Lie Groupoids and Poisson Geometry
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Generalized Connections and Higgs Fields on Lie Algebroids Nijmegen, April 5, 2016
Generalized connections and Higgs fields on Lie algebroids Nijmegen, April 5, 2016 Thierry Masson Centre de Physique Théorique Campus de Luminy, Marseille Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Motivations Discovery of Higgs particle in 2012. ➙ need for a mathematical validation of the Higgs sector in the SM. No clue from “traditional” schemes and tools. NCG: Higgs field is part of a “generalized connection”. Dubois-Violette, M., Kerner, R., and Madore, J. (1990). Noncommutative Differential Geometry and New Models of Gauge Theory. J. Math. Phys. 31, p. 323 Connes, A. and Lott, J. (1991). Particle models and noncommutative geometry. Nucl. Phys. B Proc. Suppl. 18.2, pp. 29–47 Models in NCG can reproduce the Standard Model up to the excitement connected to the diphoton resonance at 750 GeV “seen” by ATLAS and CMS! Mathematical structures difficult to master by particle physicists. Transitive Lie algebroids: ➙ generalized connections, gauge symmetries, Yang-Mills-Higgs models… ➙ Direct filiation from Dubois-Violette, Kerner, and Madore (1990). Mathematics close to “usual” mathematics of Yang-Mills theories. No realistic theory yet. 2 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy How to construct a gauge field theory? The basic ingredients are: 1 A space of local symmetries (space-time dependence): ➙ a gauge group. 2 An implementation of the symmetry on matter fields: ➙ a representation theory. 3 A notion of derivation: ➙ some differential structures. 4 A (gauge compatible) replacement of ordinary derivations: ➙ a covariant derivative. 5 A way to write a gauge invariant Lagrangian density: ➙ action functional. -
Geometric Quantization of Poisson Manifolds Via Symplectic Groupoids
Geometric Quantization of Poisson Manifolds via Symplectic Groupoids Daniel Felipe Bermudez´ Montana~ Advisor: Prof. Alexander Cardona Gu´ıo, PhD. A Dissertation Presented in Candidacy for the Degree of Bachelor of Science in Mathematics Universidad de los Andes Department of Mathematics Bogota,´ Colombia 2019 Abstract The theory of Lie algebroids and Lie groupoids is a convenient framework for study- ing properties of Poisson Manifolds. In this work we approach the problem of geometric quantization of Poisson Manifolds using the theory of symplectic groupoids. We work the obstructions of geometric prequantization and show how they can be understood as Lie algebroid's integrability obstructions. Furthermore, by examples, we explore the complete quantization scheme using polarisations and convolution algebras of Fell line bundles. ii Acknowledgements Words fall short for the the graditute I have for people I will mention. First of all I want to thank my advisors Alexander Cardona Guio and Andres Reyes Lega. Alexander Cardona, his advice and insighfull comments guided through this research. I am extremely thankfull and indebted to him for sharing his expertise and knowledge with me. Andres Reyes, as my advisor has tough me more than I could ever ever give him credit for here. By example he has shown me what a good scientist, educator and mentor should be. To my parents, my fist advisors: Your love, support and caring have provided me with the moral and emotional support to confront every aspect of my life; I owe it all to you. To my brothers: the not that productive time I spent with you have always brought me a lot joy - thank you for allowing me the time to research and write during this work. -
Coisotropic Embeddings in Poisson Manifolds 1
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 7, July 2009, Pages 3721–3746 S 0002-9947(09)04667-4 Article electronically published on February 10, 2009 COISOTROPIC EMBEDDINGS IN POISSON MANIFOLDS A. S. CATTANEO AND M. ZAMBON Abstract. We consider existence and uniqueness of two kinds of coisotropic embeddings and deduce the existence of deformation quantizations of certain Poisson algebras of basic functions. First we show that any submanifold of a Poisson manifold satisfying a certain constant rank condition, already con- sidered by Calvo and Falceto (2004), sits coisotropically inside some larger cosymplectic submanifold, which is naturally endowed with a Poisson struc- ture. Then we give conditions under which a Dirac manifold can be embedded coisotropically in a Poisson manifold, extending a classical theorem of Gotay. 1. Introduction The following two results in symplectic geometry are well known. First: a sub- manifold C of a symplectic manifold (M,Ω) is contained coisotropically in some symplectic submanifold of M iff the pullback of Ω to C has constant rank; see Marle’s work [17]. Second: a manifold endowed with a closed 2-form ω can be em- bedded coisotropically into a symplectic manifold (M,Ω) so that i∗Ω=ω (where i is the embedding) iff ω has constant rank; see Gotay’s work [15]. In this paper we extend these results to the setting of Poisson geometry. Recall that P is a Poisson manifold if it is endowed with a bivector field Π ∈ Γ(∧2TP) satisfying the Schouten-bracket condition [Π, Π] = 0. A submanifold C of (P, Π) is coisotropic if N ∗C ⊂ TC, where the conormal bundle N ∗C is defined as the anni- ∗ hilator of TC in TP|C and : T P → TP is the contraction with the bivector Π. -
Lectures on the Geometry of Poisson Manifolds, by Izu Vaisman, Progress in Math- Ematics, Vol
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 33, Number 2, April 1996 Lectures on the geometry of Poisson manifolds, by Izu Vaisman, Progress in Math- ematics, vol. 118, Birkh¨auser, Basel and Boston, 1994, vi + 205 pp., $59.00, ISBN 3-7643-5016-4 For a symplectic manifold M, an important construct is the Poisson bracket, which is defined on the space of all smooth functions and satisfies the following properties: 1. f,g is bilinear with respect to f and g, 2. {f,g} = g,f (skew-symmetry), 3. {h, fg} =−{h, f }g + f h, g (the Leibniz rule of derivation), 4. {f, g,h} {+ g,} h, f { + }h, f,g = 0 (the Jacobi identity). { { }} { { }} { { }} The best-known Poisson bracket is perhaps the one defined on the function space of R2n by: n ∂f ∂g ∂f ∂g (1) f,g = . { } ∂q ∂p − ∂p ∂q i=1 i i i i X Poisson manifolds appear as a natural generalization of symplectic manifolds: a Poisson manifold isasmoothmanifoldwithaPoissonbracketdefinedonits function space. The first three properties of a Poisson bracket imply that in local coordinates a Poisson bracket is of the form given by n ∂f ∂g f,g = πij , { } ∂x ∂x i,j=1 i j X where πij may be seen as the components of a globally defined antisymmetric contravariant 2-tensor π on the manifold, called a Poisson tensor or a Poisson bivector field. The Jacobi identity can then be interpreted as a nonlinear first-order differential equation for π with a natural geometric meaning. The Poisson tensor # π of a Poisson manifold P induces a bundle map π : T ∗P TP in an obvious way. -
Hausdorff Morita Equivalence of Singular Foliations
Hausdorff Morita Equivalence of singular foliations Alfonso Garmendia∗ Marco Zambony Abstract We introduce a notion of equivalence for singular foliations - understood as suitable families of vector fields - that preserves their transverse geometry. Associated to every singular foliation there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980's. Contents Introduction 2 1 Background on singular foliations and pullbacks 4 1.1 Singular foliations and their pullbacks . .4 1.2 Relation with pullbacks of Lie groupoids and Lie algebroids . .6 2 Hausdorff Morita equivalence of singular foliations 7 2.1 Definition of Hausdorff Morita equivalence . .7 2.2 First invariants . .9 2.3 Elementary examples . 11 2.4 Examples obtained by pushing forward foliations . 12 2.5 Examples obtained from Morita equivalence of related objects . 15 3 Morita equivalent holonomy groupoids 17 3.1 Holonomy groupoids . 17 arXiv:1803.00896v1 [math.DG] 2 Mar 2018 3.2 Morita equivalent holonomy groupoids: the case of projective foliations . 21 3.3 Pullbacks of foliations and their holonomy groupoids . 21 3.4 Morita equivalence for open topological groupoids . 27 3.5 Holonomy transformations . 29 3.6 Further invariants . 30 3.7 A second look at Hausdorff Morita equivalence of singular foliations . 31 4 Further developments 33 4.1 An extended equivalence for singular foliations . 33 ∗KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium. -
Symplectic Operad Geometry and Graph Homology
1 SYMPLECTIC OPERAD GEOMETRY AND GRAPH HOMOLOGY SWAPNEEL MAHAJAN Abstract. A theorem of Kontsevich relates the homology of certain infinite dimensional Lie algebras to graph homology. We formulate this theorem using the language of reversible operads and mated species. All ideas are explained using a pictorial calculus of cuttings and matings. The Lie algebras are con- structed as Hamiltonian functions on a symplectic operad manifold. And graph complexes are defined for any mated species. The general formulation gives us many examples including a graph homology for groups. We also speculate on the role of deformation theory for operads in this setting. Contents 1. Introduction 1 2. Species, operads and reversible operads 6 3. The mating functor 10 4. Overview of symplectic operad geometry 13 5. Cuttings and Matings 15 6. Symplectic operad theory 19 7. Examples motivated by PROPS 23 8. Graph homology 26 9. Graph homology for groups 32 10. Graph cohomology 34 11. The main theorem 38 12. Proof of the main theorem-Part I 39 13. Proof of the main theorem-Part II 43 14. Proof of the main theorem-Part III 47 Appendix A. Deformation quantisation 48 Appendix B. The deformation map on graphs 52 References 54 1. Introduction This paper is my humble tribute to the genius of Maxim Kontsevich. Needless to say, the credit for any new ideas that occur here goes to him, and not me. For how I got involved in this wonderful project, see the historical note (1.3). In the papers [22, 23], Kontsevich defined three Lie algebras and related their homology with classical invariants, including the homology of the group of outer 1 2000 Mathematics Subject Classification. -
Arxiv:1705.00466V2 [Math.DG] 1 Nov 2017 Bet,I Atclrqoins Si Hywr Moh Hsi Bec Is This Smooth
ORBISPACES AS DIFFERENTIABLE STRATIFIED SPACES MARIUS CRAINIC AND JOAO˜ NUNO MESTRE Abstract. We present some features of the smooth structure, and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures - it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek’s framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we ex- tend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complemen- tary exposition to those available, and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions. Contents 1. Introduction 1 2. Background 5 3. Orbispaces as differentiable spaces 13 4. Orbispaces as stratified spaces 27 5. Orbispaces as differentiable stratified spaces 43 References 47 1. Introduction Lie groupoids are geometric objects that generalize Lie groups and smooth man- arXiv:1705.00466v2 [math.DG] 1 Nov 2017 ifolds, and permit a unified approach to the study of several objects of interest in differential geometry, such as Lie group actions, foliations and principal bundles (see for example [9, 37, 44] and the references therein). They have found wide use in Poisson and Dirac geometry (e.g. -
Integrability of Lie Brackets
Annals of Mathematics, 157 (2003), 575–620 Integrability of Lie brackets By Marius Crainic and Rui Loja Fernandes* Abstract In this paper we present the solution to a longstanding problem of dif- ferential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As ap- plications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications. Contents 0. Introduction 1. A-paths and homotopy 1.1. A-paths 1.2. A-paths and connections 1.3. Homotopy of A-paths 1.4. Representations and A-paths 2. The Weinstein groupoid 2.1. The groupoid G(A) 2.2. Homomorphisms 2.3. The exponential map 3. Monodromy 3.1. Monodromy groups 3.2. A second-order monodromy map 3.3. Computing the monodromy 3.4. Measuring the monodromy ∗ The first author was supported in part by NWO and a Miller Research Fellowship. The second author was supported in part by FCT through program POCTI and grant POCTI/1999/MAT/33081. Key words and phrases. Lie algebroid, Lie groupoid. 576 MARIUS CRAINIC AND RUI LOJA FERNANDES 4. Obstructions to integrability 4.1. The main theorem 4.2. The Weinstein groupoid as a leaf space 4.3. Proof of the main theorem 5. Examples and applications 5.1. Local integrability 5.2. Integrability criteria 5.3. Tranversally parallelizable foliations Appendix A. Flows A.1. Flows and infinitesimal flows A.2. -
Integrability of Lie Algebroids by Proper Groupoids
Integrability of Lie algebroids by proper Lie groupoids Lu´ıs Alexandre Meira Fernandes Alves Pereira Disserta¸c˜ao para obten¸c˜aodo Grau de Mestre em Matem´atica e Aplica¸c˜oes J´uri Presidente: Prof. Doutor Miguel Tribolet de Abreu Orientador: Prof. Doutor Rui Ant´onio Loja Fernandes Vogais: Prof. Doutor Gustavo Rui de Oliveira Granja Julho de 2008 Agradecimentos Este trabalho foi apoiado por uma Bolsa de Inicia¸c˜aoCient´ıfica da Funda¸c˜ao para a Ciˆenciae a Tecnologia. 1 Resumo Um crit´eriocl´assico diz-nos que uma ´algebra de Lie (real) g ´ea ´algebra de Lie de um grupo de Lie G compacto se e s´ose existe um produto interno em g que ´einvariante para a ac¸c˜aoadjunta de g em si mesma. O objectivo deste trabalho foi o de investigar se este resultado poderia ser extendido para algebr´oides de Lie, obtendo-se um crit´erioan´alogo caracterizando quando ´eque um algebr´oide de Lie A ´eo algebr´oidede Lie de um grup´oidede Lie G pr´oprio. Teve-se como base a formula¸c˜aode uma conjectura de trabalho afirmando que a existˆenciade um produto interno em A satisfazendo uma certa propriedade de invariˆancia seria uma condi¸c˜aonecess´ariae suficiente para que A fosse in- tegrado por um grup´oide pr´oprio. O trabalho consistiu ent˜aoem decidir se a conjectura era v´alida, e caso contr´ario, porquˆee como falhava. Numa direc¸c˜ao,prov´amosque a existˆenciade um grup´oidede Lie pr´oprio integrando A e satisfazendo algumas condi¸c˜oes razo´aveis implica a existˆencia de um produto interno em A nas condi¸c˜oesda nossa conjectura. -
The Extension Problem for Lie Algebroids on Schemes
The extension problem for Lie algebroids on schemes Ugo Bruzzo SISSA (International School for Advanced Studies), Trieste Universidade Federal da Para´ıba, Jo~aoPessoa, Brazil S~aoPaulo, November 14th, 2019 2nd Workshop of the S~aoPaulo Journal of Mathematical Sciences: J.-L. Koszul in S~aoPaulo, His Work and Legacy Ugo Bruzzo Extensions of Lie algebroids 1/26 Lie algebroids X : a differentiable manifold, or complex manifold, or a noetherian separated scheme over an algebraically closed field | of characteristic zero. Lie algebroid: a vector bundle/coherent sheaf C with a morphism of OX -modules a: C ! ΘX and a |-linear Lie bracket on the sections of C satisfying [s; ft] = f [s; t] + a(s)(f ) t for all sections s; t of C and f of OX . a is a morphism of sheaves of Lie |-algebras ker a is a bundle of Lie OX -algebras Ugo Bruzzo Extensions of Lie algebroids 2/26 Examples A sheaf of Lie algebras, with a = 0 ΘX , with a = id More generally, foliations, i.e., a is injective 1 π Poisson structures ΩX −! ΘX , Poisson-Nijenhuis bracket f!; τg = Lieπ(!)τ − Lieπ(τ)! − dπ(!; τ) Jacobi identity , [[π; π]] = 0 Ugo Bruzzo Extensions of Lie algebroids 3/26 Lie algebroid morphisms 0 f : C ! C a morphism of OX -modules & sheaves of Lie k-algebras f C / C 0 0 a a ΘX ) ker f is a bundle of Lie algebras Ugo Bruzzo Extensions of Lie algebroids 4/26 Derived functors A an abelian category, A 2 Ob(A) Hom(−; A): !Ab is a (contravariant) left exact functor, i.e., if 0 ! B0 ! B ! B00 ! 0 (∗) is exact, then 0 ! Hom(B00; A) ! Hom(B; A) ! Hom(B0; A) is exact Definition I 2 Ob(A) is injective if Hom(−; I ) is exact, i.e., for every exact sequence (*), 0 ! Hom(B00; I ) ! Hom(B; I ) ! Hom(B0; I ) ! 0 is exact Ugo Bruzzo Extensions of Lie algebroids 5/26 Definition The category A has enough injectives if every object in A has an injective resolution 0 ! A ! I 0 ! I 1 ! I 2 ! ::: A abelian category with enough injectives F : A ! B left exact functor Derived functors Ri F : A ! B Ri F (A) = Hi (F (I •)) Example: Sheaf cohomology. -
On Lie Algebroid Actions and Morphisms Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 37, No 4 (1996), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES TAHAR MOKRI On Lie algebroid actions and morphisms Cahiers de topologie et géométrie différentielle catégoriques, tome 37, no 4 (1996), p. 315-331 <http://www.numdam.org/item?id=CTGDC_1996__37_4_315_0> © Andrée C. Ehresmann et les auteurs, 1996, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ C-4/IIERS DE TOPOLOGIE ET Volume XXXVII-4 (1996) GEOJIETRIE DIFFERENTIELLE CATEGORIQUES ON LIE ALGEBROID ACTIONS AND MORPHISMS by Tahar MOKRI Resume: Si AG est l’algebroïde de Lie d’un groupoide de Lie G, nous d6montrons que toute action infinit6simale de AG sur une variété M, par des champs complets de vecteurs fondamentaux, se rel6ve en une unique action du groupoide de Lie G sur M, si les cx- fibres de G sont connexes et simplement connexes. Nous appliquons ensuite ce r6sultat pour int6grer des morphismes d’algebroides de Lie, lorsque la base commune de ces algebroydes est compacte. Introduction It is known [12] that if a finite-dimensional Lie algebra 9 acts on a manifold M with complete infinitesimal generators, then this action arises from a unique action of the connected and simply connected Lie group G whose Lie algebra is Q, on the manifold M. -
LIE GROUPOIDS and LIE ALGEBROIDS LECTURE NOTES, FALL 2017 Contents 1. Lie Groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Ex
LIE GROUPOIDS AND LIE ALGEBROIDS LECTURE NOTES, FALL 2017 ECKHARD MEINRENKEN Abstract. These notes are under construction. They contain errors and omissions, and the references are very incomplete. Apologies! Contents 1. Lie groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Exercises 9 2. Foliation groupoids 10 2.1. Definition, examples 10 2.2. Monodromy and holonomy 12 2.3. The monodromy and holonomy groupoids 12 2.4. Appendix: Haefliger’s approach 14 3. Properties of Lie groupoids 14 3.1. Orbits and isotropy groups 14 3.2. Bisections 15 3.3. Local bisections 17 3.4. Transitive Lie groupoids 18 4. More constructions with groupoids 19 4.1. Vector bundles in terms of scalar multiplication 20 4.2. Relations 20 4.3. Groupoid structures as relations 22 4.4. Tangent groupoid, cotangent groupoid 22 4.5. Prolongations of groupoids 24 4.6. Pull-backs and restrictions of groupoids 24 4.7. A result on subgroupoids 25 4.8. Clean intersection of submanifolds and maps 26 4.9. Intersections of Lie subgroupoids, fiber products 28 4.10. The universal covering groupoid 29 5. Groupoid actions, groupoid representations 30 5.1. Actions of Lie groupoids 30 5.2. Principal actions 31 5.3. Representations of Lie groupoids 33 6. Lie algebroids 33 1 2 ECKHARD MEINRENKEN 6.1. Definitions 33 6.2. Examples 34 6.3. Lie subalgebroids 36 6.4. Intersections of Lie subalgebroids 37 6.5. Direct products of Lie algebroids 38 7. Morphisms of Lie algebroids 38 7.1. Definition of morphisms 38 7.2.