DOCSLIB.ORG

Bundle Gerbes and the Weyl Map

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bundle Gerbes and the Weyl Map Bundle gerbes and the Weyl map Kimberly Becker September 30, 2019 Thesis submitted for the degree of Master of Philosophy in Pure Mathematics at The University of Adelaide Faculty of Engineering, Computer and Mathematical Sciences School of Mathematical Sciences ii Contents Signed Statement v Acknowledgements vii Dedication ix Abstract xi 0 Introduction 1 1 Vector bundles 5 1.1 Definitions and morphisms . .6 1.2 Connections and curvature . .9 1.3 Constructions . 11 1.4 Subbundles of the trivial bundle . 14 1.5 The first Chern class . 15 1.6 Line bundle holonomy . 17 1.7 Homogeneous vector bundles . 18 2 Bundle gerbes 23 2.1 Preliminaries . 25 2.2 Definition . 27 2.3 Constructions . 31 2.4 Morphisms and equivariance . 34 2.5 Connections and curvature . 38 2.6 The Dixmier-Douady class . 44 2.7 Holonomy . 47 2.8 Deligne cohomology . 52 3 The cup product bundle gerbe 59 3.1 The geometry of SU(n)=T ........................... 60 3.2 General cup product bundle gerbes . 63 iii iv Contents 3.2.1 Definition . 63 3.2.2 Stable isomorphisms . 64 3.2.3 Connective data . 66 3.3 The i-th cup product bundle gerbe . 68 3.4 The cup product bundle gerbe . 71 4 The basic bundle gerbe and the Weyl map 75 4.1 The Weyl map . 76 4.2 The basic bundle gerbe . 78 4.3 The pullback of the basic bundle gerbe by the Weyl map . 83 4.4 Isomorphisms of the pullback of the basic bundle gerbe . 86 5 The stable isomorphism 93 5.1 Set up of the problem . 94 5.2 Finding the stable isomorphism . 95 5.3 Comparing holonomies . 99 A Computational lemmas 103 Bibliography 105 Signed Statement I certify that this work contains no material which has been accepted for the award of any other degree or diploma in my name, in any 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. In addition, I certify that no part of this work will, in the future, be used in a submission in my name, for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Adelaide and where applicable, any partner institution responsible for the joint-award of this degree. I give permission for the digital version of my thesis to be made available on the web, via the University’s digital research repository, the Library Search and also through web search engines, unless permission has been granted by the University to restrict access for a period of time. I acknowledge the support I have received for my research through the provision of an Australian Government Research Training Program Scholarship. Signed: . Date: . v vi Signed Statement Acknowledgements First and foremost, thank you to my supervisors, Prof. Michael Murray and Dr Daniel Stevenson. Thank you for enduring the endless drafting and reference writing that I subjected you to. Most of all, thank you for your expert supervision, without which this thesis would not have been possible. Thank you to my family for encouraging and supporting me through all my pursuits. To my mother and sister, I treasure your love and support, thank you. Thank you to the Hallams, for giving me a home away from home. Thank you to the staff and students at The University of Adelaide and Kathleen Lumley College for a wonderful past couple of years. Finally, to Michael Hallam, thank you for the boundless love and joy you bring me every day, even from other side of the world. Here’s to our new chapter together. vii viii Acknowledgements Dedication To Michael Hallam ix x Dedication Abstract This thesis reviews bundle gerbe theory and the well-known basic bundle gerbe over SU(n). We introduce the cup product bundle gerbe, and show it is stably isomorphic to the pullback of the basic bundle gerbe by the Weyl map. This result enriches our understanding of the basic bundle gerbe, which has numerous applications in physics. xi xii Abstract 0 Introduction A bundle gerbe is a differential geometric object first developed by M. K. Murray in his 1996 paper Bundle Gerbes [43]. Based on J. Giraud’s theory of gerbes, bundle gerbes provide us with a geometric realisation of degree-three integer cohomology. This is similar to the geometric realisation of degree-two integer cohomology by line bundles. For this reason, bundle gerbes can be understood as ‘higher-dimensional’ analogues of line bun- dles. Just as line bundles have become foundational to differential geometry, so too have bundle gerbes, with Murray’s paper [43] marking the beginning of what would become a deeply rich and fruitful field. The applications of bundle gerbes are far reaching, permeating many fields of math- ematics and physics. For instance, they offer geometric interpretations of the ‘twists’ in twisted K-theory [6], the B-field in mathematical physics [31], and the Wess-Zumino- Witten models [44]. Other applications of bundle gerbes to physical problems appear in the studies of quantum field theory [13, 38], topological insulators [23, 16], Chern-Simons theory [12, 39], and mirror symmetry [27]. They have also been exploited to good effect in the study of D-brane charges in string theory [6, 29, 32, 40, 41]. These constitute just a few of the many applications of bundle gerbes that we will explore throughout this thesis. Roughly, a bundle gerbe is a triple of manifolds (P; Y; M) for Y ! M a surjective submersion and P a hermitian line bundle over Y [2] ⊂ Y 2, the fibre product of Y with itself. Additionally, we require an associative bundle gerbe multiplication on fibres of P m : P(y1;y2) ⊗ P(y2;y3) ! P(y1;y3): Just as with line bundles, there are notions of bundle gerbe constructions (pullback, product, dual, etc.), connections, curvature, and holonomy. Unlike line bundles, to define the three-curvature and holonomy of a bundle gerbe connection, we require additional data besides the connection, namely a curving or B-field in the physics literature. We call the pair consisting of a connection and curving connective data. There are several types of bundle gerbe morphisms, including isomorphisms, stable isomorphisms, and D-stable isomorphisms. Crucially, stable isomorphism (respectively D-stable isomorphism) classes 1 2 0. Introduction 3 3 of bundle gerbes over M are in bijective correspondence with H (M; Z) (H (M; Z(3)D)) via the Dixmier-Douady class (Deligne class). Moreover, the real part of the Dixmier- Douady class of a bundle gerbe can be represented by a multiple of its three-curvature, and the Deligne class of a bundle gerbe can be related to its three-curvature and holonomy. An in-depth treatment of each of these topics will be provided in Chapter 2. In this work, we will focus on two specific bundle gerbes. The first of these bundle gerbes we call the cup product bundle gerbe, defined over T × SU(n)=T for T a maximal torus of SU(n). This construction is based on the work of S. Johnson in [30], allowing us to realise certain wedge products of differential forms as the three-curvature of a bundle gerbe. The second bundle gerbe of interest to us is the popularly studied basic bundle gerbe introduced in [15] and described in greater detail in [50]. This is the classic example of a bundle gerbe defined over a compact, simple, simply connected Lie group G. It is particularly useful to physicists [24], allowing them to realise the Wess-Zumino-Witten term of a Sigma model as the holonomy of a geometric object. Our work will be concerned with the basic bundle gerbe over SU(n) constructed in [42]. The cup and basic bundle gerbes are commonly studied in isolation in the literature. We will show that there is a powerful result relating the cup and basic bundle gerbes, thereby deepening our understanding of these objects and enriching the field as a whole. In this thesis, we aim to prove the following, for T the subgroup of SU(n) consisting of diagonal matrices which acts on SU(n) by left multiplication. The pullback of the basic bundle gerbe over SU(n) by the Weyl map p : T × SU(n)=T ! SU(n); (t; gT ) 7! gtg−1; is SU(n)-stably isomorphic to the cup product bundle gerbe over T × SU(n)=T . The crude interpretation of this result is that, under a transformation by the Weyl map, the basic and cup product bundle gerbes are ‘the same’ as bundle gerbes enriched with an SU(n)-action. In addition to the above statement, we will also prove that the cup and basic bundle gerbes are not D-stably isomorphic with respect to their naturally induced connective data (Theorem 5.10). While this result is significant in of itself, it also opens up many avenues for further research. These include replacing the group SU(n) by an arbitrary compact, simple, simply-connected Lie group G, considering this result in the holomorphic category, or proving an analogous result for gerbes in the sense of Giraud [25]. These and other possible extensions will be discussed in Chapter 5. There is a straightforward reason as to why we expect our main result to be true. As 1 −1 3 we will see, the three-curvature of the basic bundle gerbe is 2πiν for ν := − 24π2 tr(g dg) the basic 3-form on SU(n).
Load more

Recommended publications
  • Arxiv:Gr-Qc/0611154 V1 30 Nov 2006 Otx Fcra Geometry
    MacDowell–Mansouri Gravity and Cartan Geometry Derek K. Wise Department of Mathematics University of California Riverside, CA 92521, USA email: [email protected] November 29, 2006 Abstract 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 geomet- ric 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 recent reformulation in terms of BF theory, in the arXiv:gr-qc/0611154 v1 30 Nov 2006 context of Cartan geometry. 1 Contents 1 Introduction 3 2 Homogeneous spacetimes and Klein geometry 8 2.1 Kleingeometry ................................... 8 2.2 MetricKleingeometry ............................. 10 2.3 Homogeneousmodelspacetimes. ..... 11 3 Cartan geometry 13 3.1 Ehresmannconnections . .. .. .. .. .. .. .. .. 13 3.2 Definition of Cartan geometry . ..... 14 3.3 Geometric interpretation: rolling Klein geometries . .............. 15 3.4 ReductiveCartangeometry . 17 4 Cartan-type gauge theory 20 4.1 Asequenceofbundles .............................
    [Show full text]
  • Schematic Homotopy Types and Non-Abelian Hodge Theory
    Compositio Math. 144 (2008) 582–632 doi:10.1112/S0010437X07003351 Schematic homotopy types and non-abelian Hodge theory L. Katzarkov, T. Pantev and B. To¨en Abstract We use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor X → (X ⊗ C)sch, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on (X ⊗ C)sch. This Hodge decomposition is encoded in an action of the discrete group C×δ on the object (X ⊗ C)sch and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro- algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold. Contents Introduction 583 1 Review of the schematization functor 589 1.1 Affine stacks and schematic homotopy types ................. 589 1.2 Equivariant stacks ............................... 594 1.3 Equivariant co-simplicial algebras and equivariant affine stacks ...... 595 1.4 An explicit model for (X ⊗ C)sch ....................... 597 2 The Hodge decomposition 598 2.1 Review of the non-abelian Hodge correspondence .............
    [Show full text]
  • Twenty-Five Years of Two-Dimensional Rational Conformal Field Theory
    ZMP-HH/09-21 Hamburger Beitr¨age zur Mathematik Nr. 349 October 2009 TWENTY-FIVE YEARS OF TWO-DIMENSIONAL RATIONAL CONFORMAL FIELD THEORY J¨urgen Fuchs a, Ingo Runkel b and Christoph Schweigert b a Teoretisk fysik, Karlstads Universitet Universitetsgatan 21, S–65188 Karlstad b Organisationseinheit Mathematik, Universit¨at Hamburg Bereich Algebra und Zahlentheorie Bundesstraße 55, D–20146 Hamburg A review for the 50th anniversary of the Journal of Mathematical Physics. 1 Introduction In this article we try to give a condensed panoramic view of the development of two-dimen- sional rational conformal field theory in the last twenty-five years. Given its limited length, our contribution can be, at best, a collection of pointers to the literature. Needless to say, the arXiv:0910.3145v2 [hep-th] 15 Jan 2010 exposition is highly biased, by the taste and the limited knowledge of the authors. We present in advance our apologies to everyone whose work has been inappropriately represented or even omitted. The study of conformal field theories in two dimensions has its origin in several distinct areas of physics: • In the attempt to describe the strong interactions in elementary particle physics in the frame- work of dual models. • Under the label string theory, these models were reinterpreted as a perturbation series containing a gravitational sector. Conformal field theories appear as theories defined on the world sheet that is swept out by the string [56]. • In statistical mechanics, conformal field theory plays a fundamental role in the theory of two- dimensional critical systems [189, 92] by describing fixed points of the renormalization group.
    [Show full text]
  • Fundamental Algebraic Geometry
    http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.
    [Show full text]
  • Arxiv:1401.2824V1 [Math.AT]
    ZMP-HH/14-2 Hamburger Beitr¨age zur Mathematik Nr. 499 January 2014 A Serre-Swan theorem for gerbe modules on ´etale Lie groupoids Christoph Schweigert a, Christopher Tropp b, Alessandro Valentino a a Fachbereich Mathematik, Universit¨at Hamburg Bereich Algebra und Zahlentheorie Bundesstraße 55, D – 20 146 Hamburg b Mathematisches Institut, WWU M¨unster Einsteinstr. 62, D – 48149 M¨unster Abstract Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact ´etale Lie groupoid M, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on M. This result can be seen as an equivariant Serre-Swan theorem for twisted vector bundles. 1 Introduction The celebrated Serre-Swan theorem relates the category of vector bundles over a compact smooth manifold M to the category of finite rank projective modules over the algebra of smooth functions C∞(M, C) of M (see [GBV, Mor] for the Serre-Swan theorem in the smooth category). It relates geometric and algebraic notions and is, in particular, the starting point for the definition of vector bundles in non-commutative geometry. arXiv:1401.2824v1 [math.AT] 13 Jan 2014 A bundle gerbe on M can be seen as a geometric realization of its Dixmier-Douady class, which is a class in H3(M; Z). To such a geometric realization, a twisted K-theory group can be associated. Gerbe modules have been introduced to obtain a geometric description of twisted K-theory [BCMMS].
    [Show full text]
  • APPLICATIONS of BUNDLE MAP Theoryt1)
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 171, September 1972 APPLICATIONS OF BUNDLE MAP THEORYt1) BY DANIEL HENRY GOTTLIEB ABSTRACT. This paper observes that the space of principal bundle maps into the universal bundle is contractible. This fact is added to I. M. James' Bundle map theory which is slightly generalized here. Then applications yield new results about actions on manifolds, the evaluation map, evaluation subgroups of classify- ing spaces of topological groups, vector bundle injections, the Wang exact sequence, and //-spaces. 1. Introduction. Let E —>p B be a principal G-bundle and let EG —>-G BG be the universal principal G-bundle. The author observes that the space of principal bundle maps from E to £G is "essentially" contractible. The purpose of this paper is to draw some consequences of that fact. These consequences give new results about characteristic classes and actions on manifolds, evaluation subgroups of classifying spaces, vector bundle injections, the evaluation map and homology, a "dual" exact sequence to the Wang exact sequence, and W-spaces. The most interesting consequence is Theorem (8.13). Let G be a group of homeomorphisms of a compact, closed, orientable topological manifold M and let co : G —> M be the evaluation at a base point x e M. Then, if x(M) denotes the Euler-Poincaré number of M, we have that \(M)co* is trivial on cohomology. This result follows from the simple relationship between group actions and characteristic classes (Theorem (8.8)). We also find out information about evaluation subgroups of classifying space of topological groups. In some cases, this leads to explicit computations, for example, of GX(B0 ).
    [Show full text]
  • Fluxes, Bundle Gerbes and 2-Hilbert Spaces
    Lett Math Phys DOI 10.1007/s11005-017-0971-x Fluxes, bundle gerbes and 2-Hilbert spaces Severin Bunk1 · Richard J. Szabo2 Received: 12 January 2017 / Revised: 13 April 2017 / Accepted: 8 May 2017 © The Author(s) 2017. This article is an open access publication Abstract We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a programme of higher geometric quantisation of closed strings in flux compactifications and of M5- branes in C-fields. We review in detail the construction of the 2-category of bundle gerbes and introduce the higher geometrical structures necessary to turn their cate- gories of sections into 2-Hilbert spaces. We work out several explicit examples of 2-Hilbert spaces in the context of closed strings and M5-branes on flat space. We also work out the prequantum 2-Hilbert space associated with an M-theory lift of closed strings described by an asymmetric cyclic orbifold of the SU(2) WZW model, pro- viding an example of sections of a torsion gerbe on a curved background. We describe the dimensional reduction of M-theory to string theory in these settings as a map from 2-isomorphism classes of sections of bundle gerbes to sections of corresponding line bundles, which is compatible with the respective monoidal structures and module actions. Keywords Bundle gerbes · Higher geometric quantisation · Fluxes in string and M-theory · Higher geometric structures Mathematics Subject Classification 81S10 · 53Z05 · 18F99 B Severin Bunk [email protected] Richard J.
    [Show full text]
  • Arxiv:2109.00850V1 [Math.AG] 2 Sep 2021 § Ncaatrsi Eo(E 1].Antrlqeto Shwt Gener to How Is Question Natural a and Bundles [17])
    TAME PARAHORIC NONABELIAN HODGE CORRESPONDENCE IN POSITIVE CHARACTERISTIC OVER ALGEBRAIC CURVES MAO LI AND HAO SUN Abstract. Let G be a reductive group, and let X be an algebraic curve over an algebraically closed field k with positive characteristic. We prove a version of nonabelian Hodge correspondence for G- local systems over X and G-Higgs bundles over the Frobenius twist X′ with first order poles. To obtain a general statement of the correspondence, we introduce the language of parahoric group schemes to establish the correspondence. 1. Introduction 1.1. Background. Let X be a smooth projective scheme over C. In [18], Simpson constructed a homeomorphism between the moduli space of semistable Higgs bundles on X and the moduli space of representations of fundamental groups of X. This correspondence between local systems and Higgs bundles is analytic in nature, it does not preserve the algebraic structure. On the other hand, the work of [14] and [7] shows that the connection between Higgs bundles and local systems is much closer in prime characteristics. The main results of [7] shows that for a smooth projective curve X over an algebraically closed field k of positive characteristic, the stack of local systems on X is a twisted version of the stack of Higgs bundles on X′, the Frobenius twist of X. This result is a crucial ingredient in the proof of the geometric Langlands conjecture in positive characteristic on curves [8]. The Simpson correspondence also works for local systems and Higgs bundles with first order poles in characteristic zero (see [17]). A natural question is how to generalize the correspondence to G-Higgs bundles and G-local systems for reductive groups G (with first order poles).
    [Show full text]
  • Math 704: Part 1: Principal Bundles and Connections
    MATH 704: PART 1: PRINCIPAL BUNDLES AND CONNECTIONS WEIMIN CHEN Contents 1. Lie Groups 1 2. Principal Bundles 3 3. Connections and curvature 6 4. Covariant derivatives 12 References 13 1. Lie Groups A Lie group G is a smooth manifold such that the multiplication map G × G ! G, (g; h) 7! gh, and the inverse map G ! G, g 7! g−1, are smooth maps. A Lie subgroup H of G is a subgroup of G which is at the same time an embedded submanifold. A Lie group homomorphism is a group homomorphism which is a smooth map between the Lie groups. The Lie algebra, denoted by Lie(G), of a Lie group G consists of the set of left-invariant vector fields on G, i.e., Lie(G) = fX 2 X (G)j(Lg)∗X = Xg, where Lg : G ! G is the left translation Lg(h) = gh. As a vector space, Lie(G) is naturally identified with the tangent space TeG via X 7! X(e). A Lie group homomorphism naturally induces a Lie algebra homomorphism between the associated Lie algebras. Finally, the universal cover of a connected Lie group is naturally a Lie group, which is in one to one correspondence with the corresponding Lie algebras. Example 1.1. Here are some important Lie groups in geometry and topology. • GL(n; R), GL(n; C), where GL(n; C) can be naturally identified as a Lie sub- group of GL(2n; R). • SL(n; R), O(n), SO(n) = O(n) \ SL(n; R), Lie subgroups of GL(n; R).
    [Show full text]
  • A Twisted Nonabelian Hodge Correspondence
    A TWISTED NONABELIAN HODGE CORRESPONDENCE Alberto Garc´ıa-Raboso A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2013 Tony Pantev David Harbater Professor of Mathematics Professor of Mathematics Supervisor of Dissertation Graduate Group Chairperson Dissertation Committee: Tony Pantev, Professor of Mathematics Ron Y. Donagi, Professor of Mathematics Jonathan L. Block, Professor of Mathematics UMI Number: 3594796 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3594796 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 A TWISTED NONABELIAN HODGE CORRESPONDENCE COPYRIGHT 2013 Alberto Garc´ıa-Raboso Acknowledgements I want to thank my advisor, Tony Pantev, for suggesting the problem to me and for his constant support and encouragement; the University of Pennsylvania, for its generous financial support in the form of a Benjamin Franklin fellowship; Carlos Simpson, for his continued interest in my work; Urs Schreiber, for patiently listening to me, and for creating and maintaining a resource as useful as the nLab; Marc Hoyois, for ever mentioning the words 1-localic 1-topoi and for answering some of my questions; Angelo Vistoli, for his help with Lemma 5.2.2; Tyler Kelly, Dragos, Deliu, Umut Isik, Pranav Pandit and Ana Pe´on-Nieto, for many helpful conversations.
    [Show full text]
  • A Construction of String 2-Group Modelsusing a Transgression
    A Construction of String 2-Group Models using a Transgression-Regression Technique Konrad Waldorf Fakult¨at f¨ur Mathematik Universit¨at Regensburg Universit¨atsstraße 31 D–93053 Regensburg [email protected] Dedicated to Steven Rosenberg on the occasion of his 60th birthday Abstract In this note we present a new construction of the string group that ends optionally in two different contexts: strict diffeological 2-groups or finite-dimensional Lie 2-groups. It is canonical in the sense that no choices are involved; all the data is written down and can be looked up (at least somewhere). The basis of our construction is the basic gerbe of Gaw¸edzki-Reis and Meinrenken. The main new insight is that under a transgression- regression procedure, the basic gerbe picks up a multiplicative structure coming from the Mickelsson product over the loop group. The conclusion of the construction is a relation between multiplicative gerbes and 2-group extensions for which we use recent work of Schommer-Pries. MSC 2010: Primary 22E67; secondary 53C08, 81T30, 58H05 arXiv:1201.5052v2 [math.DG] 4 May 2012 1 Introduction The string group String(n) is a topological group defined up to homotopy equivalence as the 3-connected cover of Spin(n), for n =3 or n> 4. Concrete models for String(n) have been provided by Stolz [Sto96] and Stolz-Teichner [ST04]. In order to understand, e.g. the differential geometry of String(n), the so-called “string geometry”, it is necessary to have models in better categories than topological groups. Its 3-connectedness implies that String(n) is a K(Z, 2)-fibration over Spin(n), so that it cannot be a (finite-dimensional) Lie group.
    [Show full text]
  • Notes on Principal Bundles and Classifying Spaces
    Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG].
    [Show full text]