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The Kodama state is unique in being an exact solution to all the constraints of quantum gravity that also has a well defined semi-classical interpretation as the quantum version of a classical spacetime, namely de Sitter or anti-de sitter space. Despite this, the state fails to pass some of the key tests of a physically realistic quantum state. In an attempt to resolve this problem, we track down the root of the problem to a choice for a particular parameter: the Immirzi parameter. The Kodama state takes this parameter to be complex, whereas modern formulations of canonical quantum gravity require that the parameter is real. We generalize the Kodama state to real values of the Immirzi parameter, and find that the generalization opens up a large Hilbert space of states, one of which can be directly interpreted as particular slicing of de Sitter space. We then show that these states resolve, or are expected to resolve many of the problems associated with the original version of Kodama state. In order to resolve the interpretation of the multitude of states, we develop a new model of covariant classical and quantum gravity where the full Lorentz group is retained as a local symmetry group, and the canonical evolution generated by the constraints has a close relation to a larger group: that de Sitter group. This formalism gives strong evidence that the multitude of generalized Kodama states can be unified into a single quantum state that is quantum de Sitter space. / text

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Gauge/gravity duality is an important tool for learning about strongly coupled gauge theories. This thesis explores a set of examples of this duality in which the field theories have SU(2|4) supersymmetry and discrete sets of vacuum solutions. Specifically, we use the duality to propose Lagrangian definitions of type IIA Little String Theory on S⁵ as double-scaling limits of the Plane-Wave Matrix Model, maximally supersymmetric Yang-Mills theory on R x S² and N=4 supersymmetric Yang-Mills theory on R×S³/Zk. We find the supergravity solutions dual to generic vacua of the Plane-Wave Matrix Model and maximally supersymmetric Yang-Mills theory on R×S². We use the supergravity duals to calculate new instanton amplitudes for the Plane-Wave Matrix Model at strong coupling. Finally, we study a natural coarse-graining of the vacua, and find that the associated geometries are singular. We define an entropy functional that vanishes for regular geometries, is non-zero for singular geometries, and is maximized by the thermal state.

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