DOCSLIB.ORG

On the Relation Between Quadratic and Linear Curvature Lagrangians in Poincar E Gauge Gravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Relation Between Quadratic and Linear Curvature Lagrangians in Poincar E Gauge Gravity ON THE RELATION BETWEEN QUADRATIC AND LINEAR CURVATURE LAGRANGIANS IN POINCAREGAUGE GRAVITY y Yuri N. Obukhov and Friedrich W. Hehl Institute for Theoretical Physics, University of Cologne, D{50923 Koln, Germany Abstract We discuss the choice of the Lagrangian in the Poincare gauge theory of gravity. Drawing analogies to earlier de Sitter gauge mo dels, we p oint out the p ossibility of deriving the Einstein-Cartan Lagrangian without cosmological term from a modi ed quadratic curvature invariantof topological typ e. PACS no.: 04.50.+h; 04.20.Cv Typ eset using REVT X E In: New Ideas in the Theory of Fundamental Interactions. Pro ceedings of the Second German{ Polish Symp osium, Zakopane 1995. H-D. Do ebner, M. Paw lowski, R. Raczk a (eds.) to b e published. y On leave from: Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia. 1 1. INTRODUCTION One of the main achievements of the gauge approachtogravity (see [1] [5] and references therein) lies in a b etter understanding of the deep relations b etween the symmetry groups of spacetime and the nature of the sources of the gravitational eld. At the same time a satisfactory kinematical picture of gauge gravity emerges which sp eci es metric, coframe, and connection as the fundamental gravitational eld variables. In contrast, the dynamical asp ect of the gravitational gauge theory is far less develop ed. In general, the choice of a dynamical scheme, i.e. of the gravitational eld Lagrangian, ranges from the simplest Einstein-Cartan mo del with the Hilb ert typ e Lagrangian (linear in curvature) to the 15-parameter theory with Lagrangian quadratic in torsion and curvature, or even to non-p olynomial mo dels. Some progress was achieved in gauge theories based on the de Sitter group [6] [11] which is, in a sense, the closest semi-simple \relative" of the (non-semi-simple) Poincare group. The main idea b ehind the derivation of the gravitational eld Lagrangian was to consider it as emerging, via a certain sp ontaneous breakdown symmetry mechanism, from a unique invariant, the Chern-Pontryagin or the Euler invariant, e.g., whichhave the meaning of top ological charges. Recently this approach has b een reanalyzed in [12]. In this pap er we rep ort on an attempt to exploit the analogy with the de Sitter gauge approach. In a Riemann-Cartan spacetime, we construct a gravitational Lagrangian by starting from a top ologica l invariant quadratic in curvature, deform it suitably, and arrive, apart from an exact form, at an Einstein-Cartan Lagrangian (linear in curvature). Whereas in the traditional approach of (4.6) the emergence of a cosmological term cannot b e pre- vented, our new metho d, see our main result (4.9), yields a pure Einstein-Cartan Lagrangian without cosmological term. 2 2. TWOFOUR-DIMENSIONAL TOPOLOGICAL INVARIANTS The spacetime whichwe consider ob eys a Riemann-Cartan geometry with orthonormal i coframe # , a metric g = o # # , and a Lorentz connection = = dx . Here i ^ ^ ^ ^ the anholonomic frame indices are denoted by ; ;::: = 0;1;2;3, the holonomic co ordinate indices by i;j;::: =0;1;2;3, and o = diag (1; +1; +1; +1) is the lo cal Minkowski metric (with the help of whichwe raise and lower Greek indices). As it is well known, in four dimensions there are two top ological invariants which are constructed from the (in general, Riemann-Cartan) curvature two-form R = d ^ . These are the Euler invariant de ned by the four-form 1 ? E := R ^ R = R ^ R ; (2.1) 2 and the Chern{Pontryagin invariant describ ed by the four-form P := R ^ R = R ^ R : (2.2) Both forms (2.1) and (2.2) are functionals of the Lorentz connection and of the lo cal metric o . The is the Levi-Civita tensor, and no any other geometrical variables are ? involved. The right star denotes the so-called Lie dual with resp ect to the Lie algebra indices. The Gauss-Bonnet theorem states that an integral of (2.1), with a prop er normalization constant, over a compact manifold without a b oundary describ es its Euler characteristics (the alternating sum of the Betti numb ers which count the simplexes in an arbitrary triangulatio n of the manifold). As for the integral of (2.2), also intro ducing prop er normalization, this represents the familiar \instanton" numb er sp ecialized to the gravitational gauge case. 3. ORDINARY AND TWISTED DEFORMATIONS OF THE CURVATURE Due to the p eculiar prop erties of the Lie algebra of the de Sitter group, a new ob ject app ears within the framework of de Sitter gauge gravity part of the generalized SO(1; 4) or SO(2; 3) curvature, a two-form 3 1 := R # ; with # := # ^ # : (3.1) 2 ` Wemay call it a deformation of the original curvature form by a sp eci c contribution constructed from the translational gauge p otentials, namely the coframe one-forms # . The constant ` with the dimension of length provides the correct dimension. If we recall that the curvature of a Riemann-Cartan spacetime can b e decomp osed into six irreducible pieces (N ) R , with N =1;:::;6, see [4], then we nd that # is prop ortional to the sixth pieces (6) R , the curvature scalar, that is, in (3.1) we subtracted a certain constant scalar curvature piece from the total curvature. Similarly to (3.1), by means of the Ho dge star, we can de ne another deformation 1 1 R := R = R (# ^ # ) : (3.2) 2 2 ` ` Wemay call this a twisted translational deformation, since has the opp osite parity b ehavior compared to R . In fact, the term is prop ortional to a constant pseudoscalar (3) piece R of the curvature or, in comp onents, to R . In other words, in (3.2) a constant [ ] (3) pseudoscalar curvature piece is subtracted out. Note that R vanishes together with the torsion since, by means of the rst Bianchi identity, (3) DT = R ^ # or DT ^ # = R ^ # ^ # = R ^ # ^ # : (3.3) (3) Hence, in a Riemannian spacetime, R vanishes identically. (M ) Using also the irreducible decomp osition of the torsion into three pieces T , with M =1;2;3, the last equation can b e rewritten as (3) R ^ # ^ # = d (# ^ T ) T ^ T (3) (1) (1) (2) (3) = d # ^ T T ^ T 2 T ^ T : (3.4) For a pro of of this equation see [4] Eq.(B.2.19). Before we consider some Lagrangians in the next section, we develop some algebra for the quadratic expressions of # and .We nd: 4 # ^ # = (# ^ # ) ^ # ^ # =0: (3.5) Moreover, for anytwo-form , wehave =. Consequently,we nd # ^ # = ( # ) ^ # = # ^ # = ^ (3.6) or ^ =0: (3.7) The mixed term can b e expanded as follows: 1 # ^ = # ^ # =12: (3.8) 2 Here is, as usually, the volume four-form. If we transform the Ho dge star into the Lie ? star ,wehave ? = # = # : (3.9) Eventually,we take the Lie star of the last equation: ? = # : (3.10) 4. DEFORMED TOPOLOGICAL INVARIANTS AND THE EINSTEIN-CARTAN LAGRANGIAN Let us calculate the Euler and Pontryagin four-forms with the curvature replaced by the deformed curvature. We denote the Lagrangian of the Einstein-Cartan theory by 1 ^ R : (4.1) L := EC 2 2` Later we will meet similar Lagrangians with substituted by # .We do the corresp onding algebra rst: 5 1 # ^ R = R # ^ # 2 1 (3) = R # ^ # ^ # ^ # = R ^ # ^ # ; (4.2) [ ] 2 2 ^ R = 2` L ; (4.3) EC ? ? 2 # ^ R = # ^ R = ^ R = 2` L ; (4.4) EC ? ? (3) ^ R = ^ R = # ^ R = R ^ # ^ # : (4.5) In the formulas (4.2) and (4.5) it is of course p ossible to substitute the rst Bianchi identity (3.4) in order to splitt o a b oundary term, if desireable. For the deformations (3.1) and (3.2) one nds, resp ectively, the following generalized Euler forms: 12 ? ; (4.6) V = ^ = E +4L + Eu EC 4 ` 2 12 0 ? (3) V = R ^R = E + ; (4.7) R ^ # ^ # Eu 2 4 ` ` 1 00 ? (3) V = ^R = E +2L + R ^ # ^ # : (4.8) EC Eu 2 ` It is interesting to note that the translational Chern{Simons term # ^ T [13], via (3.4), ? app ears as b oundary term in (4.7) and (4.8). The other mixed term, R ^ ,isthe same as that in (4.8), since the Lie star can b e moved to R . Three more generalized top ological Lagrangians are de ned according to the Chern- Pontryagin pattern as follows: V = R ^R = P +4L ; (4.9) Po EC 2 0 (3) = ^ = P V R ^ # ^ # ; (4.10) Po 2 ` 1 12 00 (3) = ^R = P +2L V : (4.11) R ^ # ^ # + EC Po 2 4 ` ` As we can see, b oth deformed curvatures, (3.1) and (3.2), generate the Einstein-Cartan Lagrangian (4.1) from the top ological typ e invariants, since the variational derivatives of E and P are identically zero. In the case of one should use the Euler typ e form (4.6), while for R the Chern-Pontryagin typ e invariant (4.9) suggests itself. Actually, the case 6 (4.6) was studied in the work of MacDowell and Mansouri [6] (see also [7] [12]). The 4 problem of this de Sitter gauge approachwas a very large cosmological constant 1=` which is generated simultaneously with the Einstein-Cartan Lagrangian. To the b est of our knowledge, the p ossibili ty (4.9) of using the twisted deformation of the curvature was not rep orted in the literature, even if Mielke [14] had somewhat related thoughts, see his Eq.(9.8). A nice improvement of the usual de Sitter result is then the absence of the cosmological term in (4.9).
Load more

Recommended publications
  • Kähler Manifolds, Ricci Curvature, and Hyperkähler Metrics
    K¨ahlermanifolds, Ricci curvature, and hyperk¨ahler metrics Jeff A. Viaclovsky June 25-29, 2018 Contents 1 Lecture 1 3 1.1 The operators @ and @ ..........................4 1.2 Hermitian and K¨ahlermetrics . .5 2 Lecture 2 7 2.1 Complex tensor notation . .7 2.2 The musical isomorphisms . .8 2.3 Trace . 10 2.4 Determinant . 11 3 Lecture 3 11 3.1 Christoffel symbols of a K¨ahler metric . 11 3.2 Curvature of a Riemannian metric . 12 3.3 Curvature of a K¨ahlermetric . 14 3.4 The Ricci form . 15 4 Lecture 4 17 4.1 Line bundles and divisors . 17 4.2 Hermitian metrics on line bundles . 18 5 Lecture 5 21 5.1 Positivity of a line bundle . 21 5.2 The Laplacian on a K¨ahlermanifold . 22 5.3 Vanishing theorems . 25 6 Lecture 6 25 6.1 K¨ahlerclass and @@-Lemma . 25 6.2 Yau's Theorem . 27 6.3 The @ operator on holomorphic vector bundles . 29 1 7 Lecture 7 30 7.1 Holomorphic vector fields . 30 7.2 Serre duality . 32 8 Lecture 8 34 8.1 Kodaira vanishing theorem . 34 8.2 Complex projective space . 36 8.3 Line bundles on complex projective space . 37 8.4 Adjunction formula . 38 8.5 del Pezzo surfaces . 38 9 Lecture 9 40 9.1 Hirzebruch Signature Theorem . 40 9.2 Representations of U(2) . 42 9.3 Examples . 44 2 1 Lecture 1 We will assume a basic familiarity with complex manifolds, and only do a brief review today. Let M be a manifold of real dimension 2n, and an endomorphism J : TM ! TM satisfying J 2 = −Id.
    [Show full text]
  • Quantum Dynamical Manifolds - 3A
    Quantum Dynamical Manifolds - 3a. Cold Dark Matter and Clifford’s Geometrodynamics Conjecture D.F. Scofield ApplSci, Inc. 128 Country Flower Rd. Newark, DE, 19711 scofield.applsci.com (June 11, 2001) The purpose of this paper is to explore the concepts of gravitation, inertial mass and spacetime from the perspective of the geometrodynamics of “mass-spacetime” manifolds (MST). We show how to determine the curvature and torsion 2-forms of a MST from the quantum dynamics of its constituents. We begin by reformulating and proving a 1876 conjecture of W.K. Clifford concerning the nature of 4D MST and gravitation, independent of the standard formulations of general relativity theory (GRT). The proof depends only on certain derived identities involving the Riemann curvature 2-form in a 4D MST having torsion. This leads to a new metric-free vector-tensor fundamental law of dynamical gravitation, linear in pairs of field quantities which establishes a homeomorphism between the new theory of gravitation and electromagnetism based on the mean curvature potential. Its generalization leads to Yang-Mills equations. We then show the structure of the Riemann curvature of a 4D MST can be naturally interpreted in terms of mass densities and currents. By using the trace curvature invariant, we find a parity noninvariance is possible using a de Sitter imbedding of the Poincar´egroup. The new theory, however, is equivalent to the Newtonian one in the static, large scale limit. The theory is compared to GRT and shown to be compatible outside a gravitating mass in the absence of torsion. We apply these results on the quantum scale to determine the gravitational field of leptons.
    [Show full text]
  • Differential Geometry: Curvature and Holonomy Austin Christian
    University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics.
    [Show full text]
  • A Theorem on Holonomy
    A THEOREM ON HOLONOMY BY W. AMBROSE AND I. M. SINGER Introduction. The object of this paper is to prove Theorem 2 of §2, which shows, for any connexion, how the curvature form generates the holonomy group. We believe this is an extension of a theorem stated without proof by E. Cartan [2, p. 4]. This theorem was proved after we had been informed of an unpublished related theorem of Chevalley and Koszul. We are indebted to S. S. Chern for many discussions of matters considered here. In §1 we give an exposition of some needed facts about connexions; this exposition is derived largely from an exposition of Chern [5 ] and partly from expositions of H. Cartan [3] and Ehresmann [7]. We believe this exposition does however contain one new element, namely Lemma 1 of §1 and its use in passing from H. Cartan's definition of a connexion (the definition given in §1) to E. Cartan's structural equation. 1. Basic concepts. We begin with some notions and terminology to be used throughout this paper. The term "differentiable" will always mean what is usually called "of class C00." We follow Chevalley [6] in general for the definition of tangent vector, differential, etc. but with the obvious changes needed for the differentiable (rather than analytic) case. However if <p is a differentiable mapping we use <j>again instead of d<f>and 8<(>for the induced mappings of tangent vectors and differentials. If AT is a manifold, by which we shall always mean a differentiable manifold but which is not assumed connected, and mGM, we denote the tangent space to M at m by Mm.
    [Show full text]
  • Instantons on Cylindrical Manifolds 3
    Instantons on Cylindrical Manifolds Teng Huang Abstract 2 We consider an instanton, A , with L -curvature FA on the cylindrical manifold Z = R M, where M is a closed Riemannian n-manifold, n 4. We assume M × ≥ admits a 3-form P and a 4-form Q satisfy dP = 4Q and d M Q = (n 3) M P . ∗ − ∗ Manifolds with these forms include nearly K¨ahler 6-manifolds and nearly parallel G2-manifolds in dimension 7. Then we can prove that the instanton must be a flat connection. Keywords. instantons, special holonomy manifolds, Yang-Mills connection 1 Introduction Let X be an (n + 1)-dimensional Riemannian manifold, G be a compact Lie group and E be a principal G-bundle on X. Let A denote a connection on E with the curvature FA. The instanton equation on X can be introduced as follows. Assume there is a 4-form Q on X. Then an (n 3)-form Q exists, where is the Hodge operator on X. A connection, arXiv:1501.04525v3 [math.DG] 7 Mar 2016 − ∗ ∗ A , is called an anti-self-dual instanton, when it satisfies the instanton equation F + Q F =0 (1.1) ∗ A ∗ ∧ A When n + 1 > 4, these equations can be defined on the manifold X with a special holonomy group, i.e. the holonomy group G of the Levi-Civita connection on the tangent bundle TX is a subgroup of the group SO(n +1). Each solution of equation(1.1) satisfies the Yang-Mills equation. The instanton equation (1.1) is also well-defined on a manifold X with non-integrable G-structures, but equation (1.1) implies the Yang-Mills equation will have torsion.
    [Show full text]
  • Cartan Connection and Curvature Forms
    CARTAN CONNECTION AND CURVATURE FORMS Syafiq Johar syafi[email protected] Contents 1 Connections and Riemannian Curvature 1 1.1 Affine Connection . .1 1.2 Levi-Civita Connection and Riemannian Curvature . .2 1.3 Riemannian Curvature . .3 2 Covariant External Derivative 4 2.1 Connection Form . .4 2.2 Curvature Form . .5 2.3 Explicit Formula for E = TM ..................................5 2.4 Levi-Civita Connection and Riemannian Curvature Revisited . .6 3 An Application: Uniformisation Theorem via PDEs 6 1 Connections and Riemannian Curvature Cartan formula is a way of generalising the Riemannian curvature for a Riemannian manifold (M; g) of dimension n to a differentiable manifold. We start by defining the Riemannian manifold and curvatures. We begin by defining the usual notion of connection and the Levi-Civita connection. 1.1 Affine Connection Definition 1.1 (Affine 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) with the 1 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. D is tensorial over TpM i.e. DfV +W σ = fDV σ + DW σ: 2. D is R-linear in Γ(E) i.e. DV (σ + τ) = DV σ + DV τ: 3. D satisfies the Leibniz rule over Γ(E) i.e. if f 2 C1(M), we have DV (fσ) = df(V ) · σ + fDV σ: Note that we did not require any condition of the metric g on the manifold, nor the bundle metric on E in the abstract definition of a connection.
    [Show full text]
  • RICCI CURVATURES on HERMITIAN MANIFOLDS Contents 1. Introduction 1 2. Background Materials 9 3. Geometry of the Levi-Civita Ricc
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0 RICCI CURVATURES ON HERMITIAN MANIFOLDS KEFENG LIU AND XIAOKUI YANG Abstract. In this paper, we introduce the first Aeppli-Chern class for com- plex manifolds and show that the (1, 1)- component of the curvature 2-form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-K¨ahler Calabi-Yau manifolds by using the first Aeppli- Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct 2n−1 1 explicit Levi-Civita Ricci-flat metrics on Hopf manifolds S × S . We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and nonnegative, but their Rie- mannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian man- ifolds with nonnegative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature. Contents 1. Introduction 1 2. Background materials 9 3. Geometry of the Levi-Civita Ricci curvature 12 4. Curvature relations on Hermitian manifolds 22 5. Special metrics on Hermitian manifolds 26 6. Levi-Civita Ricci-flat and constant negative scalar curvature metrics on Hopf manifolds 29 7. Appendix: the Riemannian Ricci curvature and ∗-Ricci curvature 35 References 37 1.
    [Show full text]
  • Connections with Skew-Symmetric Ricci Tensor on Surfaces
    Connections with skew-symmetric Ricci tensor on surfaces Andrzej Derdzinski Abstract. Some known results on torsionfree connections with skew-symmet- ric Ricci tensor on surfaces are extended to connections with torsion, and Wong’s canonical coordinate form of such connections is simplified. Mathematics Subject Classification (2000). 53B05; 53C05, 53B30. Keywords. Skew-symmetric Ricci tensor, locally homogeneous connection, left-invariant connection on a Lie group, curvature-homogeneity, neutral Ein- stein metric, Jordan-Osserman metric, Petrov type III. 1. Introduction This paper generalizes some results concerning the situation where ∇ is a connection on a surface Σ, and the Ricci (1.1) tensor ρ of ∇ is skew symmetric at every point. What is known about condition (1.1) can be summarized as follows. Norden [18], [19, §89] showed that, for a torsionfree connection ∇ on a surface, skew-sym- metry of the Ricci tensor is equivalent to flatness of the connection obtained by projectivizing ∇, and implies the existence of a fractional-linear first integral for the geodesic equation. Wong [25, Theorem 4.2] found three coordinate expressions which, locally, represent all torsionfree connections ∇ with (1.1) such that ρ 6= 0 everywhere in Σ. Kowalski, Opozda and Vl´aˇsek [13] used an approach different from Wong’s to classify, locally, all torsionfree connections ∇ satisfying (1.1) that are also locally homogeneous, while in [12] they proved that, for real-analytic tor- sionfree connections ∇ with (1.1), third-order curvature-homogeneity implies local homogeneity (but one cannot replace the word ‘third’ with ‘second’). Blaˇzi´cand Bokan [2] showed that the torus T 2 is the only closed surface Σ admitting both a torsionfree connection ∇ with (1.1) and a ∇-parallel almost complex structure.
    [Show full text]
  • Torsion Constraints in Supergeometry
    Commun. Math. Phys. 133, 563-615(1990) Communications ΪΠ Mathematical Physics ©Springer-Verlagl990 Torsion Constraints in Supergeometry John Lott* I.H.E.S., F-91440 Bures-sur-Yvette, France Received November 9, 1989; in revised form April 2, 1990 Abstract. We derive the torsion constraints for superspace versions of supergravity theories by means of the theory of G-stmctures. We also discuss superconformal geometry and superKahler geometry. I. Introduction Supersymmetry is a now well established topic in quantum field theory [WB, GGRS]. The basic idea is that one can construct actions in ordinary spacetime which involve both even commuting fields and odd anticommuting fields, with a symmetry which mixes the two types of fields. These actions can then be interpreted as arising from actions in a superspace with both even and odd coordinates, upon doing a partial integration over the odd coordinates. A mathematical framework to handle the differential topology of supermanifolds, manifolds with even and odd coordinates, was developed by Berezin, Kostant and others. A very readable account of this theory is given in the book of Manin [Ma]. The right notion of differential geometry for supermanifolds is less clear. Such a geometry is necessary in order to write supergravity theories in superspace. One could construct a supergeometry by ΊL2 grading what one usually does in (pseudo) Riemannian geometry, to have supermetrics, super Levi-Civita connections, etc. The local frame group which would take the place of the orthogonal group in standard geometry would be the orthosymplectic group. However, it turns out that this would be physically undesirable. Such a program would give more fields than one needs for a minimal supergravity theory, i.e.
    [Show full text]
  • Math in Moscow Program Differential Geometry Problem Sets
    Math in Moscow program Differential Geometry Problem Sets by Prof. Maxim Kazarian Spring 2020 Differential Geometry Problem Set I 14.02.2020 @ @ 1. Find the expression for the basic vector fields @x and @y , and the basic differential 1-forms dx, dy in the polar coordinates on the plane. 2. Draw the phase portrait and find the phase flow for the following vector fields on the plane @ (a) @x ; @ @ (b) x @x + y @y ; @ @ (c) x @y − y @y ; @ @ (d) x @y − y @x ; @ @ (e) x @y + y @x . 3. Let N and E be the unit length vector fields on the sphere directed to the North and to the East, respectively. Compare the commutator of their flows with the flow of their commutator. 4. Check that the distribution defined by the following 1-form α is integrable. Find a func- tion f such that df = g α, g 6= 0. (a) α = z dx + z dy + 2(1 − x − y) dz; (b) α = 2 y z dx + 2 x z dy + (1 − xy) dz. P 5. Find the coordinate expression for the commutator of the vector fields A = ai@xi and P n B = bi@xi in R . 6. The coordinateless definition of the differential of a 1-form is du(A; B) = @A(u(B)) − @B(u(A)) − u([A; B]): Check that the expression on the right-hand side is bilinear with respect to the multiplica- tion of the fields A and B by functions. Differential Geometry Problem Set II 21.02.2020 2 Parametrization of a plane curve τ 7! γ(τ) 2 R is called natural if jγ_ j ≡ 1.
    [Show full text]
  • Connections and Curvature
    C Connections and Curvature Introduction In this appendix we present results in differential geometry that serve as a useful background for material in the main body of the book. Material in 1 on connections is somewhat parallel to the study of the natural connec- tion§ on a Riemannian manifold made in 11 of Chapter 1, but here we also study the curvature of a connection. Material§ in 2 on second covariant derivatives is connected with material in Chapter 2 on§ the Laplace operator. Ideas developed in 3 and 4, on the curvature of Riemannian manifolds and submanifolds, make§§ contact with such material as the existence of com- plex structures on two-dimensional Riemannian manifolds, established in Chapter 5, and the uniformization theorem for compact Riemann surfaces and other problems involving nonlinear, elliptic PDE, arising from studies of curvature, treated in Chapter 14. Section 5 on the Gauss-Bonnet theo- rem is useful both for estimates related to the proof of the uniformization theorem and for applications to the Riemann-Roch theorem in Chapter 10. Furthermore, it serves as a transition to more advanced material presented in 6–8. In§§ 6 we discuss how constructions involving vector bundles can be de- rived§ from constructions on a principal bundle. In the case of ordinary vector fields, tensor fields, and differential forms, one can largely avoid this, but it is a very convenient tool for understanding spinors. The principal bundle picture is used to construct characteristic classes in 7. The mate- rial in these two sections is needed in Chapter 10, on the index§ theory for elliptic operators of Dirac type.
    [Show full text]
  • Riemannian Submersions and Instantons
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ,Mrrrlirmot,w/ ModP//r,t~. vol. 1. pp. 381-393. 1980 0?70-0?55/80/040381-13 $0?.0010 Prmted in the U.S.A. All rights reserved. Copyright G 1981 Pergamon Press Ltd. RIEMANNIAN SUBMERSIONS AND INSTANTONS BILL WATSON* Department of Mathematics College of Charleston Charleston, South Carolina 29401. USA 1. INTRODUCTION Certainly the mbst important aspects of theoretical particle physics during the 1970’s have been the insights and extensions provided by nonabelian gauge theories. It has been known for some time that the quantized Dirac field, locally invariant under U(l), leads to a presentation of Maxwell’s equations in a concise and illuminating form. The exten- sion of the electromagnetic theory to nonabelian groups [e.g., SU(2)l was accomplished in the seminal paper of Yang and Mills [ll. The fibre bundle model for classical gauge theories probably had its origins in the work of Lubkin [2]. A review of the later devel- opments may be found in Mayer [3]. Continuing the theme of the foundational paper of Atiyah, Hitchin, and Singer [41, bringing the power of real differential geometry to bear on the problem of finding self- dual solutions of the Yang-Mills field equations, the present study outlines a new, totally real formalism for instantons. 2. FIBRE SPACES AND FIBRE BUNDLES Since our principal differential geometric objects will be smooth locally trival Riemann- ian fibre spaces, we recapture here some of the pertinent ideas.
    [Show full text]