On Dirac's Incomplete Analysis of Gauge Transformations
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Vector Bundles on Projective Space
Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X. -
Stable Isomorphism Vs Isomorphism of Vector Bundles: an Application to Quantum Systems
Alma Mater Studiorum · Universita` di Bologna Scuola di Scienze Corso di Laurea in Matematica STABLE ISOMORPHISM VS ISOMORPHISM OF VECTOR BUNDLES: AN APPLICATION TO QUANTUM SYSTEMS Tesi di Laurea in Geometria Differenziale Relatrice: Presentata da: Prof.ssa CLARA PUNZI ALESSIA CATTABRIGA Correlatore: Chiar.mo Prof. RALF MEYER VI Sessione Anno Accademico 2017/2018 Abstract La classificazione dei materiali sulla base delle fasi topologiche della materia porta allo studio di particolari fibrati vettoriali sul d-toro con alcune strut- ture aggiuntive. Solitamente, tale classificazione si fonda sulla nozione di isomorfismo tra fibrati vettoriali; tuttavia, quando il sistema soddisfa alcune assunzioni e ha dimensione abbastanza elevata, alcuni autori ritengono in- vece sufficiente utilizzare come relazione d0equivalenza quella meno fine di isomorfismo stabile. Scopo di questa tesi `efissare le condizioni per le quali la relazione di isomorfismo stabile pu`osostituire quella di isomorfismo senza generare inesattezze. Ci`onei particolari casi in cui il sistema fisico quantis- tico studiato non ha simmetrie oppure `edotato della simmetria discreta di inversione temporale. Contents Introduction 3 1 The background of the non-equivariant problem 5 1.1 CW-complexes . .5 1.2 Bundles . 11 1.3 Vector bundles . 15 2 Stability properties of vector bundles 21 2.1 Homotopy properties of vector bundles . 21 2.2 Stability . 23 3 Quantum mechanical systems 28 3.1 The single-particle model . 28 3.2 Topological phases and Bloch bundles . 32 4 The equivariant problem 36 4.1 Involution spaces and general G-spaces . 36 4.2 G-CW-complexes . 39 4.3 \Real" vector bundles . 41 4.4 \Quaternionic" vector bundles . -
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). -
TRANSVERSE STRUCTURE of LIE FOLIATIONS 0. Introduction This
TRANSVERSE STRUCTURE OF LIE FOLIATIONS BLAS HERRERA, MIQUEL LLABRES´ AND A. REVENTOS´ Abstract. We study if two non isomorphic Lie algebras can appear as trans- verse algebras to the same Lie foliation. In particular we prove a quite sur- prising result: every Lie G-flow of codimension 3 on a compact manifold, with basic dimension 1, is transversely modeled on one, two or countable many Lie algebras. We also solve an open question stated in [GR91] about the realiza- tion of the three dimensional Lie algebras as transverse algebras to Lie flows on a compact manifold. 0. Introduction This paper deals with the problem of the realization of a given Lie algebra as transverse algebra to a Lie foliation on a compact manifold. Lie foliations have been studied by several authors (cf. [KAH86], [KAN91], [Fed71], [Mas], [Ton88]). The importance of this study was increased by the fact that they arise naturally in Molino’s classification of Riemannian foliations (cf. [Mol82]). To each Lie foliation are associated two Lie algebras, the Lie algebra G of the Lie group on which the foliation is modeled and the structural Lie algebra H. The latter algebra is the Lie algebra of the Lie foliation F restricted to the clousure of any one of its leaves. In particular, it is a subalgebra of G. We remark that although H is canonically associated to F, G is not. Thus two interesting problems are naturally posed: the realization problem and the change problem. The realization problem is to know which pairs of Lie algebras (G, H), with H subalgebra of G, can arise as transverse and structural Lie algebras, respectively, of a Lie foliation F on a compact manifold M. -
Functor and Natural Operators on Symplectic Manifolds
Mathematical Methods and Techniques in Engineering and Environmental Science Functor and natural operators on symplectic manifolds CONSTANTIN PĂTRĂŞCOIU Faculty of Engineering and Management of Technological Systems, Drobeta Turnu Severin University of Craiova Drobeta Turnu Severin, Str. Călugăreni, No.1 ROMANIA [email protected] http://www.imst.ro Abstract. Symplectic manifolds arise naturally in abstract formulations of classical Mechanics because the phase–spaces in Hamiltonian mechanics is the cotangent bundle of configuration manifolds equipped with a symplectic structure. The natural functors and operators described in this paper can be helpful both for a unified description of specific proprieties of symplectic manifolds and for finding links to various fields of geometric objects with applications in Hamiltonian mechanics. Key-words: Functors, Natural operators, Symplectic manifolds, Complex structure, Hamiltonian. 1. Introduction. diffeomorphism f : M → N , a vector bundle morphism The geometrics objects (like as vectors, covectors, T*f : T*M→ T*N , which covers f , where −1 tensors, metrics, e.t.c.) on a smooth manifold M, x = x :)*)(()*( x * → xf )( * NTMTfTfT are the elements of the total spaces of a vector So, the cotangent functor, )(:* → VBmManT and bundles with base M. The fields of such geometrics objects are the section in corresponding vector more general ΛkT * : Man(m) → VB , are the bundles. bundle functors from Man(m) to VB, where Man(m) By example, the vectors on a smooth manifold M (subcategory of Man) is the category of m- are the elements of total space of vector bundles dimensional smooth manifolds, the morphisms of this (TM ,π M , M); the covectors on M are the elements category are local diffeomorphisms between of total space of vector bundles (T*M , pM , M). -
Foliations Transverse to Fibers of a Bundle
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 42, Number 2, February 1974 FOLIATIONS TRANSVERSE TO FIBERS OF A BUNDLE J. F. PLANTE Abstract. Consider a fiber bundle where the base space and total space are compact, connected, oriented smooth manifolds and the projection map is smooth. It is shown that if the fiber is null-homologous in the total space, then the existence of a foliation of the total space which is transverse to each fiber and such that each leaf has the same dimension as the base implies that the fundamental group of the base space has exponential growth. Introduction. Let (E, p, B) be a locally trivial fiber bundle where /»:£-*-/? is the projection, E and B are compact, connected, oriented smooth manifolds and p is a smooth map. (By smooth we mean C for some r^l and, henceforth, all maps are assumed smooth.) Let b denote the dimension of B and k denote the dimension of the fiber (thus, dim E= b+k). By a section of the fibration we mean a smooth map a:B-+E such that p o a=idB. It is well known that if a section exists then the fiber over any point in B represents a nontrivial element in Hk(E; Z) since the image of a section is a compact orientable manifold which has intersection number one with the fiber over any point in B. The notion of a section may be generalized as follows. Definition. A polysection of (E,p,B) is a covering projection tr:B-*B together with a map Ç:B-*E such that the following diagram commutes. -
FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. -
Lecture Notes on Foliation Theory
INDIAN INSTITUTE OF TECHNOLOGY BOMBAY Department of Mathematics Seminar Lectures on Foliation Theory 1 : FALL 2008 Lecture 1 Basic requirements for this Seminar Series: Familiarity with the notion of differential manifold, submersion, vector bundles. 1 Some Examples Let us begin with some examples: m d m−d (1) Write R = R × R . As we know this is one of the several cartesian product m decomposition of R . Via the second projection, this can also be thought of as a ‘trivial m−d vector bundle’ of rank d over R . This also gives the trivial example of a codim. d- n d foliation of R , as a decomposition into d-dimensional leaves R × {y} as y varies over m−d R . (2) A little more generally, we may consider any two manifolds M, N and a submersion f : M → N. Here M can be written as a disjoint union of fibres of f each one is a submanifold of dimension equal to dim M − dim N = d. We say f is a submersion of M of codimension d. The manifold structure for the fibres comes from an atlas for M via the surjective form of implicit function theorem since dfp : TpM → Tf(p)N is surjective at every point of M. We would like to consider this description also as a codim d foliation. However, this is also too simple minded one and hence we would call them simple foliations. If the fibres of the submersion are connected as well, then we call it strictly simple. (3) Kronecker Foliation of a Torus Let us now consider something non trivial. -
Reduced Phase Space Quantization and Dirac Observables
Reduced Phase Space Quantization and Dirac Observables T. Thiemann∗ MPI f. Gravitationsphysik, Albert-Einstein-Institut, Am M¨uhlenberg 1, 14476 Potsdam, Germany and Perimeter Institute f. Theoretical Physics, Waterloo, ON N2L 2Y5, Canada Abstract In her recent work, Dittrich generalized Rovelli’s idea of partial observables to construct Dirac observables for constrained systems to the general case of an arbitrary first class constraint algebra with structure functions rather than structure constants. Here we use this framework and propose how to implement explicitly a reduced phase space quantization of a given system, at least in principle, without the need to compute the gauge equivalence classes. The degree of practicality of this programme depends on the choice of the partial observables involved. The (multi-fingered) time evolution was shown to correspond to an automorphism on the set of Dirac observables so generated and interesting representations of the latter will be those for which a suitable preferred subgroup is realized unitarily. We sketch how such a programme might look like for General Relativity. arXiv:gr-qc/0411031v1 6 Nov 2004 We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for General Relativity such that they are spatially diffeomorphism invariant. This has the important consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed Master constraint programme. 1 Introduction It is often stated that there are no Dirac observables known for General Relativity, except for the ten Poincar´echarges at spatial infinity in situations with asymptotically flat boundary conditions. -
Theory of Systems with Constraints
Theory of systems with Constraints Javier De Cruz P´erez Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Abstract: We discuss systems subject to constraints. We review Dirac's classical formalism of dealing with such problems and motivate the definition of objects such as singular and nonsingular Lagrangian, first and second class constraints, and the Dirac bracket. We show how systems with first class constraints can be considered to be systems with gauge freedom. We conclude by studing a classical example of systems with constraints, electromagnetic field I. INTRODUCTION II. GENERAL CONSIDERATIONS Consider a system with a finite number, k , of degrees of freedom is described by a Lagrangian L = L(qs; q_s) that don't depend on t. The action of the system is [3] In the standard case, to obtain the equations of motion, Z we follow the usual steps. If we use the Lagrangian for- S = dtL(qs; q_s) (1) malism, first of all we find the Lagrangian of the system, calculate the Euler-Lagrange equations and we obtain the If we apply the least action principle action δS = 0, we accelerations as a function of positions and velocities. All obtain the Euler-Lagrange equations this is possible if and only if the Lagrangian of the sys- tems is non-singular, if not so the system is non-standard @L d @L − = 0 s = 1; :::; k (2) and we can't apply the standard formalism. Then we say @qs dt @q_s that the Lagrangian is singular and constraints on the ini- tial data occur. -
Commentary on Thurston's Work on Foliations
COMMENTARY ON FOLIATIONS* Quoting Thurston's definition of foliation [F11]. \Given a large supply of some sort of fabric, what kinds of manifolds can be made from it, in a way that the patterns match up along the seams? This is a very general question, which has been studied by diverse means in differential topology and differential geometry. ... A foliation is a manifold made out of striped fabric - with infintely thin stripes, having no space between them. The complete stripes, or leaves, of the foliation are submanifolds; if the leaves have codimension k, the foliation is called a codimension k foliation. In order that a manifold admit a codimension- k foliation, it must have a plane field of dimension (n − k)." Such a foliation is called an (n − k)-dimensional foliation. The first definitive result in the subject, the so called Frobenius integrability theorem [Fr], concerns a necessary and sufficient condition for a plane field to be the tangent field of a foliation. See [Spi] Chapter 6 for a modern treatment. As Frobenius himself notes [Sa], a first proof was given by Deahna [De]. While this work was published in 1840, it took another hundred years before a geometric/topological theory of foliations was introduced. This was pioneered by Ehresmann and Reeb in a series of Comptes Rendus papers starting with [ER] that was quickly followed by Reeb's foundational 1948 thesis [Re1]. See Haefliger [Ha4] for a detailed account of developments in this period. Reeb [Re1] himself notes that the 1-dimensional theory had already undergone considerable development through the work of Poincare [P], Bendixson [Be], Kaplan [Ka] and others. -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG].