Littérature scientifique sur le sujet « Prequantisation »

Abstract This is a mostly self-contained survey article about bundle gerbes and some of their recent applications in geometry, field theory, and quantisation. We cover the definition of bundle gerbes with connection and their morphisms, and explain the classification of bundle gerbes with connection in terms of differential cohomology. We then survey how the surface holonomy of bundle gerbes combines with their transgression line bundles to yield a smooth bordism-type field theory. Finally, we exhibit the use of bundle gerbes in geometric quantisation of 2-plectic as well as 1- and 2-shifted symplectic forms. This generalises earlier applications of gerbes to the prequantisation of quasi-symplectic groupoids.

Abstract We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We start by briefly recalling the construction of the 2-category of bundle gerbes, with minor alterations that allow us to endow morphisms with additive structures. The morphisms in the resulting 2-categories are investigated in detail. We introduce a direct sum on morphism categories of bundle gerbes and show that these categories are cartesian monoidal and abelian. Endomorphisms of the trivial bundle gerbe, or higher functions, carry the structure of a rig-category, a categorised ring, and we show that generic morphism categories of bundle gerbes form module categories over this rig-category. We continue by presenting a categorification of the hermitean bundle metric on a hermitean line bundle. This is achieved by introducing a functorial dual that extends the dual of vector bundles to morphisms of bundle gerbes, and constructing a two-variable adjunction for the aforementioned rig-module category structure on morphism categories. Its right internal hom is the module action, composed by taking the dual of the acting higher functions, while the left internal hom is interpreted as a bundle gerbe metric. Sections of bundle gerbes are defined as morphisms from the trivial bundle gerbe to the bundle gerbe under consideration. We show that the resulting categories of sections carry a rig-module structure over the category of nite-dimensional Hilbert spaces with its canonical direct sum and tensor product. A suitable definition of 2-Hilbert spaces is given, modifying previous definitions by the use of two-variable adjunctions. We prove that the category of sections of a bundle gerbe, with its additive and module structures, fits into this framework, thus obtaining a 2-Hilbert space of sections. In particular, this can be constructed for prequantum bundle gerbes in problems of higher geometric quantisation. We define a dimensional reduction functor and show that the categorical structures introduced on the 2-category of bundle gerbes naturally reduce to their counterparts on hermitean line bundles with connections. In several places in this thesis, we provide examples, making 2-Hilbert spaces of sections and dimensional reduction very explicit.

Une variété 'n-plectique' est un couple constitué d'une variété et d'une (n+1)-forme fermée et non-dégénérée. Ces variétés généralisent le cas symplectique (1-plectique) et donnent un cadre naturel aux théories géométriques des champs classiques (comme les variétés symplectiques sont l'arène naturel de la mécanique classique). Les variétés n-plectiques, déjà étudiées depuis des années 70, sont devenues très importantes à cause de leur rôle dans l'approche dite 'supérieure' à la géométrie et topologie différentielle, c'est-à-dire les structures subtiles, de type catégorique, récemment découvertes. Dans ce projet de thèse, l'accent sera mis sur le cas 2-plectique, notamment sur l'étude des sous-variétés distinguées (Lagrangiennes, co-isotropes, ...), la dynamique des systèmes Hamiltoniens et des symétries des variétés 2-plectiques, ainsi que sur la préquantification de celles-ci
An ‘n-plectic manifold’ is a couple formed by a manifold and a closed, non-degenerate differentiable form of degree (n+1). These manifolds generalize the symplectic case (1-plectic) and give a natural framework for studying geometric classical field theories (as well as symplectic manifolds give a natural framework for studying classical mechanics). N-plectic manifolds, already studied since the 70’s, became paramount because of their role in the so-called ‘higher’ approach to differential geometry and topology, subtle structures related to category theory, freshly discovered. In this PhD thesis, we will study almost exclusively 2-plectic manifolds, notably distinguished submanifolds (Lagrangian, co-isotropic…), the dynamic of Hamiltonian systems and symetries of 2-plectic manifolds, as well as their prequantisation