A Construction of String 2-Group Modelsusing a Transgression
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Some Stuff About the String Gerbe
Some Stuff about the String Gerbe Danny Stevenson Department of Mathematics, University of California, Riverside Riverside, CA 92521, USA email: [email protected] March 14, 2006 1 Introduction Let G be a compact, simple and simply connected Lie group and let π : P → M be a principal G bundle over a smooth manifold M. Let ν denote the universally transgressive generator of H3(G; Z) = Z and let c ∈ H4(BG; Z) = H3(G; Z) be the transgression of ν. By regarding H4(BG; Z) as a lattice in (S2g∗)G we will allow ourselves to confuse c and the basic inner product h , i on g. Recall [23] that h ,√i is the Killing form on g, normalised so that the longest root θ has length 2. In the physics literature, M is said to be string, or admit a string structure, if a certain characteristic class in H3(LM; Z) vanishes (here LM denotes the free loop space of M). This characteristic class is the obstruction to lifting the structure group of the principal LG- bundle LP → LM to LGc — the Kac-Moody group. As has been observed by several authors [12, 13, 19] the obstruction in H3(LM; Z) is closely related to the characteristic class c ∈ H4(M; Z): if M is 2-connected a lift of the structure group to LGc exists precisely when the map c: M → K(Z, 4) is null- homotopic. As is well known, if G = Spin(n) then 2c = p1. This obstruction problem on LM can be phrased in the language of homotopy theory down on M. -
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. -
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]. -
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. -
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. -
Supersymmetric Field Theories and Orbifold Cohomology
SUPERSYMMETRIC FIELD THEORIES AND ORBIFOLD COHOMOLOGY A Dissertation Submitted to the Graduate School of the University of Notre Dame in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Augusto Stoffel Stephan Stolz, Director Graduate Program in Mathematics Notre Dame, Indiana April 2016 SUPERSYMMETRIC FIELD THEORIES AND ORBIFOLD COHOMOLOGY Abstract by Augusto Stoffel Using the Stolz–Teichner framework of supersymmetric Euclidean field theories (EFTs), we provide geometric interpretations of some aspects of the algebraic topology of orbifolds. We begin with a classification of 0j1-dimensional twists for EFTs over an orbifold X, and show that the collection of concordance classes of twisted EFTs over the inertia ΛX is in natural bijection with the delocalized twisted cohomology of X (which is isomorphic to its complexified K-theory). Then, turning to 1j1-dimensional considerations, we construct a (partial) twist functor over X taking as input a class in H3(X; Z). Next, we define a dimensional reduction procedure relating the 0j1-dimensional Euclidean bordism category over ΛX and its 1j1-dimensional counterpart over X, and explore some applications. As a basic example, we show that dimensional reduction of untwisted EFTs over a global quotient orbifold X==G recovers the equivariant Chern character. Finally, we describe the dimensional reduction of the 1j1-twist built earlier, showing that it has the expected relation to twisted K-theory. CONTENTS ACKNOWLEDGMENTS . iv CHAPTER 1: INTRODUCTION . 1 1.1 Supersymmetric field theories and cohomology theories . 1 1.2 Field theories over orbifolds . 4 1.3 Outline of the dissertation . 7 CHAPTER 2: STACKS IN DIFFERENTIAL GEOMETRY . -
On the String 2-Group
On the String 2-Group Urs Schreiber∗ May 3, 2006 Oberseminar Topologie Universit¨atBonn, May 2, 2006 Abstract We try to convey the main idea for 1. • what the String-group StringG is • and how it is the nerve of a 2-group StrG; 2. as well as • what a StrG-2-bundle is • and how it is “the same” as a StringG-bundle. The first point is due to [10, 9], which will be reviewed in section 2. The second point has been addressed in [8] using the language of bundle gerbes. In section 3 we review this, using a 2-functorial language which is natural with respect to the 2-group nature of StrG. A commented bibliography is given in section 1. Contents 1 Motivation 2 2 The String 2-Group 3 2.1 2-Groups . 4 2.2 StringG ................................ 6 3 String 2-Bundles 9 3.1 2-Bundles . 9 3.2 Transition Bundles . 15 4 Outlook: String Connections 19 ∗e-mail: urs.schreiber at math.uni-hamburg.de 1 1 Motivation The main motivation for the following discussion has its origin in theoretical physics. Elementary particles with spin are described by sections of spin bundles. From the physical point of view, the necessity of a spin structure on spacetime may be deduced from a certain global anomaly for the path integral of a single, point-like, fermion. The path integral (albeit a somewhat heuristic device) can be regarded as a single valued function on the space of configurations of the particle, only if the (first and) second Stiefel-Whitney class of spacetime vanishes. -
String Structures Associated to Indefinite Lie Groups
String structures associated to indefinite Lie groups Hisham Sati1 and Hyung-bo Shim2 1Division of Science and Mathematics, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi, UAE 2Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA April 2, 2019 Abstract String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signature orthogonal group O(n) to the indefinite-signature orthogonal group O(p, q), i.e. from the Riemannian to the pseudo-Riemannian setting. This requires that we work at the unstable level, which makes the discussion more subtle than the stable case. Similar, but much simpler, constructions hold for other noncompact Lie groups such as the unitary group U(p, q) and the symplectic group Sp(p, q). This extension provides a starting point for an abundance of constructions in (higher) geometry and applications in physics. Contents 1 Introduction 2 2 The Postnikov tower, Whitehead tower, and variants 4 2.1 ThePostnikovtower ............................... ..... 4 2.2 TheWhiteheadtower ............................... .... 5 2.3 Avariantview .................................... ... 7 2.4 Computationalaspects. ....... 8 2.5 Special case: higher connected covers of classifying spaces and obstructions . 10 3 Applications 13 arXiv:1504.02088v2 [math-ph] 31 Mar 2019 3.1 Special orthogonal groups SO(n) and SO(p,q)...................... 13 3.2 IndefiniteSpingroups ............................. ...... 14 3.3 IndefiniteStringgroups . ....... 17 3.4 String structure associated to indefinite unitary and symplectic groups . 21 3.5 Relation to twisted structures . -
Higher Gauge Theory – I
Higher Gauge Theory { I John C. Baez joint work with: Toby Bartels, Alissa Crans, James Dolan, Aaron Lauda, Urs Schreiber, Danny Stevenson. Barrett Lectures Saturday April 29th, 2006 1 G 3+ f1 D f2 Notes and references at: http://math.ucr.edu/home/baez/barrett/ 1 The Big Idea Using n-categories, instead of starting with a set of things: ¢ ¡¢ ¡ ¦ ¥¦ ¥ ¤ £¤ £ we can now start with a category of things and processes: ¨ §¨ § © © or a 2-category of things, processes, and processes be- tween processes: ... and so on. 2 I'll illustrate this with examples from higher gauge the- ory. This describes not only how particles transform as they move along paths in spacetime: but also how strings transform as they trace out surfaces: ... and so on. Where ordinary gauge theory uses groups, which are special categories: higher gauge theory uses 2-groups: which are special 2-categories. Where ordinary gauge theory uses bundles, higher gauge theory uses 2-bundles. Everything gets ‘categorified’! But first let's back up a bit.... 3 The Fundamental Groupoid Defining the fundamental group of a space X requires us to pick a basepoint ∗ 2 X. This is a bit ad hoc, and no good when X has several components. Sometimes it's better to use the fundamental groupoid of X. This is the category Π1(X) where: • objects are points of X: •x • morphisms are homotopy classes of paths in X: γ x • ( • y We compose homotopy classes of paths in the obvious way. Composition is associative, and every point has an identity path 1x : x ! x. -
Bundle Gerbes Examples Stable Isomorphism Other Things
Informal definition of p-gerbes Background Bundle gerbes Examples Stable isomorphism Other things Bundle gerbes Michael Murray University of Adelaide http://www.maths.adelaide.edu.au/˜mmurray Principal Bundles, Gerbes and Stacks 2007 Informal definition of p-gerbes Background Bundle gerbes Examples Stable isomorphism Other things Outline 1 Informal definition of p-gerbes 2 Background 3 Bundle gerbes 4 Examples 5 Stable isomorphism 6 Other things Informal definition of p-gerbes Background Bundle gerbes Examples Stable isomorphism Other things Informal definition of p-gerbe Very informally a p-gerbe is a geometric object representing p + 2 dimensional cohomology. The motivating example is p = 0 and two dimensional cohomology. The geometric objects are then U(1) principal bundles or hermitian line bundles. A little less informally a p-gerbe P is an object of some category C. There is a functor C → Man. If M is the image of P under this functor we say P is (a p-gerbe) over M and write P ⇒ M. This functor has to admit pullbacks. If f : N → M is smooth and P ⇒ M there is a unique p-gerbe f ∗(P) ⇒ N and a commuting diagram f ∗(P) → P ⇓ ⇓ f N → M Informal definition of p-gerbes Background Bundle gerbes Examples Stable isomorphism Other things If P and Q are p-gerbes there is a dual P∗ and a product P ⊗ Q. These have to be well-behaved with respect to pullback. Associated to every p-gerbe P ⇒ M there is a characteristic class c(P) ∈ Hp+2(M, Z) which is natural with respect to pullbacks. -
A Note on Bundle Gerbes and Infinite-Dimensionality
J. Aust. Math. Soc. 90 (2011), 81–92 doi:10.1017/S1446788711001078 A NOTE ON BUNDLE GERBES AND INFINITE-DIMENSIONALITY MICHAEL MURRAY ˛ and DANNY STEVENSON (Received 18 August 2010; accepted 25 October 2010) Communicated by V. Mathai Dedicated to Alan Carey, on the occasion of his 60th birthday Abstract Let .P; Y / be a bundle gerbe over a fibre bundle Y ! M. We show that if M is simply connected and the fibres of Y ! M are connected and finite-dimensional, then the Dixmier–Douady class of .P; Y / is torsion. This corrects and extends an earlier result of the first author. 2010 Mathematics subject classification: primary 53C08. Keywords and phrases: bundle gerbes, Dixmier–Douady class, infinite-dimensionality. 1. Introduction The idea of bundle gerbes T9U had its original motivation in attempts by the first author and Alan Carey to geometrise degree-three cohomology classes. This, in turn, arose from a shared interest in anomalies in quantum field theory resulting from nontrivial cohomology classes in the space of connections modulo gauge transformations. Even in the earliest of their joint papers on anomalies T6U, which demonstrates that the Wess– Zumino–Witten term can be understood as holonomy for a line bundle on the loop group, there is a bundle gerbe, at that time unnoticed, lurking in the background. It was not until some time later that they realised that a better interpretation of the Wess– Zumino–Witten term for a map of a surface into a compact Lie group is as the surface holonomy of the pullback of the basic bundle gerbe over that group T5U. -
UNIVERSITY of CALIFORNIA RIVERSIDE Division Algebras
UNIVERSITY OF CALIFORNIA RIVERSIDE Division Algebras, Supersymmetry and Higher Gauge Theory A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by John Gmerek Huerta June 2011 Dissertation Committee: Professor John Baez, Chairperson Professor Vyjayanthi Chari Professor Stefano Vidussi Copyright by John Gmerek Huerta 2011 The Dissertation of John Gmerek Huerta is approved: Committee Chairperson University of California, Riverside Acknowledgements A dissertation is the capstone to a doctoral program, and the acknowledgements provide a useful place to thank the countless people who have helped out along the way, both personally and professionally. First, of course, I thank my advisor, John Baez. It is hard to imagine a better advisor, and no one deserves more credit for my mathematical and professional growth during this program. “Thanks” does not seem sufficient, but it is all I have to give. Also deserving special mention is John’s collaborator, James Dolan. I am convinced there is no subject in mathematics for which Jim does not have some deep insight, and I thank him for sharing a few of these insights with me. Together, John and Jim are an unparalleled team: there are no two better people with whom to talk about mathematics, and no two people more awake to the joy of mathematics. I would also like to thank Geoffrey Dixon, Tevian Dray, Robert Helling, Corinne Manogue, Chris Rogers, Hisham Sati, James Stasheff, and Riccardo Nicoletti for helpful conversations and correspondence. I especially thank Urs Schreiber for many discussions of higher gauge theory and L1-superalgebras, smooth 1-groups, and supergeometry.