Lecture 2: Homotopy Invariance We Give Two Proofs of the Following
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Stable Higgs Bundles on Ruled Surfaces 3
STABLE HIGGS BUNDLES ON RULED SURFACES SNEHAJIT MISRA Abstract. Let π : X = PC(E) −→ C be a ruled surface over an algebraically closed field k of characteristic 0, with a fixed polarization L on X. In this paper, we show that pullback of a (semi)stable Higgs bundle on C under π is a L-(semi)stable Higgs bundle. Conversely, if (V,θ) ∗ is a L-(semi)stable Higgs bundle on X with c1(V ) = π (d) for some divisor d of degree d on C and c2(V ) = 0, then there exists a (semi)stable Higgs bundle (W, ψ) of degree d on C whose pullback under π is isomorphic to (V,θ). As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on X whenever g(C) ≥ 2 and the base field is C. 1. Introduction A Higgs bundle on an algebraic variety X is a pair (V, θ) consisting of a vector bundle V 1 over X together with a Higgs field θ : V −→ V ⊗ ΩX such that θ ∧ θ = 0. Higgs bundle comes with a natural stability condition (see Definition 2.3 for stability), which allows one to study the moduli spaces of stable Higgs bundles on X. Higgs bundles on Riemann surfaces were first introduced by Nigel Hitchin in 1987 and subsequently, Simpson extended this notion on higher dimensional varieties. Since then, these objects have been studied by many authors, but very little is known about stability of Higgs bundles on ruled surfaces. -
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). -
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]. -
Group Invariant Solutions Without Transversality 2 in Detail, a General Method for Characterizing the Group Invariant Sections of a Given Bundle
GROUP INVARIANT SOLUTIONS WITHOUT TRANSVERSALITY Ian M. Anderson Mark E. Fels Charles G. Torre Department of Mathematics Department of Mathematics Department of Physics Utah State University Utah State University Utah State University Logan, Utah 84322 Logan, Utah 84322 Logan, Utah 84322 Abstract. We present a generalization of Lie’s method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations. The char- acterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics, harmonic maps, and general relativity. This work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions. Keywords. Lie symmetry reduction, group invariant solutions, kinematic reduction diagram, dy- namic reduction diagram. arXiv:math-ph/9910015v2 13 Apr 2000 February , Research supported by NSF grants DMS–9403788 and PHY–9732636 1. Introduction. Lie’s method of symmetry reduction for finding the group invariant solutions to partial differential equations is widely recognized as one of the most general and effective methods for obtaining exact solutions of non-linear partial differential equations. -
Composable Geometric Motion Policies Using Multi-Task Pullback Bundle Dynamical Systems
Composable Geometric Motion Policies using Multi-Task Pullback Bundle Dynamical Systems Andrew Bylard, Riccardo Bonalli, Marco Pavone Abstract— Despite decades of work in fast reactive plan- ning and control, challenges remain in developing reactive motion policies on non-Euclidean manifolds and enforcing constraints while avoiding undesirable potential function local minima. This work presents a principled method for designing and fusing desired robot task behaviors into a stable robot motion policy, leveraging the geometric structure of non- Euclidean manifolds, which are prevalent in robot configuration and task spaces. Our Pullback Bundle Dynamical Systems (PBDS) framework drives desired task behaviors and prioritizes tasks using separate position-dependent and position/velocity- dependent Riemannian metrics, respectively, thus simplifying individual task design and modular composition of tasks. For enforcing constraints, we provide a class of metric-based tasks, eliminating local minima by imposing non-conflicting potential functions only for goal region attraction. We also provide a geometric optimization problem for combining tasks inspired by Riemannian Motion Policies (RMPs) that reduces to a simple least-squares problem, and we show that our approach is geometrically well-defined. We demonstrate the 2 PBDS framework on the sphere S and at 300-500 Hz on a manipulator arm, and we provide task design guidance and an open-source Julia library implementation. Overall, this work Fig. 1: Example tree of PBDS task mappings designed to move a ball along presents a fast, easy-to-use framework for generating motion the surface of a sphere to a goal while avoiding obstacles. Depicted are policies without unwanted potential function local minima on manifolds representing: (black) joint configuration for a 7-DoF robot arm general manifolds. -
4 Fibered Categories (Aaron Mazel-Gee) Contents
4 Fibered categories (Aaron Mazel-Gee) Contents 4 Fibered categories (Aaron Mazel-Gee) 1 4.0 Introduction . .1 4.1 Definitions and basic facts . .1 4.2 The 2-Yoneda lemma . .2 4.3 Categories fibered in groupoids . .3 4.3.1 ... coming from co/groupoid objects . .3 4.3.2 ... and 2-categorical fiber product thereof . .4 4.0 Introduction In the same way that a sheaf is a special sort of functor, a stack will be a special sort of sheaf of groupoids (or a special special sort of groupoid-valued functor). It ends up being advantageous to think of the groupoid associated to an object X as living \above" X, in large part because this perspective makes it much easier to study the relationships between the groupoids associated to different objects. For this reason, we use the language of fibered categories. We note here that throughout this exposition we will often say equal (as opposed to isomorphic), and we really will mean it. 4.1 Definitions and basic facts φ Definition 1. Let C be a category. A category over C is a category F with a functor p : F!C. A morphism ξ ! η p(φ) in F is called cartesian if for any other ζ 2 F with a morphism ζ ! η and a factorization p(ζ) !h p(ξ) ! p(η) of φ p( ) in C, there is a unique morphism ζ !λ η giving a factorization ζ !λ η ! η of such that p(λ) = h. Pictorially, ζ - η w - w w 9 w w ! w w λ φ w w w w - w w w w ξ w w w w w w p(ζ) w p(η) w w - w h w ) w (φ w p w - p(ξ): In this case, we call ξ a pullback of η along p(φ). -
Arxiv:1505.02430V1 [Math.CT] 10 May 2015 Bundle Functors and Fibrations
Bundle functors and fibrations Anders Kock Introduction The notions of bundle, and bundle functor, are useful and well exploited notions in topology and differential geometry, cf. e.g. [12], as well as in other branches of mathematics. The category theoretic set up relevant for these notions is that of fibred category, likewise a well exploited notion, but for certain considerations in the context of bundle functors, it can be carried further. In particular, we formalize and develop, in terms of fibred categories, some of the differential geometric con- structions: tangent- and cotangent bundles, (being examples of bundle functors, respectively star-bundle functors, as in [12]), as well as jet bundles (where the for- mulation of the functorality properties, in terms of fibered categories, is probably new). Part of the development in the present note was expounded in [11], and is repeated almost verbatim in the Sections 2 and 4 below. These sections may have interest as a piece of pure category theory, not referring to differential geometry. 1 Basics on Cartesian arrows We recall here some classical notions. arXiv:1505.02430v1 [math.CT] 10 May 2015 Let π : X → B be any functor. For α : A → B in B, and for objects X,Y ∈ X with π(X) = A and π(Y ) = B, let homα (X,Y) be the set of arrows h : X → Y in X with π(h) = α. The fibre over A ∈ B is the category, denoted XA, whose objects are the X ∈ X with π(X) = A, and whose arrows are arrows in X which by π map to 1A; such arrows are called vertical (over A). -
Lecture 3: Field Theories Over a Manifold, and Smooth Field Theories
Notre Dame Graduate Student Topology Seminar, Spring 2018 Lecture 3: Field theories over a manifold, and smooth field theories As mentioned last time, the goal is to show that n ∼ n 0j1-TFT (X) = Ωcl(X); n ∼ n 0j1-TFT [X] = HdR(X) Last time, we showed that: Ω∗(X) =∼ C1(ΠTX) ∼ 1 0j1 = C (SMan(R ;X) Today I want to introduce two other ingredients we'll need: • field theories over a manifold • smoothness of field theories via fibered categories 3.1 Field theories over a manifold Field theories over a manifold X were introduced by Segal. The idea is to give a family of field theories parametrized by X. Note that this is a general mathematical move that is familiar; for example, instead of just considering vector spaces, we consider families of vector spaces, i.e. vector bundles, which are families of vector spaces parametrized by a space, satisfying certain additional conditions. In the vector bundle example, we can recover a vector space from a vector bundle E over a space X by taking the fiber Ex over a point x 2 X. The first thing we will discuss is how to extract a field theory as the “fiber over a x 2 X" from a family of field theories parametrized by some manifold X. Recall that in Lecture 1, we discussed the non-linear sigma-model as giving a path integral motivation for the axioms of functorial field theories. The 1-dimensional non-linear sigma model with target M n did the following: σM : 1 − RBord −! V ect pt 7−! C1(M) [0; t] 7−! e−t∆M : C1(M) ! C1(M) The Feynman-Kan formula gave us a path integral description of this operator: for x 2 M, we had Z eS(φ)Dφ (e−t∆f)(x) = f(φ(0)) Z fφ:[0;t]!Xjφ(t)=xg If instead of having just one target manifold M, how does this change when one has a whole family of manifolds fMxg? 1 A first naive approach might be to simply take this X-family of Riemannian manifolds, fMxg. -
Chapter 3 Connections
Chapter 3 Connections Contents 3.1 Parallel transport . 69 3.2 Fiber bundles . 72 3.3 Vector bundles . 75 3.3.1 Three definitions . 75 3.3.2 Christoffel symbols . 80 3.3.3 Connection 1-forms . 82 3.3.4 Linearization of a section at a zero . 84 3.4 Principal bundles . 88 3.4.1 Definition . 88 3.4.2 Global connection 1-forms . 89 3.4.3 Frame bundles and linear connections . 91 3.1 The idea of parallel transport A connection is essentially a way of identifying the points in nearby fibers of a bundle. One can see the need for such a notion by considering the following question: Given a vector bundle π : E ! M, a section s : M ! E and a vector X 2 TxM, what is meant by the directional derivative ds(x)X? If we regard a section merely as a map between the manifolds M and E, then one answer to the question is provided by the tangent map T s : T M ! T E. But this ignores most of the structure that makes a vector 69 70 CHAPTER 3. CONNECTIONS bundle interesting. We prefer to think of sections as \vector valued" maps on M, which can be added and multiplied by scalars, and we'd like to think of the directional derivative ds(x)X as something which respects this linear structure. From this perspective the answer is ambiguous, unless E happens to be the trivial bundle M × Fm ! M. In that case, it makes sense to think of the section s simply as a map M ! Fm, and the directional derivative is then d s(γ(t)) − s(γ(0)) ds(x)X = s(γ(t)) = lim (3.1) dt t!0 t t=0 for any smooth path with γ_ (0) = X, thus defining a linear map m ds(x) : TxM ! Ex = F : If E ! M is a nontrivial bundle, (3.1) doesn't immediately make sense because s(γ(t)) and s(γ(0)) may be in different fibers and cannot be added. -
Surface Bundles in Topology, Algebraic Geometry, and Group Theory
Surface Bundles in Topology, Algebraic Geometry, and Group Theory Nick Salter and Bena Tshishiku Surface Bundles 푆 and whose structure group is the group Diff(푆) of dif- A surface is one of the most basic objects in topology, but feomorphisms of 푆. In particular, 퐵 is covered by open −1 the mathematics of surfaces spills out far beyond its source, sets {푈훼} on which the bundle is trivial 휋 (푈훼) ≅ 푈훼 × 푆, penetrating deeply into fields as diverse as algebraic ge- and local trivializations are glued by transition functions ometry, complex analysis, dynamics, hyperbolic geometry, 푈훼 ∩ 푈훽 → Diff(푆). geometric group theory, etc. In this article we focus on the Although the bundle is locally trivial, any nontrivial mathematics of families of surfaces: surface bundles. While bundle is globally twisted, similar in spirit to the Möbius the basics belong to the study of fiber bundles, we hope strip (Figure 1). This twisting is recorded in an invariant to illustrate how the theory of surface bundles comes into called the monodromy representation to be discussed in the close contact with a broad range of mathematical ideas. section “Monodromy.” We focus here on connections with three areas—algebraic topology, algebraic geometry, and geometric group theory—and see how the notion of a surface bundle pro- vides a meeting ground for these fields to interact in beau- tiful and unexpected ways. What is a surface bundle? A surface bundle is a fiber bun- dle 휋 ∶ 퐸 → 퐵 whose fiber is a 2-dimensional manifold Nick Salter is a Ritt Assistant Professor at Columbia University.