Conformal Gauge Theories, Cartan Geometry and Transitive Lie Algebroids

Abstract : Our current knowledge about Universe rests on the existence of four fundamental interactions. These are : gravitation, electromagnetism, weak interaction and strong interaction. They have formed the conceptual basis of modern physics since half a century. I am interested in the classical aspect of the underlying physical theories : « gauge theories ». My approach is that of a mathematical physicist. First, this consists in studying gauge theories in their mathematical formulation, in order to enlighten some underlying geometric and algebraic structures. Second, new mathematical frameworks are proposed to formulate gauge theories, generalizing the previous ones. In this aim, we explored conformal geometry and its associated conformal gauge theories. These are gravitational gauge theories for which one passes from the Lorentz group to the conformal as structure group. The whole work is formulated in the language of Cartan geometry. Applying the dressing field method, which consists in reducing the gauge symmetry of a theory by a mere change of variables, we recover some objects usually defined in this geometry, as Tractors and Twistors. The bonus is that we get a deeper understanding of their geometric nature. We also present the theory of transitive Lie algebroids, and different ways of formulating gauge theories in this framework. First, we develop a notion of tensors on Lie algebroids, with an adapted basis which is fundamental in order to facilitate computations. It is possible, as N. Boroojerdian already did, to describe in a unique lagrangian General Relativity with cosmological constant together with Yang-Mills theories for other interactions. We recast this work in our clearer notations. The work of C. Fournel is also presented, in which the notion of generalized connection on Lie algebroids allows to write a lagrangian which contains both Yang-Mills theory and a Higgs term embedded in a quartic potential. Finally, we present a recent work which consists in combining Cartan geometry and transitive Lie algebroids. For this, we write Atiyah Lie sequences corresponding to both principal bundle related to a Cartan geometry, and then we give the definition of a Cartan connection in this language. We show the equivalence of this definition with the usual one on principal fibre bundles. We also compare our approach with that of M. Crampin and D. Saunders, also quite recent.
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Submitted on : Tuesday, December 4, 2018 - 9:04:36 PM
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Jeremy Attard. Conformal Gauge Theories, Cartan Geometry and Transitive Lie Algebroids. Mathematical Physics [math-ph]. Aix Marseille Université, 2018. English. ⟨tel-01944804⟩



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