The Atiyah-Singer Index Theorem*
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Realizing Cohomology Classes As Euler Classes 1
Ó DOI: 10.2478/s12175-012-0057-2 Math. Slovaca 62 (2012), No. 5, 949–966 REALIZING COHOMOLOGY CLASSES AS EULER CLASSES Aniruddha C Naolekar (Communicated by J´ulius Korbaˇs ) s ◦ ABSTRACT. For a space X,letEk(X), Ek(X)andEk (X) denote respectively the set of Euler classes of oriented k-plane bundles over X, the set of Euler classes of stably trivial k-plane bundles over X and the spherical classes in Hk(X; Z). s ◦ We prove some general facts about the sets Ek(X), Ek(X)andEk (X). We also compute these sets in the cases where X is a projective space, the Dold manifold P (m, 1) and obtain partial computations in the case that X is a product of spheres. c 2012 Mathematical Institute Slovak Academy of Sciences 1. Introduction Given a topological space X, we address the question of which classes x ∈ Hk(X; Z) can be realized as the Euler class e(ξ)ofanorientedk-plane bundle ξ over X. We also look at the question of which integral cohomology classes are spherical. It is convenient to make the following definition. s º ≥ ÒØÓÒ 1.1 For a space X and an integer k 1, the sets Ek(X), Ek(X) ◦ and Ek (X) are defined to be k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is an oriented k-plane bundle over X s k Ek(X)= e(ξ) ∈ H (X; Z) | ξ is a stably trivial k-plane bundle over X ◦ k k ∗ Ek (X)= x ∈ H (X; Z) | there exists f : S −→ X with f (x) =0 . -
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. -
Fermionic Finite-Group Gauge Theories and Interacting Symmetric
Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms Meng Guo1;2;3, Kantaro Ohmori4, Pavel Putrov4;5, Zheyan Wan6, Juven Wang4;7 1Department of Mathematics, Harvard University, Cambridge, MA 02138, USA 2Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada 3Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada 4School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA 5 ICTP, Trieste 34151, Italy 6School of Mathematical Sciences, USTC, Hefei 230026, China 7Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA, USA Abstract We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging G-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group G symmetry. We use the fact that the latter are classified by Pontryagin duals to spin-bordism groups of the classifying space BG. We also consider unoriented analogues, that is G-equivariant invertible pin±-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian G using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in dimensions 3, 4, and other. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of 't Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary deconfined spin-TQFTs (surface fermionic topological orders). -
K-Theoryand Characteristic Classes
MATH 6530: K-THEORY AND CHARACTERISTIC CLASSES Taught by Inna Zakharevich Notes by David Mehrle [email protected] Cornell University Fall 2017 Last updated November 8, 2018. The latest version is online here. Contents 1 Vector bundles................................ 3 1.1 Grassmannians ............................ 8 1.2 Classification of Vector bundles.................... 11 2 Cohomology and Characteristic Classes................. 15 2.1 Cohomology of Grassmannians................... 18 2.2 Characteristic Classes......................... 22 2.3 Axioms for Stiefel-Whitney classes................. 26 2.4 Some computations.......................... 29 3 Cobordism.................................. 35 3.1 Stiefel-Whitney Numbers ...................... 35 3.2 Cobordism Groups.......................... 37 3.3 Geometry of Thom Spaces...................... 38 3.4 L-equivalence and Transversality.................. 42 3.5 Characteristic Numbers and Boundaries.............. 47 4 K-Theory................................... 49 4.1 Bott Periodicity ............................ 49 4.2 The K-theory spectrum........................ 56 4.3 Some properties of K-theory..................... 58 4.4 An example: K-theory of S2 ..................... 59 4.5 Power Operations............................ 61 4.6 When is the Hopf Invariant one?.................. 64 4.7 The Splitting Principle........................ 66 5 Where do we go from here?........................ 68 5.1 The J-homomorphism ........................ 70 5.2 The Chern Character and e invariant............... -
Arxiv:1910.04634V1 [Math.DG] 10 Oct 2019 ˆ That E Sdnt by Denote Us Let N a En H Subbundle the Define Can One of Points Bundle
SPIN FRAME TRANSFORMATIONS AND DIRAC EQUATIONS R.NORIS(1)(2), L.FATIBENE(2)(3) (1) DISAT, Politecnico di Torino, C.so Duca degli Abruzzi 24, I-10129 Torino, Italy (2)INFN Sezione di Torino, Via Pietro Giuria 1, I-10125 Torino, Italy (3) Dipartimento di Matematica – University of Torino, via Carlo Alberto 10, I-10123 Torino, Italy Abstract. We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer re- quirements, transformations allowed by spin frames are more general than usual spin transformations and they usually do not preserve the induced metric structures. We study how these new transformations affect connections both on the spin bundle and on the frame bundle and how this reflects on the Dirac equations. 1. Introduction Dirac equations provide an important tool to study the geometric structure of manifolds, as well as to model the behaviour of a class of physical particles, namely fermions, which includes electrons. The aim of this paper is to generalise a key item needed to formulate Dirac equations, the spin structures, in order to extend the range of allowed transformations. Let us start by first reviewing the usual approach to Dirac equations. Let (M,g) be an orientable pseudo-Riemannian manifold with signature η = (r, s), such that r + s = m = dim(M). R arXiv:1910.04634v1 [math.DG] 10 Oct 2019 Let us denote by L(M) the (general) frame bundle of M, which is a GL(m, )-principal fibre bundle. -
Characteristic Classes and K-Theory Oscar Randal-Williams
Characteristic classes and K-theory Oscar Randal-Williams https://www.dpmms.cam.ac.uk/∼or257/teaching/notes/Kthy.pdf 1 Vector bundles 1 1.1 Vector bundles . 1 1.2 Inner products . 5 1.3 Embedding into trivial bundles . 6 1.4 Classification and concordance . 7 1.5 Clutching . 8 2 Characteristic classes 10 2.1 Recollections on Thom and Euler classes . 10 2.2 The projective bundle formula . 12 2.3 Chern classes . 14 2.4 Stiefel–Whitney classes . 16 2.5 Pontrjagin classes . 17 2.6 The splitting principle . 17 2.7 The Euler class revisited . 18 2.8 Examples . 18 2.9 Some tangent bundles . 20 2.10 Nonimmersions . 21 3 K-theory 23 3.1 The functor K ................................. 23 3.2 The fundamental product theorem . 26 3.3 Bott periodicity and the cohomological structure of K-theory . 28 3.4 The Mayer–Vietoris sequence . 36 3.5 The Fundamental Product Theorem for K−1 . 36 3.6 K-theory and degree . 38 4 Further structure of K-theory 39 4.1 The yoga of symmetric polynomials . 39 4.2 The Chern character . 41 n 4.3 K-theory of CP and the projective bundle formula . 44 4.4 K-theory Chern classes and exterior powers . 46 4.5 The K-theory Thom isomorphism, Euler class, and Gysin sequence . 47 n 4.6 K-theory of RP ................................ 49 4.7 Adams operations . 51 4.8 The Hopf invariant . 53 4.9 Correction classes . 55 4.10 Gysin maps and topological Grothendieck–Riemann–Roch . 58 Last updated May 22, 2018. -
Introduction
Introduction Spin Geometry is the hidden facet of Riemannian Geometry. It arises from the representation theory of the special orthogonal group SOn, more precisely, from the spinor representation, a certain representation of the Lie algebra son which is not a representation of SOn. Spinors can always be constructed locally on a given Riemannian manifold, but globally there are topological obstructions for their exis- tence. Spin Geometry lies therefore at the cross-road of several subelds of modern Mathematics. Algebra, Geometry, Topology, and Analysis are subtly interwoven in the theory of spinors, both in their denition and in their applications. Spinors have also greatly inuenced Theoretical Physics, which is nowadays one of the main driv- ing forces fostering their formidable development. The Noncommutative Geometry of Alain Connes has at its core the Dirac operator on spinors. The same Dirac operator is at the heart of the Atiyah–Singer index formula for elliptic operators on compact manifolds, linking in a spectacular way the topology of a manifold to the space of solutions to elliptic equations. Signicantly, the classical Riemann–Roch formula and its generalization by Hirzebruch; the Gauß–Bonnet formula and its extension by Chern; and nally Hirzebruch’s topological signature theorem, provide most of the examples in the index formula, but it was the Dirac operator acting on spinors which turned out to be the keystone of the index formula, both in its formulation and in its subsequent developments. The Dirac operator appears to be the primordial example of an elliptic operator, while spinors, although younger than differential forms or tensors, illustrate once again that aux âmes bien nées, la valeur n’attend point le nombre des années. -
Combinatorial Topology and Applications to Quantum Field Theory
Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Vivek Shende, Chair Professor Ian Agol Professor Constantin Teleman Professor Joel Moore Fall 2018 Abstract Combinatorial Topology and Applications to Quantum Field Theory by Ryan George Thorngren Doctor of Philosophy in Mathematics University of California, Berkeley Professor Vivek Shende, Chair Topology has become increasingly important in the study of many-body quantum mechanics, in both high energy and condensed matter applications. While the importance of smooth topology has long been appreciated in this context, especially with the rise of index theory, torsion phenomena and dis- crete group symmetries are relatively new directions. In this thesis, I collect some mathematical results and conjectures that I have encountered in the exploration of these new topics. I also give an introduction to some quantum field theory topics I hope will be accessible to topologists. 1 To my loving parents, kind friends, and patient teachers. i Contents I Discrete Topology Toolbox1 1 Basics4 1.1 Discrete Spaces..........................4 1.1.1 Cellular Maps and Cellular Approximation.......6 1.1.2 Triangulations and Barycentric Subdivision......6 1.1.3 PL-Manifolds and Combinatorial Duality........8 1.1.4 Discrete Morse Flows...................9 1.2 Chains, Cycles, Cochains, Cocycles............... 13 1.2.1 Chains, Cycles, and Homology.............. 13 1.2.2 Pushforward of Chains.................. 15 1.2.3 Cochains, Cocycles, and Cohomology......... -
Arxiv:1404.7224V2 [Hep-Th] 21 Jun 2014 Singer Upre Npr Ytentoa Cec Onainudrg Under Foundation Science Physics
ANOMALIES AND INVERTIBLE FIELD THEORIES DANIEL S. FREED Abstract. We give a modern geometric viewpoint on anomalies in quantum field theory and illustrate it in a 1-dimensional theory: supersymmetric quantum mechanics. This is background for the resolution of worldsheet anomalies in orientifold superstring theory. Contents 1. Introduction 1 2. Anomalies 2 2.1. Fields and field theories: formal view 2 2.2. Anomalies: traditional view 4 2.3. Anomalies: modern view 5 3. Supersymmetric quantum mechanics 6 3.1. Lagrangian anomaly 7 3.2. Hamiltonian anomaly 9 3.3. Trivializing the lagrangian and hamiltonian anomalies 10 3.4. The anomaly as an invertible field theory 12 4. Central simple algebras and topology 13 4.1. Some ko-modules 14 4.2. Some maps between ko-modules 16 5. Supersymmetric QM with a general target 17 References 20 1. Introduction arXiv:1404.7224v2 [hep-th] 21 Jun 2014 The subject of anomalies in quantum field theories is an old one, and it is well-trodden. There is a huge physics literature on this topic of anomalies, for which one entree is [Be]. Important work in the early 1980s [AS1, AgW, AgG, ASZ] tied the study of local anomalies to the Atiyah- Singer topological index theorem, and extensions to global anomalies [W1, W2] were not far behind. These ideas were quickly fit in to geometric invariants in index theory, such as the determinant line bundle and the η-invariant. Indeed, many developments in geometric index theory at that time were directly motivated by the physics. A geometric picture of anomalies emerged from this interaction [F1, 1]. -
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. -
Bundles with Spherical Euler Class
Pacific Journal of Mathematics BUNDLES WITH SPHERICAL EULER CLASS Luis Guijarro, Thomas Schick, and Gerard Walschap Volume 207 No. 2 December 2002 PACIFIC JOURNAL OF MATHEMATICS Vol. 207, No. 2, 2002 BUNDLES WITH SPHERICAL EULER CLASS Luis Guijarro, Thomas Schick, and Gerard Walschap In this note, we show that the holonomy group of a Rie- mannian connection on a k-dimensional Euclidean vector bun- dle is transitive on the unit sphere bundle whenever the Eu- ler class χ is spherical. We extract several consequences from this, among them that this is always the case as long as χ does not vanish, and the base of the bundle is simply connected and rationally (k + 1)/2-connected. 1. Introduction and statement of main results. Recall that every complete, noncompact Riemannian manifold M with non- negative sectional curvature contains a compact, totally geodesic (hence also nonnegatively curved) submanifold, called a soul. The soul theorem of Cheeger and Gromoll ([2]) states that M is then diffeomorphic to the normal bundle of the soul in M. When studying the converse, namely the question of whether a vector bundle over a closed nonnegatively curved manifold admits a complete met- ric which has also nonnegative sectional curvature, there appear to be con- straints on the topology of the bundle which imply that any such metric must be rigid. One such is the following: Definition 1.1. A Euclidean vector bundle with Riemannian connection is said to have transitive holonomy if the holonomy group associated to the connection acts transitively on (any fiber of) the corresponding sphere bundle. -
Spinc GEOMETRY of K¨AHLER MANIFOLDS and the HODGE
SPINc GEOMETRY OF KAHLER¨ MANIFOLDS AND THE HODGE LAPLACIAN ON MINIMAL LAGRANGIAN SUBMANIFOLDS O. HIJAZI, S. MONTIEL, AND F. URBANO Abstract. From the existence of parallel spinor fields on Calabi- Yau, hyper-K¨ahleror complex flat manifolds, we deduce the ex- istence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the sub- manifolds are compact, we obtain sharp estimates on their Betti numbers. When the ambient manifold is K¨ahler-Einstein with pos- itive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of K¨ahlerian Killing spinor fields for some particular spinc struc- tures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. Applications on the Morse index of minimal Lagrangian submanifolds are obtained. 1. Introduction Recently, connections between the spectrum of the classical Dirac operator on submanifolds of a spin Riemannian manifold and its ge- ometry were investigated. Even when the submanifold is spin, many problems appear. In fact, it is known that the restriction of the spin bundle of a spin manifold M to a spin submanifold is a Hermitian bun- dle given by the tensorial product of the intrinsic spin bundle of the submanifold and certain bundle associated with the normal bundle of the immersion ([2, 3, 6]). In general, it is not easy to have a control on such a Hermitian bundle. Some results have been obtained ([2, 24, 25]) when the normal bundle of the submanifold is trivial, for instance for hypersurfaces.