### Abstract:

Classically, the thermodynamics of near extremal black holes fails below a certain temperature. To avoid this problem, the existence of a “thermodynamic mass gap” between the extremal state and the lightest near extremal state was conjectured. At the same time, for a small Hawking temperature T, temperature dependent quantum corrections to the classical thermodynamic variables can be determined. The effect of these corrections in resolving the existence of a mass gap was studied in recent literature. This thesis is motivated by the problem of understanding the zero temperature limit of the quantum corrected partition function Z and consequently, the entropy S to study the statistical mechanics of extremal black hole states. We motivate the thermodynamic mass gap problem with relevant background material in Chapter 1. We describe how the Jackiw-Teitelboim (JT) theory of 2D gravity, and the resulting Schwarzian action describe the dynamics of the dimensionally reduced Einstein-Maxwell theory of near extremal black holes, in the near horizon region. In Chapter 2, we discuss the method of coadjoint orbit quantization for quantizing the Schwarzian action. We then discuss how to set up a path integral for Z for a canonical ensemble of non rotating, non supersymmetric, near extremal black holes with a fixed charge Q, and subsequently evaluate the path integral. This is achieved by reducing the theory to an effective 1D action at the boundary of the near horizon region, at the level of the path integral. This is then accompanied by a brief account of multi black hole solutions to Einstein-Hilbert and Einstein-Maxwell actions and the method of heat kernel that comes useful in evaluating one loop functional determinants. In Chapter 3, we attempt to look for different effective boundary theories that could possibly rectify the divergence in the entropy in the zero temperature limit. We discuss why the Schwarzian action correctly describes a perturbed JT-boundary theory, where the perturbation takes the extremal Reissner-Nordström (RN) solution to the near extremal RN solution. This is followed by our efforts at resolving the difficulties in determining the contribution of quadratic fluctuations around multi black hole saddles to the path integral. We discuss the applicability of existing methods in the literature in calculating heat kernel coefficients on manifolds with conical singularities, where the metric is conformally related to a coordinate separable one. The possibility of such non perturbative corrections in rectifying the behaviour of Z and S in the zero temperature limit is discussed. We present some conclusions in Chapter 4.