LECTURE 23: the STOKES FORMULA 1. Volume Forms Last
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Introduction to Gauge Theory Arxiv:1910.10436V1 [Math.DG] 23
Introduction to Gauge Theory Andriy Haydys 23rd October 2019 This is lecture notes for a course given at the PCMI Summer School “Quantum Field The- ory and Manifold Invariants” (July 1 – July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg–Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too. Contents 1 Introduction2 2 Bundles and connections4 2.1 Vector bundles . .4 2.1.1 Basic notions . .4 2.1.2 Operations on vector bundles . .5 2.1.3 Sections . .6 2.1.4 Covariant derivatives . .6 2.1.5 The curvature . .8 2.1.6 The gauge group . 10 2.2 Principal bundles . 11 2.2.1 The frame bundle and the structure group . 11 2.2.2 The associated vector bundle . 14 2.2.3 Connections on principal bundles . 16 2.2.4 The curvature of a connection on a principal bundle . 19 arXiv:1910.10436v1 [math.DG] 23 Oct 2019 2.2.5 The gauge group . 21 2.3 The Levi–Civita connection . 22 2.4 Classification of U(1) and SU(2) bundles . 23 2.4.1 Complex line bundles . 24 2.4.2 Quaternionic line bundles . 25 3 The Chern–Weil theory 26 3.1 The Chern–Weil theory . 26 3.1.1 The Chern classes . 28 3.2 The Chern–Simons functional . 30 3.3 The modui space of flat connections . 32 3.3.1 Parallel transport and holonomy . -
Recognizing Surfaces
RECOGNIZING SURFACES Ivo Nikolov and Alexandru I. Suciu Mathematics Department College of Arts and Sciences Northeastern University Abstract The subject of this poster is the interplay between the topology and the combinatorics of surfaces. The main problem of Topology is to classify spaces up to continuous deformations, known as homeomorphisms. Under certain conditions, topological invariants that capture qualitative and quantitative properties of spaces lead to the enumeration of homeomorphism types. Surfaces are some of the simplest, yet most interesting topological objects. The poster focuses on the main topological invariants of two-dimensional manifolds—orientability, number of boundary components, genus, and Euler characteristic—and how these invariants solve the classification problem for compact surfaces. The poster introduces a Java applet that was written in Fall, 1998 as a class project for a Topology I course. It implements an algorithm that determines the homeomorphism type of a closed surface from a combinatorial description as a polygon with edges identified in pairs. The input for the applet is a string of integers, encoding the edge identifications. The output of the applet consists of three topological invariants that completely classify the resulting surface. Topology of Surfaces Topology is the abstraction of certain geometrical ideas, such as continuity and closeness. Roughly speaking, topol- ogy is the exploration of manifolds, and of the properties that remain invariant under continuous, invertible transforma- tions, known as homeomorphisms. The basic problem is to classify manifolds according to homeomorphism type. In higher dimensions, this is an impossible task, but, in low di- mensions, it can be done. Surfaces are some of the simplest, yet most interesting topological objects. -
The Divergence Theorem Cartan's Formula II. for Any Smooth Vector
The Divergence Theorem Cartan’s Formula II. For any smooth vector field X and any smooth differential form ω, LX = iX d + diX . Lemma. Let x : U → Rn be a positively oriented chart on (M,G), with volume j ∂ form vM , and X = Pj X ∂xj . Then, we have U √ 1 n ∂( gXj) L v = di v = √ X , X M X M g ∂xj j=1 and √ 1 n ∂( gXj ) tr DX = √ X . g ∂xj j=1 Proof. (i) We have √ 1 n LX vM =diX vM = d(iX gdx ∧···∧dx ) n √ =dX(−1)j−1 gXjdx1 ∧···∧dxj ∧···∧dxn d j=1 n √ = X(−1)j−1d( gXj) ∧ dx1 ∧···∧dxj ∧···∧dxn d j=1 √ n ∂( gXj ) = X dx1 ∧···∧dxn ∂xj j=1 √ 1 n ∂( gXj ) =√ X v . g ∂xj M j=1 ∂X` ` k ∂ (ii) We have D∂/∂xj X = P` ∂xj + Pk Γkj X ∂x` , which implies ∂X` tr DX = X + X Γ` Xk. ∂x` k` ` k Since 1 Γ` = X g`r{∂ g + ∂ g − ∂ g } k` 2 k `r ` kr r k` r,` 1 == X g`r∂ g 2 k `r r,` √ 1 ∂ g ∂ g = k = √k , 2 g g √ √ ∂X` X` ∂ g 1 n ∂( gXj ) tr DX = X + √ k } = √ X . ∂x` g ∂x` g ∂xj ` j=1 Typeset by AMS-TEX 1 2 Corollary. Let (M,g) be an oriented Riemannian manifold. Then, for any X ∈ Γ(TM), d(iX dvg)=tr DX =(div X)dvg. Stokes’ Theorem. Let M be a smooth, oriented n-dimensional manifold with boundary. Let ω be a compactly supported smooth (n − 1)-form on M. -
2 Non-Orientable Surfaces §
2 NON-ORIENTABLE SURFACES § 2 Non-orientable Surfaces § This section explores stranger surfaces made from gluing diagrams. Supplies: Glass Klein bottle • Scarf and hat • Transparency fish • Large pieces of posterboard to cut • Markers • Colored paper grid for making the room a gluing diagram • Plastic tubes • Mobius band templates • Cube templates from Exploring the Shape of Space • 24 Mobius Bands 2 NON-ORIENTABLE SURFACES § Mobius Bands 1. Cut a blank sheet of paper into four long strips. Make one strip into a cylinder by taping the ends with no twist, and make a second strip into a Mobius band by taping the ends together with a half twist (a twist through 180 degrees). 2. Mark an X somewhere on your cylinder. Starting at the X, draw a line down the center of the strip until you return to the starting point. Do the same for the Mobius band. What happens? 3. Make a gluing diagram for a cylinder by drawing a rectangle with arrows. Do the same for a Mobius band. 4. The gluing diagram you made defines a virtual Mobius band, which is a little di↵erent from a paper Mobius band. A paper Mobius band has a slight thickness and occupies a small volume; there is a small separation between its ”two sides”. The virtual Mobius band has zero thickness; it is truly 2-dimensional. Mark an X on your virtual Mobius band and trace down the centerline. You’ll get back to your starting point after only one trip around! 25 Multiple twists 2 NON-ORIENTABLE SURFACES § 5. -
GEOMETRY FINAL 1 (A): If M Is Non-Orientable and P ∈ M, Is M
GEOMETRY FINAL CLAY SHONKWILER 1 (a): If M is non-orientable and p ∈ M, is M − {p} orientable? Answer: No. Suppose M − {p} is orientable, and let (Uα, xα) be an atlas that gives an orientation on M − {p}. Now, let (V, y) be a coordinate chart on M (in some atlas) containing the point p. If (U, x) is a coordinate chart such that U ∩ V 6= ∅, then, since x and y are both smooth maps with smooth inverses, the composition x ◦ y−1 : y(U ∩ V ) → x(U ∩ V ) is a smooth map. Hence, the collection {(Uα, xα), (V, α)} gives an atlas on M. Now, since M − {p} is orientable, i ! ∂xα det j > 0 ∂xβ for all α, β. However, since M is non-orientable, there must exist coordinate charts where the Jacobian is negative on the overlap; clearly, one of each such pair must be (V, y). Hence, there exists β such that ! ∂yi det j ≤ 0. ∂xβ Suppose this is true for all β such that Uβ ∩V 6= ∅. Then, since y(q) = (y1(q), . , yn(q)), lety ˜ be such thaty ˜(p) = (y2(p), y1(p), y3(p), y4(p), . , yn(p)). Theny ˜ : V → y˜(V ) is a diffeomorphism, so (V, y˜) gives a coordinate chart containing p and ! ! ∂y˜i ∂yi det j = − det j ≥ 0 ∂xβ ∂xβ for all β such that Uβ ∩ V 6= ∅. Hence, unless equality holds in some case, {(Uα, xα), (V, y˜)} defines an orientation on M, contradicting the fact that M is non-orientable. Additionally, it’s clear that equality can never hold, since y : Uβ ∩ V → y(Uβ ∩ V ) is a diffeomorphism and, thus, has full rank. -
3. Introducing Riemannian Geometry
3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4). -
Hodge Theory
HODGE THEORY PETER S. PARK Abstract. This exposition of Hodge theory is a slightly retooled version of the author's Harvard minor thesis, advised by Professor Joe Harris. Contents 1. Introduction 1 2. Hodge Theory of Compact Oriented Riemannian Manifolds 2 2.1. Hodge star operator 2 2.2. The main theorem 3 2.3. Sobolev spaces 5 2.4. Elliptic theory 11 2.5. Proof of the main theorem 14 3. Hodge Theory of Compact K¨ahlerManifolds 17 3.1. Differential operators on complex manifolds 17 3.2. Differential operators on K¨ahlermanifolds 20 3.3. Bott{Chern cohomology and the @@-Lemma 25 3.4. Lefschetz decomposition and the Hodge index theorem 26 Acknowledgments 30 References 30 1. Introduction Our objective in this exposition is to state and prove the main theorems of Hodge theory. In Section 2, we first describe a key motivation behind the Hodge theory for compact, closed, oriented Riemannian manifolds: the observation that the differential forms that satisfy certain par- tial differential equations depending on the choice of Riemannian metric (forms in the kernel of the associated Laplacian operator, or harmonic forms) turn out to be precisely the norm-minimizing representatives of the de Rham cohomology classes. This naturally leads to the statement our first main theorem, the Hodge decomposition|for a given compact, closed, oriented Riemannian manifold|of the space of smooth k-forms into the image of the Laplacian and its kernel, the sub- space of harmonic forms. We then develop the analytic machinery|specifically, Sobolev spaces and the theory of elliptic differential operators|that we use to prove the aforementioned decom- position, which immediately yields as a corollary the phenomenon of Poincar´eduality. -
Equivariant Orientation Theory
EQUIVARIANT ORIENTATION THEORY S.R. COSTENOBLE, J.P. MAY, AND S. WANER Abstract. We give a long overdue theory of orientations of G-vector bundles, topological G-bundles, and spherical G-fibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equivariant orien- tation such that every orientable G-vector bundle admits an orientation. Our focus here is on the geometric and homotopical aspects, rather than the coho- mological aspects, of orientation theory. Orientations are described in terms of functors defined on equivariant fundamental groupoids of base G-spaces, and the essence of the theory is to construct an appropriate universal tar- get category of G-vector bundles over orbit spaces G=H. The theory requires new categorical concepts and constructions that should be of interest in other subjects, such as algebraic geometry. Contents Introduction 2 Part I. Fundamental groupoids and categories of bundles 5 1. The equivariant fundamental groupoid 5 2. Categories of G-vector bundles and orientability 6 3. The topologized fundamental groupoid 8 4. The topologized category of G-vector bundles over orbits 9 Part II. Categorical representation theory and orientations 11 5. Bundles of Groupoids 11 6. Skeletal, faithful, and discrete bundles of groupoids 13 7. Representations and orientations of bundles of groupoids 16 8. Saturated and supersaturated representations 18 9. Universal orientable representations 23 Part III. Examples of universal orientable representations 26 10. Cyclic groups of prime order 26 11. Orientations of V -dimensional G-bundles 30 12. -
ON a THEOREM by FARB and MASUR Let Σg Be the Closed
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 12, Pages 3463{3464 S 0002-9939(00)05450-2 Article electronically published on June 7, 2000 ON A THEOREM BY FARB AND MASUR KOJI FUJIWARA (Communicated by Ronald A. Fintushel) Abstract. Farb and Masur showed that an irreducible lattice in a semisimple Lie group of rank at least two always has finite image by a homomorphism into the outer automorphism group of a closed, orientable surface group. We point out that their theorem extends to the outer automorphism groups of a certain class of torsion-free, freely indecomposable word-hyperbolic groups. Let Σg be the closed, orientable surface of genus g and let Mod(Σg)denotethe mapping class group of Σg. Farb and Masur proved the following in [FM]. Theorem 1 (Farb-Masur). Let Γ be an irreducible lattice in a semisimple Lie group of rank at least two. Let h :Γ! Mod(Σg) be a homomorphism. Then Im(h) is finite. Their proof uses a deep work on the Poisson boundary of mapping class groups by Kaimanovich-Masur [KM]. Let G be a torsion-free, freely indecomposable word-hyperbolic group [G]. By Sela, there is a graph of groups decomposition of G such that the edge groups are isomorphic to Z and the vertex groups are in two families; one is a collection of the fundamental groups of punctured surfaces fSigi, which are called CMQ subgroups, and the other is rigid in the sense that the outer automorphism groups are finite. This graph decomposition, which is called JSJ-decomposition of G, is unique in a certain sense; in particular the set fSigi is an invariant of G (Theorem 1.7 of [S]). -
Gauge Theory
Preprint typeset in JHEP style - HYPER VERSION 2018 Gauge Theory David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html [email protected] Contents 0. Introduction 1 1. Topics in Electromagnetism 3 1.1 Magnetic Monopoles 3 1.1.1 Dirac Quantisation 4 1.1.2 A Patchwork of Gauge Fields 6 1.1.3 Monopoles and Angular Momentum 8 1.2 The Theta Term 10 1.2.1 The Topological Insulator 11 1.2.2 A Mirage Monopole 14 1.2.3 The Witten Effect 16 1.2.4 Why θ is Periodic 18 1.2.5 Parity, Time-Reversal and θ = π 21 1.3 Further Reading 22 2. Yang-Mills Theory 26 2.1 Introducing Yang-Mills 26 2.1.1 The Action 29 2.1.2 Gauge Symmetry 31 2.1.3 Wilson Lines and Wilson Loops 33 2.2 The Theta Term 38 2.2.1 Canonical Quantisation of Yang-Mills 40 2.2.2 The Wavefunction and the Chern-Simons Functional 42 2.2.3 Analogies From Quantum Mechanics 47 2.3 Instantons 51 2.3.1 The Self-Dual Yang-Mills Equations 52 2.3.2 Tunnelling: Another Quantum Mechanics Analogy 56 2.3.3 Instanton Contributions to the Path Integral 58 2.4 The Flow to Strong Coupling 61 2.4.1 Anti-Screening and Paramagnetism 65 2.4.2 Computing the Beta Function 67 2.5 Electric Probes 74 2.5.1 Coulomb vs Confining 74 2.5.2 An Analogy: Flux Lines in a Superconductor 78 { 1 { 2.5.3 Wilson Loops Revisited 85 2.6 Magnetic Probes 88 2.6.1 't Hooft Lines 89 2.6.2 SU(N) vs SU(N)=ZN 92 2.6.3 What is the Gauge Group of the Standard Model? 97 2.7 Dynamical Matter 99 2.7.1 The Beta Function Revisited 100 2.7.2 The Infra-Red Phases of QCD-like Theories 102 2.7.3 The Higgs vs Confining Phase 105 2.8 't Hooft-Polyakov Monopoles 109 2.8.1 Monopole Solutions 112 2.8.2 The Witten Effect Again 114 2.9 Further Reading 115 3. -
A Primer on Exterior Differential Calculus
Theoret. Appl. Mech., Vol. 30, No. 2, pp. 85-162, Belgrade 2003 A primer on exterior differential calculus D.A. Burton¤ Abstract A pedagogical application-oriented introduction to the cal- culus of exterior differential forms on differential manifolds is presented. Stokes’ theorem, the Lie derivative, linear con- nections and their curvature, torsion and non-metricity are discussed. Numerous examples using differential calculus are given and some detailed comparisons are made with their tradi- tional vector counterparts. In particular, vector calculus on R3 is cast in terms of exterior calculus and the traditional Stokes’ and divergence theorems replaced by the more powerful exte- rior expression of Stokes’ theorem. Examples from classical continuum mechanics and spacetime physics are discussed and worked through using the language of exterior forms. The nu- merous advantages of this calculus, over more traditional ma- chinery, are stressed throughout the article. Keywords: manifolds, differential geometry, exterior cal- culus, differential forms, tensor calculus, linear connections ¤Department of Physics, Lancaster University, UK (e-mail: [email protected]) 85 86 David A. Burton Table of notation M a differential manifold F(M) set of smooth functions on M T M tangent bundle over M T ¤M cotangent bundle over M q TpM type (p; q) tensor bundle over M ΛpM differential p-form bundle over M ΓT M set of tangent vector fields on M ΓT ¤M set of cotangent vector fields on M q ΓTpM set of type (p; q) tensor fields on M ΓΛpM set of differential p-forms -
Integration on Manifolds
CHAPTER 4 Integration on Manifolds §1 Introduction Until now we have studied manifolds from the point of view of differential calculus. The last chapter introduces to our study the methods of integraI calculus. As the new tools are developed in the next sections, the reader may be somewhat puzzied about their reievancy to the earlierd material. Hopefully, the puzziement will be resolved by the chapter's en , but a pre liminary ex ampIe may be heIpful. Let p(z) = zm + a1zm-1 + + a ... m n, be a complex polynomial on p.a smooth compact region in the pIane whose boundary contains no zero of In Sectionp 3 of the previous chapter we showed that the number of zeros of inside n, counting multiplicities, equals the degree of the map argument principle, A famous theorem in complex variabie theory, called the asserts that this may also be calculated as an integraI, f d(arg p). òO 151 152 CHAPTER 4 TNTEGRATION ON MANIFOLDS You needn't understand the precise meaning of the expression in order to appreciate our point. What is important is that the number of zeros can be computed both by an intersection number and by an integraI formula. r Theo ems like the argument principal abound in differential topology. We shall show that their occurrence is not arbitrary orStokes fo rt uitoustheorem. but is a geometrie consequence of a generaI theorem known as Low dimensionaI versions of Stokes theorem are probably familiar to you from calculus : l. Second Fundamental Theorem of Calculus: f: f'(x) dx = I(b) -I(a).