 2-category of bundle gerbes, with minor alter- ations that allow us to endow morphisms with additive structures. The morphisms in the resulting 2-categories are investigated in detail. We introduce a direct sum on morphism categories of bundle gerbes and show that these categories are cartesian monoidal and abelian. Endomorphisms of the trivial bundle gerbe, or higher functions, carry the struc- ture of a rig-category, a categorified ring, and we show that generic morphism categories of bundle gerbes form module categories over this rig-category. We continue by presenting a categorification of the hermitean bundle metric on a hermitean line bundle. This is achieved by introducing a functorial dual that extends the dual of vector bundles to morphisms of bundle gerbes, and constructing a two-variable adjunction for the aforementioned rig-module category structure on morphism categories. Its right internal hom is the module action, composed by taking the dual of the acting higher functions, while the left internal hom is interpreted as a bundle gerbe metric. Sections of bundle gerbes are defined as morphisms from the trivial bundle gerbe to the bundle gerbe under consideration. We show that the resulting categories of sections carry a rig-module structure over the category of finite-dimensional Hilbert spaces with its canonical direct sum and tensor product. A suitable definition of 2-Hilbert spaces is given, modifying previous definitions by the use of two-variable adjunctions. We prove that the category of sections of a bundle gerbe, with its additive and module structures, fits into this framework, thus obtaining a 2-Hilbert space of sections. In particular, this can be constructed for prequantum bundle gerbes in problems of higher geometric quantisation. We define a dimensional reduction functor and show that the categorical structures introduced on the 2-category of bundle gerbes naturally reduce to their counterparts on hermitean line bundles with connections. In several places in this thesis, we provide examples, making 2-Hilbert spaces of sections and dimensional reduction very explicit. i ii To my parents. iv Acknowledgements First and foremost, I would like to thank Richard Szabo for his supervision during the course of my PhD programme. I am grateful for all the time, support and advice he has given me over the last three years, for always taking my ideas and concerns seriously, for our often long discussions, and for always being frank with me. I would like to thank Christian S¨amannfor being my second supervisor, but even more for his input into this project and being approachable, caring and supportive throughout my PhD. I am grateful to Alexander Schenkel for countless discussions, his interest in my progress and career, and for giving me the opportunity to speak in Nottingham and at his MFO Mini-Workshop. During the second half of my PhD I have had the pleasure of collaborating with Konrad Waldorf. I would like to thank him in this place for inviting me to Greifswald, for the insightful discussions we have had, and for his support outside of our collaboration. Moreover, I would like to thank Michael Murray and Danny Stevenson for several interesting discussions on bundle gerbes and their modules, Branislav Jurˇcoand Jos´e Figueroa-O'Farrill for being my examiners, and Des Johnston for agreeing to watch over my viva. I would like to thank David Jordan for running his TQFT seminar, Gwendolyn Barnes for running a category theory reading group, and Tim Weelinck, Matt Booth, Lukas M¨uller and Jenny August for their enthusiasm and their contributions to the homotopy theory and higher categories reading group. Finally, I gratefully acknowledge support granted by Heriot-Watt University through a James-Watt Scholarship. v vi ACADEMIC REGISTRY Research Thesis Submission Name: School: Version: (i.e. First, Degree Sought: Resubmission, Final) Declaration In accordance with the appropriate regulations I hereby submit my thesis and I declare that: 1) the thesis embodies the results of my own work and has been composed by myself 2) where appropriate, I have made acknowledgement of the work of others and have made reference to work carried out in collaboration with other persons 3) the thesis is the correct version of the thesis for submission and is the same version as any electronic versions submitted*. 4) my thesis for the award referred to, deposited in the Heriot-Watt University Library, should be made available for loan or photocopying and be available via the Institutional Repository, subject to such conditions as the Librarian may require 5) I understand that as a student of the University I am required to abide by the Regulations of the University and to conform to its discipline. 6) I confirm that the thesis has been verified against plagiarism via an approved plagiarism detection application e.g. Turnitin. * Please note that it is the responsibility of the candidate to ensure that the correct version of the thesis is submitted. Signature of Date: Candidate: Submission Submitted By (name in capitals): Signature of Individual Submitting: Date Submitted: For Completion in the Student Service Centre (SSC) Received in the SSC by (name in capitals): Method of Submission (Handed in to SSC; posted through internal/external mail): E-thesis Submitted (mandatory for final theses) Signature: Date: Please note this form should be bound into the submitted thesis. Academic Registry/Version (1) August 2016 viii Contents 1 Introduction 1 1.1. Quantisation and categorification 1 1.2. A glance at bundle gerbes and higher quantisation 3 1.3. Outline and main results 6 2 Bundle gerbes 11 2.1. Preliminaries on coverings and sheaves of categories 11 2.2. The 2-category of bundle gerbes 16 2.3. Deligne cohomology and higher categories 26 3 Categorical structures on morphisms of bundle gerbes 35 3.1. Additive structures on morphisms 35 3.2. Pairings of morphisms { closed structures 44 3.3. Examples 50 4 2-Hilbert spaces from bundle gerbes 57 4.1. Higher geometric quantisation 57 4.2. Higher geometric structures 61 4.3. 2-Hilbert spaces 65 4.4. The 2-Hilbert space of a bundle gerbe 69 4.5. Example: Bundle gerbes on R3 73 4.6. Example: A decomposable bundle gerbe 76 4.7. Remarks on the torsion constraint 81 5 Transgression and dimensional reduction 85 5.1. Diffeological spaces and bundles 85 ix 5.2. The transgression functor 87 5.3. Transgression of categorical structures 95 5.4. Unparameterised loops, orientation, and Reality 99 5.5. Dimensional reduction 102 6 Conclusions 109 A Special morphisms of bundle gerbes 113 A.1. Remarks on monoidal structures and strictness 113 A.2. Morphisms of bundle gerbes and descent 114 A.3. Mutual surjective submersions 118 B Proofs 123 B.1. Proof of Theorem 3.14 123 B.2. Proof of Theorem 3.49 127 References 131 x Notation • We use the term `2-category' to refer to what is sometimes called a weak 2-category, or bicategory, i.e. we allow for non-trivial associators and unitors. If we explicitly refer to 2-categories where these are trivial, we will use the term `strict 2-category'. • For a category C and objects a; b 2 C, we denote the collection of morphisms from a to b in C by C(a; b). Similarly, if C is a 2-category and a; b 2 C are objects, the category of morphisms from a to b in C is denoted C(a; b). For two 1-morphisms φ, : a ! b in C, the collection of 2-morphisms from φ to is C(φ, ). • In a 2-category C, we denote the horizontal composition of 2-morphisms by ◦1, while the vertical composition is denoted ◦2. • For an n-category C, we denote its underlying n-groupoid by C∼, i.e. C∼ is the n- category obtained from C by discarding all non-invertible k-morphisms for k = 1; : : : ; n, while π0C denotes the collection of objects of C modulo the equivalence relation a ∼ b if there exists a zig-zag of morphisms between a and b. • Mfd is the category of smooth manifolds and smooth maps. We we define manifolds to be second countable and Hausdorff. • DfgSp is the category of diffeological spaces and diffeological maps. • HVBdl is the relative sheaf of categories (see Section 2.1) of smooth hermitean vec- tor bundles, with smooth fibrewise linear maps as morphisms, with respect to the Grothendieck topology given by smooth surjective submersions on Mfd (cf. [75]). Here, we include non-invertible morphisms of vector bundles. HLBdl is the full sub-sheaf of categories of hermitean line bundles on Mfd. • For a morphism 2 HVBdl(M)(E; F ), we write ∗ : F ! E for its adjoint, defined ∗ by hE( (f); e) = hF (f; (e)) via the hermitean metrics hE and hF on E and F , respectively, with e 2 E and f 2 F . The transpose of is denoted t : F ∗ ! E∗ and given by t(φ)(e) = φ( (e)) for e 2 E and φ 2 F ∗.
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