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In this thesis we examine some of the interesting aspects of the Wess-Zumino- Novikov-Witten model when this model has been gauged and its energy tensor twisted by the addition of the derivative of one of its Cartan subalgebra valued currents. Specifically we consider the group valued model with the group taken as 5^(211) which is the Lie super group used to describe N = 2 supersymmetry. This model is advocated as being a good and natural description of the N = 2 superstring (also known as the charged spinning string, or N = 2 fermionic string) when it tensors an additional topological system of ghosts. The evidence for this assertion is presented by gauging and twisting the model and then extracting the N = 2 super Liouville action by the method of Hamiltonian reduction. The connection between the 5L (2|1)/5L (2|1) Wess-Zumino-Novikov-Witten model and field theory is made through its current algebra. As is true of many super groups there exists more than one interpretation of the Dynkin diagram for the algebra of 5L(2|1) and this results in more that one set of currents for this model. The classical and quantum currents in free field form are found in both cases, as is the highly non-linear transformation by which the two sets of currents are related. An analysis of a section of the cohomology of physical states of the model is undertaken. It is shown that the additional topological ghost system that tensors the gauged, twisted SL (2\l) model when it describes the N = 2 string only contributes a vacuum state to the overall cohomology, so reducing the analysis. As the 5L(2|1)/5L(2|1) Wess-Zumino-Novikov-Witten model is a topological field theory its spectrum of physical states lie in the cohomology class defined with respect to the BRST charge. The spectrum formed from the free field currents composes the so called Wakimoto module and this is calculated via the BRST formalism.

Brown and York's gravitational microcanonical density of states is extended to general spacetime dimension and shown to be dependent upon features of the 4 dimensional gravitational action for its interpretation. Black hole entropy is calculated from the density of states path integral in general spacetime dimension, and the interpretation is shown to be likewise dependent upon the dimension of spacetime. The entropy of de Sitter and Rindler horizons are calculated using the black hole density of states and the notion of local horizon entropy density is shown to be supported. The applicability of the microcanonical ensemble to black hole mechanics is discussed at a fundamental level focussing on the absence of angular velocity as an external parameter in the gravitational Hamiltonian. The rotational ensemble and a new ensemble - the angular momentum ensemble - are introduced following Jaynes' information theory approach to statistical mechanics and proposed as more compelling candidates to calculate black hole entropy as a function of state. A program to calculate the density of states path integral non-perturbatively using causal dynamical triangulations is initiated. Regge calculus expressions for extrinsic curvature are extended to the case of Lorentzian hypersurfaces and used to derive Regge calculus expressions for quasilocal energy-momentum. The Regge version of the black hole density of states action is derived and specialised to the 3d and 4d spacetime constructions of causal dynamical triangulations. Finally, the recent suggestion that entropy is observer dependent is shown to be incompatible with the Tolman law for the equilibrium temperature in a gravitational field.

Two different works in mathematical physics are presented: A construction of conformal infinity in null and spatial directions is constructed for the Rainbow-flat space-time corresponding to doubly special relativity. From this construction a definition of asymptotic DSRness is put forward which is com- patible with the correspondence principle of Rainbow gravity. Furthermore a result equating asymptotically flat space-times with asymptotically DSR spacetimes is presented. An overview of microlocality in braided ribbon networks is presented. Follow- ing this, a series of definitions are presented to explore the concept of microlocality and the topology of ribbon networks. Isolated substructure of ribbon networks are introduced, and a theorem is proven that allows them to be relocated. This is fol- lowed by a demonstration of microlocal translations. Additionally, an investigation into macrolocality and the implications of invariants in braided ribbon networks are presented.

In this thesis some equilibrium and non-equilibrium statistical methods are implied on two different versions of non-perturbative quantum gravity. Firstly, we report a novel statistical mechanics in which a class of evolutionary maps act on trivalent spin network in randomly chosen initial states and give rise to Self-organized Criticality. The result of continuously applying these maps indicate an expansion in the space-time area associated. Secondly, a previously unknown statistical mechanics in quantum gravity is introduced in the framework of two dimensional Causal Dynamical Triangulations. This provides us a useful and new tools to understand this quantum gravity in terms of effective spins. This study reveals a correspondence between the statistics of Anti-ferromagnetic systems and Causal Dynamical quantum gravity. More importantly, it provides a basis for studying anti-ferromagnetic systems in a background independent way. Thirdly, two novel properties of area operator in Loop Quantum Gravity are reported: 1) the generic degeneracy and 2) the ladder symmetry. These were not known previously for years. The first one indicates that corresponding to any eigenvalue of area operator in loop quantum gravity there exists a finite number of degenerate eigenstates. This degeneracy is shown to be one way for the explanation of black hole entropy in a microscopic way. More importantly, we reproduce Bekenstein-Hawking entropy of black hole by comparing the minimal energy of a decaying frequency from a loop quantum black hole and the extracted energy from a perturbed black hole in the highly damping mode. This consistency reveals a treasure model for describing a black hole in loop quantum gravity that does nor suffer from the restrictions of an isolated horizon. The second property indicates there exists a ladder symmetry unexpectedly in the complete spectrum of area eigenvalues. This symmetry suggests the eigenvalues of area could be classified into different evenly spaced subsets, each called a `generation.' All generations are evenly spaced; but the gap between the levels in any every generation is unique. One application of the two new properties of area operator have been considered here for introducing a generalized picture of horizon whose area cells are not restricted to the subset considered in quantum isolated horizon theory. Instead, the area cells accepts values from the complete spectrum. Such horizon in the presence of all elements of diffeomorphism group contains a number of degrees of freedom independently from the bulk freedom whose logarithm scales with the horizon area. Note that this is not the case in quantum isolated horizon when the complete elements of diffeomorphism applies. Finally, we use a simple statistical method in which no pre-assumption is made for the essence of the energy quanta radiated from the hole. We derive the effects of the black hole horizon fluctuations and reveal a new phenomenon called "quantum amplification effects" affecting black hole radiation. This effect causes unexpectedly a few un-blended radiance modes manifested in spectrum as discrete brightest lines. The frequency of these modes scales with the mass of black hole. This modification to Hawking's radiation indicates a window at which loop quantum gravity can be observationally tested at least for primordial black holes.

Two different works in mathematical physics are presented: A construction of conformal infinity in null and spatial directions is constructed for the Rainbow-flat space-time corresponding to doubly special relativity. From this construction a definition of asymptotic DSRness is put forward which is com- patible with the correspondence principle of Rainbow gravity. Furthermore a result equating asymptotically flat space-times with asymptotically DSR spacetimes is presented. An overview of microlocality in braided ribbon networks is presented. Follow- ing this, a series of definitions are presented to explore the concept of microlocality and the topology of ribbon networks. Isolated substructure of ribbon networks are introduced, and a theorem is proven that allows them to be relocated. This is fol- lowed by a demonstration of microlocal translations. Additionally, an investigation into macrolocality and the implications of invariants in braided ribbon networks are presented.

In this thesis some equilibrium and non-equilibrium statistical methods are implied on two different versions of non-perturbative quantum gravity. Firstly, we report a novel statistical mechanics in which a class of evolutionary maps act on trivalent spin network in randomly chosen initial states and give rise to Self-organized Criticality. The result of continuously applying these maps indicate an expansion in the space-time area associated. Secondly, a previously unknown statistical mechanics in quantum gravity is introduced in the framework of two dimensional Causal Dynamical Triangulations. This provides us a useful and new tools to understand this quantum gravity in terms of effective spins. This study reveals a correspondence between the statistics of Anti-ferromagnetic systems and Causal Dynamical quantum gravity. More importantly, it provides a basis for studying anti-ferromagnetic systems in a background independent way. Thirdly, two novel properties of area operator in Loop Quantum Gravity are reported: 1) the generic degeneracy and 2) the ladder symmetry. These were not known previously for years. The first one indicates that corresponding to any eigenvalue of area operator in loop quantum gravity there exists a finite number of degenerate eigenstates. This degeneracy is shown to be one way for the explanation of black hole entropy in a microscopic way. More importantly, we reproduce Bekenstein-Hawking entropy of black hole by comparing the minimal energy of a decaying frequency from a loop quantum black hole and the extracted energy from a perturbed black hole in the highly damping mode. This consistency reveals a treasure model for describing a black hole in loop quantum gravity that does nor suffer from the restrictions of an isolated horizon. The second property indicates there exists a ladder symmetry unexpectedly in the complete spectrum of area eigenvalues. This symmetry suggests the eigenvalues of area could be classified into different evenly spaced subsets, each called a `generation.' All generations are evenly spaced; but the gap between the levels in any every generation is unique. One application of the two new properties of area operator have been considered here for introducing a generalized picture of horizon whose area cells are not restricted to the subset considered in quantum isolated horizon theory. Instead, the area cells accepts values from the complete spectrum. Such horizon in the presence of all elements of diffeomorphism group contains a number of degrees of freedom independently from the bulk freedom whose logarithm scales with the horizon area. Note that this is not the case in quantum isolated horizon when the complete elements of diffeomorphism applies. Finally, we use a simple statistical method in which no pre-assumption is made for the essence of the energy quanta radiated from the hole. We derive the effects of the black hole horizon fluctuations and reveal a new phenomenon called "quantum amplification effects" affecting black hole radiation. This effect causes unexpectedly a few un-blended radiance modes manifested in spectrum as discrete brightest lines. The frequency of these modes scales with the mass of black hole. This modification to Hawking's radiation indicates a window at which loop quantum gravity can be observationally tested at least for primordial black holes.

We construct a 4d Lorentzian spin foam model capable of describing both spacelike and timelike surfaces. To do so we use a coherent state approach inspired by the Riemannian FK model. Using the coherent state method we reproduce the results of the EPRL model for Euclidean tetrahedra and extend the model to include Lorentzian tetrahedra. The coherent states of spacelike/timelike triangles are found to correspond to elements of the discrete/continuous series of SU(1,1). It is found that the area spectrum of both spacelike and timelike surfaces is quantized. A path integral for the quantum theory is defined as a product of vertex amplitudes. The states corresponding to timelike triangles are constructed in a basis diagonalised with respect to a noncompact generator. A derivation of the matrix elements of the generators of SL(2,C) in this basis is provided.

In this thesis we review the status of an approach to Quantum Gravity through lattice toy models, Quantum Graphity. In particular, we describe the two toy models introduced in the literature and describe with a certain level of details the results obtained so far. We emphasize the connection between Quantum Graphity and emergent gravity, and the relation with Variable Speed of Light theories.