The diffeomorphism field is introduced to the physics literature in [1] where it arises as a background field coupled to Polyakov’s quantum gravity in two dimensions, where Einstein’s gravity is trivial. Moreover, it is seen in many ways as the gravitational analog of the Yang-Mills field. This raises the question of whether the diffeomorphism field exists in higher dimensions, playing an essential role in gravity either by supplementing Einstein’s theory or by modifying it. With this motivation, several distinct theories governing the dynamics of the diffeomorphism field have been constructed and developed by mimicking the construction of the Yang-Mills theory from the Kac-Moody algebra. This analogy, however, is not perfect and there are many subtleties and difficulties encountered. This thesis constitutes a further development. The previously proposed theories are carefully examined; certain subtleties and problems in them have been discovered and made apparent. Some of these problems have been solved, and for others possible routes to follow have been laid down. Finally, other geometric approaches than the ones followed before are investigated.

This paper will examine observables in the bc system, a two-dimensional free conformal field theory. We begin by encoding the bc system into the BV formalism following procedures of Costello and Gwilliam. This will allow us to construct the factorization algebra of observables for the bc system. The cohomology of the factorization algebra recovers the observables themselves. In cohomology, we will compute the commutation relations and factorization algebra structure maps for observables supported on disks and annuli. These structure maps will be used to prove the equivalence of the factorization algebra and vertex algebra structures for the bc system. This proof provides a rigorous derivation of the free fermionic vertex algebra starting from the action functional of the bc system. Using this equivalence, we will provide a dictionary to translate the action of the Virasoro algebra to the language of factorization algebras. Also in this paper, we examine the bc system in four-dimensions. We construct its factorization algebra and show that its observables are anti-commutative. Lastly, we prove that the global observables of the bc system are one-dimensional on a compact manifold of complex dimension one or two.

Inspiré par l'algèbre d’échange et birapport de rang n introduit par F. Labourie, nous construisons un anneau muni de la structure de Poisson--- l’algèbre d’échangée de rang n Zn(P) pour étudier les espaces de modules de birapports . Nous prouvons que Zn(P) hérite d'une structure de Poisson provenant de l’algèbre d’échangée. Pour tenir compte des “birapports” dans l’anneau de fraction, en interprétant Zn(P) par un modèle géométrique dans l'étude de la géométrie théorie des invariants, nous montrons que Zn(P) est intègre. Ensuite, nous considérons l'anneau Bn(P) engendreré par les birapports dans l'anneau de fraction de Zn(P). Pour n = 2,3, nous trouvons un homomorphisme injectif poissonienne de l'anneau engendré par coordonnées de Fock-Goncharovde sur l'espace des configurations de drapeaux dans Rn vers Bn(P). En étudiant le système intégrable discret pour l'espace des configurations de polygones N-tordus dans RP1, à une transformation de Fourier discrète, nous rapportons asymptotiquement l'algèbre d’échangée à l'algèbre de Virasoro sur une hypersurface de MN, 1. / Inspired by the swapping algebra and the rank n cross-ratio introduced by F. Labourie, we construct a ring equipped with the swapping Poisson structure---the rank n swapping algebra Zn(P) to study the moduli spaces of cross ratios. We prove that Zn(P) inherits a Poisson structure form the swapping bracket. To consider the "cross-ratios" in the fraction ring, by interpreting Zn(P) by a geometric model in the study of geometry invariant theory, we prove that Zn(P) is an integral domain. Then we consider the ring Bn(P) generated by the cross ratios in the fraction ring of Zn(P). For n = 2,3, we embed in a Poisson way the ring generated by Fock-Goncharov coordinates for configuration space of flags in Rn into Bn(P). By studying the discrete integrable system for the configuration space MN,1 of N-twisted polygons in RP1, up to a discrete Fourier transformation, we asymptotically relate the swapping algebra to the Virasoro algebra on a hypersurface of MN,1.

Inspired by the swapping algebra and the rank n cross-ratio introduced by F. Labourie, we construct a ring equipped with the swapping Poisson structure---the rank n swapping algebra Zn(P) to study the moduli spaces of cross ratios. We prove that Zn(P) inherits a Poisson structure form the swapping bracket. To consider the "cross-ratios" in the fraction ring, by interpreting Zn(P) by a geometric model in the study of geometry invariant theory, we prove that Zn(P) is an integral domain. Then we consider the ring Bn(P) generated by the cross ratios in the fraction ring of Zn(P). For n = 2,3, we embed in a Poisson way the ring generated by Fock-Goncharov coordinates for configuration space of flags in Rn into Bn(P). By studying the discrete integrable system for the configuration space MN,1 of N-twisted polygons in RP1, up to a discrete Fourier transformation, we asymptotically relate the swapping algebra to the Virasoro algebra on a hypersurface of MN,1.