Higher Gauge Theory with String 2-Groups
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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. -
Categorical Models of Type Theory
Categorical models of type theory Michael Shulman February 28, 2012 1 / 43 Theories and models Example The theory of a group asserts an identity e, products x · y and inverses x−1 for any x; y, and equalities x · (y · z) = (x · y) · z and x · e = x = e · x and x · x−1 = e. I A model of this theory (in sets) is a particularparticular group, like Z or S3. I A model in spaces is a topological group. I A model in manifolds is a Lie group. I ... 3 / 43 Group objects in categories Definition A group object in a category with finite products is an object G with morphisms e : 1 ! G, m : G × G ! G, and i : G ! G, such that the following diagrams commute. m×1 (e;1) (1;e) G × G × G / G × G / G × G o G F G FF xx 1×m m FF xx FF m xx 1 F x 1 / F# x{ x G × G m G G ! / e / G 1 GO ∆ m G × G / G × G 1×i 4 / 43 Categorical semantics Categorical semantics is a general procedure to go from 1. the theory of a group to 2. the notion of group object in a category. A group object in a category is a model of the theory of a group. Then, anything we can prove formally in the theory of a group will be valid for group objects in any category. 5 / 43 Doctrines For each kind of type theory there is a corresponding kind of structured category in which we consider models. -
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. -
Universality of Multiplicative Infinite Loop Space Machines
UNIVERSALITY OF MULTIPLICATIVE INFINITE LOOP SPACE MACHINES DAVID GEPNER, MORITZ GROTH AND THOMAS NIKOLAUS Abstract. We establish a canonical and unique tensor product for commutative monoids and groups in an ∞-category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that En-(semi)ring objects in C give rise to En-ring spectrum objects in C. In the case that C is the ∞-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable ∞-categories. In particular, we identify preadditive and additive ∞-categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring(D ⊗ C) ≃ Ring(D) ⊗ C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in ∞-categories. Contents 0. Introduction 1 1. ∞-categories of commutative monoids and groups 4 2. Preadditive and additive ∞-categories 6 3. Smashing localizations 8 4. Commutative monoids and groups as smashing localizations 11 5. Canonical symmetric monoidal structures 13 6. More functoriality 15 7. ∞-categories of semirings and rings 17 8. Multiplicative infinite loop space theory 19 Appendix A. Comonoids 23 Appendix B. Algebraic theories and monadic functors 23 References 26 0. Introduction The Grothendieck group K0(M) of a commutative monoid M, also known as the group completion, is the universal abelian group which receives a monoid map from M. -
Seminar on Homogeneous and Symmetric Spaces
PROGRAMME SEMINAR ON HOMOGENEOUS AND SYMMETRIC SPACES GK1821 Programme Introduction1 Basic Definitions2 Talks5 1. Lie Groups6 2. Quotients6 3. Grassmannians6 4. Riemannian Symmetric Spaces7 5. The Adjoint Action8 6. Algebraic Groups9 7. Reductive Linear Algebraic Groups9 8. Quotients by Algebraic Groups9 9. Compactifications 10 10. Uniformization 11 11. Dirac Operators on Homogeneous Spaces 11 12. Period Domains 12 13. Hermitian Symmetric Spaces 12 14. Shimura Varieties 12 15. Affine Grassmannians 12 References 13 Introduction Groups arise as symmetries of objects and we study groups by studying their action as symmetries on geometric objects, such as vector spaces, manifolds and more general topological spaces. One particularly nice type of such geometric objects are homogeneous spaces. For example, the general linear group GLn and the symmetric group Sn arise naturally in many contexts and can be understood from their actions on many different spaces. Already for GLn there are several incarnations: as finite group GLn(Fq), as Lie group GLn(R) or GLn(C), as algebraic group R 7! GLn(R). As first approximation, we should think of homogeneous spaces as topological coset spaces G=H where H is a subgroup of G. A symmetric space is then a homogeneous space with the property that H is the fixed set of an involution on G. There are other, better and more precise characterizations that we’ll use. For example, a Riemannian manifold which is a symmetric space can also be characterized by a local symmetry condition. Any space with symmetries, i.e. a G-action, decomposes into its orbits, which are each homogeneous spaces. -
Math 395: Category Theory Northwestern University, Lecture Notes
Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere. -
CHAPTER IV.3. FORMAL GROUPS and LIE ALGEBRAS Contents
CHAPTER IV.3. FORMAL GROUPS AND LIE ALGEBRAS Contents Introduction 2 0.1. Why does the tangent space of a Lie group have the structure of a Lie algebra? 2 0.2. Formal moduli problems and Lie algebras 3 0.3. Inf-affineness 4 0.4. The functor of inf-spectrum and the exponential construction 5 0.5. What else is done in this chapter? 6 1. Formal moduli problems and co-algebras 7 1.1. Co-algebras associated to formal moduli problems 7 1.2. The monoidal structure 10 1.3. The functor of inf-spectrum 11 1.4. An example: vector prestacks 13 2. Inf-affineness 16 2.1. The notion of inf-affineness 16 2.2. Inf-affineness and inf-spectrum 17 2.3. A criterion for being inf-affine 18 3. From formal groups to Lie algebras 20 3.1. The exponential construction 20 3.2. Corollaries of Theorem 3.1.4 20 3.3. Lie algebras and formal moduli problems 22 3.4. The ind-nilpotent version 24 3.5. Base change 24 3.6. Extension to prestacks 25 3.7. An example: split square-zero extensions 27 4. Proof of Theorem 3.1.4 30 4.1. Step 1 30 4.2. Step 2 30 4.3. Step 3 32 5. Modules over formal groups and Lie algebras 33 5.1. Modules over formal groups 33 5.2. Relation to nil-isomorphisms 35 5.3. Compatibility with colimits 36 6. Actions of formal groups on prestacks 37 6.1. Action of groups vs. Lie algebras 37 6.2. -
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. -
Category Theory Course
Category Theory Course John Baez September 3, 2019 1 Contents 1 Category Theory: 4 1.1 Definition of a Category....................... 5 1.1.1 Categories of mathematical objects............. 5 1.1.2 Categories as mathematical objects............ 6 1.2 Doing Mathematics inside a Category............... 10 1.3 Limits and Colimits.......................... 11 1.3.1 Products............................ 11 1.3.2 Coproducts.......................... 14 1.4 General Limits and Colimits..................... 15 2 Equalizers, Coequalizers, Pullbacks, and Pushouts (Week 3) 16 2.1 Equalizers............................... 16 2.2 Coequalizers.............................. 18 2.3 Pullbacks................................ 19 2.4 Pullbacks and Pushouts....................... 20 2.5 Limits for all finite diagrams.................... 21 3 Week 4 22 3.1 Mathematics Between Categories.................. 22 3.2 Natural Transformations....................... 25 4 Maps Between Categories 28 4.1 Natural Transformations....................... 28 4.1.1 Examples of natural transformations........... 28 4.2 Equivalence of Categories...................... 28 4.3 Adjunctions.............................. 29 4.3.1 What are adjunctions?.................... 29 4.3.2 Examples of Adjunctions.................. 30 4.3.3 Diagonal Functor....................... 31 5 Diagrams in a Category as Functors 33 5.1 Units and Counits of Adjunctions................. 39 6 Cartesian Closed Categories 40 6.1 Evaluation and Coevaluation in Cartesian Closed Categories. 41 6.1.1 Internalizing Composition................. 42 6.2 Elements................................ 43 7 Week 9 43 7.1 Subobjects............................... 46 8 Symmetric Monoidal Categories 50 8.1 Guest lecture by Christina Osborne................ 50 8.1.1 What is a Monoidal Category?............... 50 8.1.2 Going back to the definition of a symmetric monoidal category.............................. 53 2 9 Week 10 54 9.1 The subobject classifier in Graph................. -
Categorical Representations of Categorical Groups
Theory and Applications of Categories, Vol. 16, No. 20, 2006, pp. 529–557. CATEGORICAL REPRESENTATIONS OF CATEGORICAL GROUPS JOHN W. BARRETT AND MARCO MACKAAY Abstract. A representation theory for (strict) categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these bicategories contain representations which are indecomposable but not irreducible. A simple example is computed in explicit detail. 1. Introduction In three-dimensional topology there is a very successful interaction between category theory, topology, algebra and mathematical physics which is reasonably well understood. Namely, monoidal categories play a central role in the construction of invariants of three- manifolds (and knots, links and graphs in three-manifolds), which can be understood using quantum groups and, from a physics perspective, the Chern-Simons functional integral. The monoidal categories determined by the quantum groups are all generalisations of the idea that the representations of a group form a monoidal category. The corresponding situation for four-manifold topology is less coherently understood and one has the feeling that the current state of knowledge is very far from complete. The complexity of the algebra increases dramatically in increasing dimension (though it might eventually stabilise). Formalisms exist for the application of categorical algebra to four- dimensional topology, for example using Hopf categories [CF], categorical groups [Y-HT] or monoidal 2-categories [CS, BL, M-S]. Since braided monoidal categories are a special type of monoidal 2-category (ones with only one object), then there are examples of the latter construction given by the representation theory of quasi-triangular Hopf algebras. This leads to the construction of the four-manifolds invariants by Crane, Yetter, Broda and Roberts which give information on the homotopy type of the four-manifold [CKY, R, R- EX].