Quantum gravity

Abstract

This research proposal intends to analyze several aspects related to quantum gravity, ranging from noncommutative theories up to the study of the AdS/CFT correspondence. Noncommutative field theories arise as a low energy limit of string theory in the background of a NS-NS two form field. The main quantum property of these theories is the mixing of ultraviolet and infrared divergences which undermines renormalizability. We discovered that the only four dimensional renormalizable theory is the supersymmetric Wess-Zumino model. In three dimensions, we have also shown that the supersymmetric nonlinear sigma models are renormalizable in the 1/N expansion. We intend to follow up the study of the several dualities that are found in three dimensions when the Chern-Simons field is present and to verify under which conditions this duality can be extended to the noncommutative case. In noncommutative theories the fields present exotic properties. The Seiberg-Witten map allows us to work with ordinary fields with their characteristics properties. In general, the formulation with commutative fields is incomplete since the dynamics is not known in explicit form. Besides, we have shown that the gauge field couples to matter, like a gravitational field giving rise to the phenomenon of emergent gravity. This shows a deep connection between noncommutative theories and gravitation which deserves to be better understood. We intend to analyze the noncommutative gravitational theories which give rise to the emergent gravitation and are related with the mixing of the ultraviolet and infrared divergences mentioned above. Still using this formulation, we found that supersymmetry can not be realized off-shell, preventing a superspace formulation. We believe that supersymmetry becomes local in the same way as the gauge field, previously discussed, turns translations into a local symmetry. This aspect will also be explored. When it was understood that the nonperturbative regime of string theory included membranes and supergravity theories in eleven dimensions, several exceptional discoveries were made. One of them is the equivalence between certain gravity and gauge theories. The main one is known as the AdS/CFT correspondence and its major property is that it includes the holographic principle. We have done several contributions to this area, including a new quantization of the scalar field in anti De Sitter space and an improved version of the correspondence. We intend to follow this line analyzing the situation when the 5-sphere in AdS_5 X S^5 is replaced by a deformed sphere in which case the corresponding gauge theory still is conformal. We intend to get several types of string solutions propagating in this deformed space. Besides, it was found recently that both the gauge theory and the string theory have integrable sectors. We intend to analyze the integrability in these deformed backgrounds. Duality has an essential role in string theory. In some models it suggests the existence of dimensions above eleven, including time like ones. Two or more times theories are troublesome since the dynamical evolution is ambiguous. However, if there exists a gauge symmetry which removes the extra time dimensions then it may turn the theory into a consistent one. We wish to analyze, using BRST-BFV techniques, a formulation of two times theories whose gauge group is Sp(2,R). We will build the path integral and show that several systems can be reached by different gauge choices. (AU)