DOCSLIB.ORG

Linear Analysis on Manifolds

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Analysis on Manifolds Linear Analysis on Manifolds Pierre Albin Contents Contents i 1 Differential operators on manifolds 3 1.1 Smooth manifolds . 3 1.2 Partitions of unity . 5 1.3 Tangent bundle . 5 1.4 Vector bundles . 8 1.5 Pull-back bundles . 9 1.6 The cotangent bundle . 11 1.7 Operations on vector bundles . 12 1.8 Sections of bundles . 14 1.9 Differential forms . 16 1.10 Orientations and integration . 18 1.11 Exterior derivative . 20 1.12 Differential operators . 21 1.13 Exercises . 23 1.14 Bibliography . 25 2 A brief introduction to Riemannian geometry 27 2.1 Riemannian metrics . 27 2.2 Parallel translation . 30 2.3 Geodesics . 34 2.4 Levi-Civita connection . 37 i ii CONTENTS 2.5 Associated bundles . 39 2.6 E-valued forms . 40 2.7 Curvature . 42 2.8 Complete manifolds of bounded geometry . 46 2.9 Riemannian volume and formal adjoint . 46 2.10 Hodge star and Hodge Laplacian . 48 2.11 Exercises . 51 2.12 Bibliography . 54 3 Laplace-type and Dirac-type operators 57 3.1 Principal symbol of a differential operator . 57 3.2 Laplace-type operators . 60 3.3 Dirac-type operators . 61 3.4 Clifford algebras . 64 3.5 Hermitian vector bundles . 69 3.6 Examples . 73 3.6.1 The Gauss-Bonnet operator . 73 3.6.2 The signature operator . 75 3.6.3 The @ operator . 76 3.6.4 The Dirac operator . 78 3.6.5 Twisted Clifford modules . 80 3.7 Supplement: Spin structures . 80 3.8 Exercises . 85 3.9 Bibliography . 88 4 Pseudodifferential operators on Rm 89 4.1 Distributions . 89 4.2 Examples . 94 4.2.1 Fundamental solutions . 94 4.2.2 Principal values and finite parts . 95 CONTENTS iii 4.3 Tempered distributions . 96 4.3.1 Convolution . 96 4.3.2 Fourier transform . 98 4.4 Pseudodifferential operators on Rm . 102 4.4.1 Symbols and oscillatory integrals . 102 4.4.2 Amplitudes and mapping properties . 108 4.5 Exercises . 116 4.6 Bibliography . 117 5 Pseudodifferential operators on manifolds 119 5.1 Symbol calculus and parametrices . 119 5.2 Hodge decomposition . 123 5.3 The index of an elliptic operator . 127 5.4 Pseudodifferential operators acting on L2 . 131 5.5 Sobolev spaces . 136 5.6 Exercises . 140 5.7 Bibliography . 144 6 Spectrum of the Laplacian 145 6.1 Spectrum and resolvent . 145 6.2 The heat kernel . 149 6.3 Weyl's law . 156 6.4 Long-time behavior of the heat kernel . 158 6.5 Supplement: Determinant of the Laplacian . 159 6.6 Exercises . 161 6.7 Bibliography . 162 7 The Atiyah-Singer index theorem 163 7.1 Getzler rescaling . 163 7.2 Atiyah-Singer for Dirac-type operators . 171 7.3 Chern-Weil theory . 176 CONTENTS 1 7.4 Examples . 180 7.5 Atiyah-Singer for elliptic operators . 181 7.6 Exercises . 185 7.7 Bibliography . 187 Introduction This text is a version of the lecture notes used during a course at the University of Illinois at Urbana-Champaign in Spring 2012 and a course at the University of Rome `La Sapienza' in Summer 2014, as well as several reading courses. Many of the participants of the original course had attended a course on differentiable manifolds but did not have any experience with Riemannian geometry. The intention of the course was to introduce the techniques of geometric analysis on manifolds and establish the Atiyah-Singer index theorem. I learned these techniques from Rafe Mazzeo and Richard Melrose and indeed the construction of the heat kernel and proof of the local index theorem are taken from Melrose's book on the Atiyah- Patodi-Singer index theorem. There are several textbooks that cover some of the same ground as these notes. Notably, John Roe, Elliptic operators, topology and asymptotic methods. Second edition. Pitman Research Notes in Mathematics Series, 395. Longman, Harlow, 1998., Nicole Berline, Ezra Getzler, Michele` Vergne, Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004., H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989., David D. Bleecker and Bernhelm Booss-Bavnbek, Index Theory with Ap- plications to Mathematics and Physics. International Press, 2013. Beyond the feedback I received from the students taking the course, I received a helpful list of corrections from Ra´ulS´anchez Gal´anand excellent suggestions from Chris Kottke whose own lecture notes on the subject can be found at his webpage http://ckottke.ncf.edu/neu/7376_sp16/. 2 CONTENTS Chapter 1 Differential operators on manifolds 1.1 Smooth manifolds The m-dimensional Euclidean space Rm; the m-dimensional sphere Sm and the m- dimensional torus Tm = S1 × ::: × S1 are all examples of smooth manifolds, the context where our studies will take place. A general manifold is a space constructed from open sets in Rm by patching them together smoothly. Formally, M is a manifold if it is a paracompact1 Hausdorff topological space with an `atlas', i.e., a covering by open sets fVαg together with homeomorphisms φα : Uα −! Vα m for some open subsets fUαg of R : The integer m is known as the dimension of M and the maps φα : Uα −! Vα (or sometimes just the sets fVαg) are known as coordinate charts. An atlas is smooth if whenever Vαβ := Vα \Vβ 6= ; the map −1 −1 −1 (1.1) αβ := φβ ◦ φα : φα (Vαβ) −! φβ (Vαβ) −1 −1 m is a smooth diffeomorphism between the open sets φα (Vαβ); φβ (Vαβ) ⊆ R : (For 1 k us smooth will always mean C :) If the maps αβ are all C -smooth, we say that the atlas is Ck: If φα : Uα −! Vα is a coordinate chart, it is typical to write −1 (y1; : : : ; ym) = φα : Vα −! Uα and refer to the yi as local coordinates of M on Vα: 1 A topological space is paracompact if every open cover has an open refinement in which each point is covered finitely many times. 3 4 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS Two smooth atlases are compatible (or equivalent) if their union is a smooth atlas. A smooth structure on M is an equivalence class of smooth atlases or, equivalently, a maximal smooth atlas on M: A manifold M together with a smooth structure is a smooth manifold. To indicate that a manifold M is m-dimensional we will sometimes write M m or, more often, we will refer to M as an m-manifold. A more general notion than manifold is that of a manifold with boundary. The former is defined in the same way as a manifold, but the local model is m m R− = f(x1; : : : ; xn) 2 R : x1 ≤ 0g: If M is a smooth manifold with boundary, its boundary @M consists of those points m in M that are in the image of @R− for some (and hence all) local coordinate charts. It is easy to check that @M is itself a smooth manifold (without boundary). The product of two manifolds with boundary is an example of a still more general notion, a manifold with corners. A map between two smooth manifolds (with or without boundary) F : M −! M 0 is smooth if it is smooth in local coordinates. That is, whenever 0 0 0 0 φα : Uα −! Vα ⊆ M; φγ : Uγ −! Vγ ⊆ M ; are coordinate charts, the map 0 −1 0 (1.2) (φγ) ◦ F ◦ φα : Uα −! Uγ is smooth (whenever it is defined). Notice that this definition is equivalent to making coordinate charts diffeomorphisms. The space of smooth maps between M and M 0 will be denoted C1(M; M 0) and when M 0 = R; simply by C1(M): Notice that a map F 2 C1(M; M 0) naturally induces a pull-back map F ∗ : C1(M 0) −! C1(M);F ∗(h)(p) = h(F (p)) for all h 2 C1(M 0); p 2 M: For us a manifold will mean a connected smooth manifold without boundary, unless otherwise specified. Similarly, unless we specify otherwise, a map between manifolds will be smooth. A subset X ⊆ M is a submanifold of M of dimension k if among the m coordinates on M at each point of X there are k that restrict to give coordinates of 1.2. PARTITIONS OF UNITY 5 X: That is, at each point p 2 X there is a coordinate chart of M; φ : U −! V with q 2 V; whose restriction to U \ Rk × f0g maps into X; φ : U\ k × f0g −! V \ X: U\Rk×{0g R These restricted charts then define a smooth k-manifold structure on X: 1.2 Partitions of unity A manifold is defined by patching together many copies of Rm: It is often useful to take a construction on Rm and transfer it to a manifold by carrying it out on each patch and then adding the pieces together. For these constructions, it is very convenient that every open covering Vα of a smooth manifold carries a partition of unity. That is, a collection !i of smooth functions !i : M −! R such that: i) At each p 2 M only finitely many !i(p) are non-zero ii) Each !i is supported inside some Vα iii) The sum of the !i is the constant function equal to one at each point p 2 M: One remarkable fact that follows easily from the existence of partitions of unity is that any smooth manifold can be identified with a smooth submanifold of RN for some N: This is easy to show using a locally finite atlas with a corresponding partition of unity.
Load more

Recommended publications
  • Connections on Bundles Md
    Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II.
    [Show full text]
  • Laplacians in Geometric Analysis
    LAPLACIANS IN GEOMETRIC ANALYSIS Syafiq Johar syafi[email protected] Contents 1 Trace Laplacian 1 1.1 Connections on Vector Bundles . .1 1.2 Local and Explicit Expressions . .2 1.3 Second Covariant Derivative . .3 1.4 Curvatures on Vector Bundles . .4 1.5 Trace Laplacian . .5 2 Harmonic Functions 6 2.1 Gradient and Divergence Operators . .7 2.2 Laplace-Beltrami Operator . .7 2.3 Harmonic Functions . .8 2.4 Harmonic Maps . .8 3 Hodge Laplacian 9 3.1 Exterior Derivatives . .9 3.2 Hodge Duals . 10 3.3 Hodge Laplacian . 12 4 Hodge Decomposition 13 4.1 De Rham Cohomology . 13 4.2 Hodge Decomposition Theorem . 14 5 Weitzenb¨ock and B¨ochner Formulas 15 5.1 Weitzenb¨ock Formula . 15 5.1.1 0-forms . 15 5.1.2 k-forms . 15 5.2 B¨ochner Formula . 17 1 Trace Laplacian In this section, we are going to present a notion of Laplacian that is regularly used in differential geometry, namely the trace Laplacian (also called the rough Laplacian or connection Laplacian). We recall the definition of connection on vector bundles which allows us to take the directional derivative of vector bundles. 1.1 Connections on Vector Bundles Definition 1.1 (Connection). Let M be a differentiable manifold and E a vector bundle over M. A connection or covariant derivative at a point p 2 M is a map D : Γ(E) ! Γ(T ∗M ⊗ E) 1 with the properties for any V; W 2 TpM; σ; τ 2 Γ(E) and f 2 C (M), we have that DV σ 2 Ep with the following properties: 1.
    [Show full text]
  • Gsm025-Endmatter.Pdf
    http://dx.doi.org/10.1090/gsm/025 Selected Titles in This Series 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A.
    [Show full text]
  • Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu
    Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes for a topics in topology course, University of Notre Dame, Spring 2004, Spring 2013. Last revision: November 15, 2013 i The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential opera- tors on a compact manifold, namely those operators P with the property that dim ker P; dim coker P < 1. In general these integers are very difficult to compute without some very precise information about P . Remarkably, their difference, called the index of P , is a “soft” quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. m n Suppose we are given a linear operator A : C ! C . From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators.
    [Show full text]
  • Â-Genus and Collapsing
    The Journal of Geometric Analysis Volume 10, Number 3, 2000 A A-Genus and Collapsing By John Lott ABSTRACT. If M is a compact spin manifold, we give relationships between the vanishing of A( M) and the possibility that M can collapse with curvature bounded below. 1. Introduction The purpose of this paper is to extend the following simple lemma. Lemma 1.I. If M is a connectedA closed Riemannian spin manifold of nonnegative sectional curvature with dim(M) > 0, then A(M) = O. Proof Let K denote the sectional curvature of M and let R denote its scalar curvature. Suppose that A(M) (: O. Let D denote the Dirac operator on M. From the Atiyah-Singer index theorem, there is a nonzero spinor field 7t on M such that D~p = 0. From Lichnerowicz's theorem, 0 = IDOl 2 dvol = IV~Pl 2 dvol + ~- I•12 dvol. (1.1) From our assumptions, R > 0. Hence V~p = 0. This implies that I~k F2 is a nonzero constant function on M and so we must also have R = 0. Then as K > 0, we must have K = 0. This implies, from the integral formula for A'(M) [14, p. 231], that A"(M) = O. [] The spin condition is necessary in Lemma 1.l, as can be seen in the case of M = CP 2k. The Ricci-analog of Lemma 1.1 is false, as can be seen in the case of M = K3. Definition 1.2. A connected closed manifold M is almost-nonnegatively-curved if for every E > 0, there is a Riemannian metric g on M such that K(M, g) diam(M, g)2 > -E.
    [Show full text]
  • NON-EXISTENCE of METRIC on Tn with POSITIVE SCALAR
    NON-EXISTENCE OF METRIC ON T n WITH POSITIVE SCALAR CURVATURE CHAO LI Abstract. In this note we present Gromov-Lawson's result on the non-existence of metric on T n with positive scalar curvature. Historical results Scalar curvature is one of the simplest invariants of a Riemannian manifold. In general dimensions, this function (the average of all sectional curvatures at a point) is a weak measure of the local geometry, hence it's suspicious that it has no relation to the global topology of the manifold. In fact, a result of Kazdan-Warner in 1975 states that on a compact manifold of dimension ≥ 3, every smooth function which is negative somewhere, is the scalar curvature of some Riemannian metric. However people know that there are manifolds which carry no metric whose scalar scalar curvature is everywhere positive. The first examples of such manifolds were given in 1962 by Lichnerowicz. It is known that if X is a compact spin manifold and A^ 6= 0 then by Lichnerowicz formula (which will be discussed later) then X doesn't carry any metric with everywhere positive scalar curvature. Note that spin assumption is essential here, since the complex projective plane has positive sectional curvature and non-zero A^-genus. Despite these impressive results, one simple question remained open: Can the torus T n, n ≥ 3, carry a metric of positive scalar curvature? This question was settled for n ≤ 7 by R. Schoen and S. T. Yau. Their method involves the existence of smooth solution of Plateau problem, so the dimension restriction n ≤ 7 is essential.
    [Show full text]
  • LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The
    LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class of fibrations given by fiber bundles. Fiber bundles play an important role in many geometric contexts. For example, the Grassmaniann varieties and certain fiber bundles associated to Stiefel varieties are central in the classification of vector bundles over (nice) spaces. The fact that fiber bundles are examples of Serre fibrations follows from Theorem ?? which states that being a Serre fibration is a local property. 1. Fiber bundles and principal bundles Definition 6.1. A fiber bundle with fiber F is a map p: E ! X with the following property: every ∼ −1 point x 2 X has a neighborhood U ⊆ X for which there is a homeomorphism φU : U × F = p (U) such that the following diagram commutes in which π1 : U × F ! U is the projection on the first factor: φ U × F U / p−1(U) ∼= π1 p * U t Remark 6.2. The projection X × F ! X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F . By definition, a fiber bundle is a map which is `locally' homeomorphic to a trivial bundle. The homeomorphism φU in the definition is a local trivialization of the bundle, or a trivialization over U. Let us begin with an interesting subclass. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R ! S1 is a covering projection with fiber Z. Suppose X is a space which is path-connected and locally simply connected (in fact, the weaker condition of being semi-locally simply connected would be enough for the following construction).
    [Show full text]
  • Hodge Decomposition
    Hodge Decomposition Daniel Lowengrub April 27, 2014 1 Introduction All of the content in these notes in contained in the book Differential Analysis on Complex Man- ifolds by Raymond Wells. The primary objective here is to highlight the steps needed to prove the Hodge decomposition theorems for real and complex manifolds, in addition to providing intuition as to how everything fits together. 1.1 The Decomposition Theorem On a given complex manifold X, there are two natural cohomologies to consider. One is the de Rham Cohomology which can be defined on a general, possibly non complex, manifold. The second one is the Dolbeault cohomology which uses the complex structure. We’ll quickly go over the definitions of these cohomologies in order to set notation but for a precise discussion I recommend Huybrechts or Wells. If X is a manifold, we can define the de Rahm Complex to be the chain complex 0 E(Ω0)(X) −d E(Ω1)(X) −d ::: Where Ω = T∗ denotes the cotangent vector bundle and Ωk is the alternating product Ωk = ΛkΩ. In general, we’ll use the notation! E(E) to! denote the sheaf! of sections associated to a vector bundle E. The boundary operator is the usual differentiation operator. The de Rham cohomology of the manifold X is defined to be the cohomology of the de Rham chain complex: d Ker(E(Ωn)(X) − E(Ωn+1)(X)) n (X R) = HdR , d Im(E(Ωn-1)(X) − E(Ωn)(X)) ! We can also consider the de Rham cohomology with complex coefficients by tensoring the de Rham complex with C in order to obtain the de Rham complex! with complex coefficients 0 d 1 d 0 E(ΩC)(X) − E(ΩC)(X) − ::: where ΩC = Ω ⊗ C and the differential d is linearly extended.
    [Show full text]
  • Gsm038-Endmatter.Pdf
    http://dx.doi.org/10.1090/gsm/038 Selected Titles in This Series 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V.
    [Show full text]
  • Dirac Operators for Matrix Algebras Converging to Coadjoint Orbits
    DIRAC OPERATORS FOR MATRIX ALGEBRAS CONVERGING TO COADJOINT ORBITS MARC A. RIEFFEL Abstract. In the high-energy physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. In the present paper we provide a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. This enables us to prove our main theorem, whose content is that, for the quantum metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance. Contents Introduction 2 1. Ergodic actions and Dirac operators 5 2. Invariance under the group actions 9 3. Charge Conjugation 11 4. Casimir operators 12 5. First facts about spinors 14 6. The C*-metrics 16 7. More about spinors 20 8. Dirac operators for matrix algebras 21 arXiv:2108.01136v1 [math.OA] 2 Aug 2021 9. The fuzzy sphere 23 10. Homogeneous spaces and the Dirac operator for G 29 11. Dirac operators for homogeneous spaces 30 12. Complex structure on coadjoint orbits 36 13. Yet more about spinors 40 14.
    [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]
  • 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]