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Generalized Connections and Higgs Fields on Lie Algebroids Nijmegen, April 5, 2016

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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. At least three mathematical schemes to construct gauge field theories: Ordinary differential geometry of principal fiber bundles. Noncommutative geometry. Transitive Lie algebroids (to be explained in this talk). 3 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Ordinary differential geometry Given a G-principal fiber bundle P over M, the ingredients are gauge group: G(P) is the group of vertical automorphisms of P. representation theory: sections of associated vector bundles. ➙ Natural action of G(P). differential structures: (ordinary) de Rham differential calculus. covariant derivative: connection 1-form ω on P. ➙ covariant derivative on sections of any associated vector bundles. action functional: integration on the base manifold M, Killing form on the Lie algebra g of G, Hodge star operator, curvature of ω. 4 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Noncommutative geometry Given an associative algebra A, the ingredients are representation theory: a right module M over A. gauge group: Aut(M), the group of automorphisms of the right module. differential structures: any differential calculus defined on topof A. ➙ many choices: spectral triples, derivations, twisted derivations… covariant derivatives: noncommutative connections on M, (need a differential calculus). action functional: depends on the differential calculus. spectral triples: spectral action… derivation-based differential calculus: noncommutative integration, Hodge star operator, curvature of the connection… 5 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Outline 1 Lie algebroids and their representations 2 Differential structures 3 Connections and covariant derivatives 4 The gauge group 5 Structures to construct an action functional 6 Gauge theories Lazzarini, S. and Masson, T. (2012). Connections on Lie algebroids and on derivation-based non-commutative geometry. J. Geom. Phys. 62, pp. 387–402 Fournel, C., Lazzarini, S., and Masson, T. (2013). Formulation of gauge theories on transitive Lie algebroids. J. Geom. Phys. 64, pp. 174–191 6 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Lie algebroids and their representations Lie algebroids and their representations 1 Lie algebroids and their representations 2 Differential structures 3 Connections and covariant derivatives 4 The gauge group 5 Structures to construct an action functional 6 Gauge theories 7 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Lie algebroids and their representations Generalities on Lie algebroids M a smooth manifold, Γ(T M) the Lie algebra and C1(M)-module of vector fields. Definition in terms of algebras and modules (as inNCG). Definition (Lie algebroids) A Lie algebroid A is a finite projective module over C1(M) equipped with a Lie bracket [−; −] and a C1(M)-linear Lie morphism ρ : A ! Γ(T M) such that [X; f Y] = f [X; Y] + (ρ(X)·f )Y for any X; Y 2 A and f 2 C1(M). ρ is the anchor of A. The usual definition uses the vector bundle A such that A = Γ(A). A is viewed as a generalization of the tangent bundle. ➙ We will never use this point of view. Natural notion of morphisms of Lie algebroids… 8 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Lie algebroids and their representations Transitive Lie algebroids ρ A Lie algebroid A ! Γ(T M) is transitive if ρ is surjective. Proposition (The kernel of a transitive Lie algebroid) Let A be a transitive Lie algebroid. L = Ker ρ is a Lie algebroid with null anchor on M. ➙ L is called the kernel of A. The vector bundle L such that L = Γ(L) is a locally trivial bundle in Lie algebras. ➙ This gives the Lie structure onL. One has the short exact sequence of Lie algebras and C1(M)-modules / ι / ρ / / 0 L A Γ(T M) 0 This short exact sequence is the key structure of what follows… Very trivial example: A = Γ(T M) ➙ L = 0. 9 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Lie algebroids and their representations Example 1: Derivations of a vector bundle E a vector bundle over M. Diff1(E) the space of first order differential operators on E. Symbol map: 1 ∗ σ : Diff (E) ! HomC1(M)(T M; End(E)) ' Γ(T M ⊗ End(E)) ⊃ Γ(T M) − D(E) = σ 1(Γ(T M)) is the transitive Lie algebroid of derivations of E: / ι / σ / / 0 A(E) D(E) Γ(T M) 0 with A(E) = Γ(End(E)) (0th-order diff. op.). A(E) is an associative algebra (Lie structure is the commutator). 10 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Lie algebroids and their representations Representation of a Lie algebroid ρ A −! Γ(T M) a Lie algebroid and E ! M a vector bundle. Definition (Representation of a Lie algebroid) A representation of A on E is a morphism of Lie algebroids φ : A ! D(E). When A is transitive, one has the commutative diagram of exact rows: / ι / ρ / / 0 L A Γ(T M) 0 φL φ / ι / σ / / 0 A(E) D(E) Γ(T M) 0 φL : L ! A(E) is a morphism of Lie algebras. 11 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Lie algebroids and their representations Example 2: Atiyah Lie algebroids π P −! M a G-principal fiber bundle, g the Lie algebra of G. Rд : P ! P, Rд(p) = p·д, the right action of G on P. ΓG (T P) = fX 2 Γ(T P)/ Rд ∗X = X for all д 2 Gg ΓG (P; g) = fv : P ! g /v(p·д) = Adд−1v(p) for all д 2 Gg Both are Lie algebras and C1(M)-modules. P P P ! M ΓG (T ) = π∗-projectable vector fields in Γ(T ) ➙ π∗ :ΓG (T ) Γ(T ). P P; ! P j ι :ΓG ( g) ΓG (T ) defined by ι(v) p = v(p)jp , P (g 3 v 7! v fundamental vector field on P). S.E.S. of Lie algebras and C1(M)-modules: / ι / π∗ / / 0 ΓG (P; g) ΓG (T P) Γ(T M) 0 ΓG (T P) is the transitive Atiyah Lie algebroid associated to P The representations of ΓG (T P) are given by the associated vector bundles to P. 12 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Lie algebroids and their representations Example 3: Trivial Lie algebroids Trivial Lie algebroid = Atiyah Lie algebroid of a trivial principal bundle M × G. Concrete description in terms of the bundle T M ⊕ (M × g): C1(M)-module: TLA(M; g) ≡ A = Γ(T M ⊕ (M × g)). Bracket: [X ⊕ γ;Y ⊕ η] = [X;Y ] ⊕ (X ·η − Y ·γ + [γ;η]) Anchor: ρ(X ⊕ γ ) = X. Kernel: L = Γ(M × g) (section of a trivial bundle). Proposition Every transitive Lie algebroid A is locally of the form TLA(U; g) for U ⊂ M open subset. Trivialization of an Atiyah Lie algebroid ΓG (T P) ↔ Trivialization of P. 13 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Differential structures Differential structures 1 Lie algebroids and their representations 2 Differential structures 3 Connections and covariant derivatives 4 The gauge group 5 Structures to construct an action functional 6 Gauge theories 14 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Differential structures Differential forms: general definition A Lie algebroid, φ : A ! D(E) a representation of A on E. Definition (Differential forms) For p 2 N, let Ωp (A; E) be the linear space of C1(M)-multilinear antisymmetric maps Ap ! Γ(E). = 0 Ω0( ; E) = Γ(E) For p , letL A . • ; E p ; E Ω (A ) = p ≥0 Ω (A ) is equipped with the natural differential pX+1 i D i+1 _ (dφωD)(X1;:::; Xp+1) = (−1) φ(Xi )·ωD(X1;::: ::::; Xp+1) i=1 X i j i+j _ _ + (−1) ωD([Xi ; Xj ]; X1;::: :::: ::::; Xp+1) 1≤i <j ≤p+1 φ(X)·ϕ is the action of the first order diff.
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