Duality and asymptotic symmetries in gravisolitons and gauge theories

The thesis deals with exotic gauge theories and gravitational solitons with a view to the problem of quantum gravity. In fact gravitational solitons and dualities in exotic gauge theories are able to shed new light on quantum gravity from various points of view: perturbative, on asymptotic observables, non-perturbative, new proposals for quantization of gravity (among all the Cornel proposal), topology/geometry changes, string theory. First, gauge theories whose gauge field is a p-form or a mixed-symmetry tensor are studied. The asymptotic charges in the case of p-forms are explicitly computed and their behavior under dualities inherited from representation theory is studied. Furthermore, an existence and uniqueness theorem is proved for the duality map linking the charges both in the case of p-forms and in the generalizing case of mixed-symmetry tensors. In both cases the result is a topological nature of the duality map. However, in the case of mixed symmetry tensors it was necessary to develop new mathematical tools such as a generalization of the de Rham cohomology, leading to the definition of a new cohomology for differentiable varieties. The first steps for the explicit computation of the asymptotic charges of gauge theories dual to the graviton gauge theory (linearized gravity) are also discussed. Then, the focus shifts to gravitational solitons. A general asymptotic expansion valid for any gravisoliton solution is developed. The leading-order asymptotic symmetries and the algebra generated by them are computed with the aim of testing the corner proposal in a non-perturbative sector of gravity. The result is positive. Two works on the geometrization of field theories from string theory are briefly discussed, whose focus slightly deviates from the main theme of the thesis but which nevertheless addresses the problem of quantum gravity.