Conformal quantum mechanics in holography and Hodge theory

Abstract

AdS2/CFT1 is a particularly interesting example of holography since it appears naturally as the near-horizon limit of extremal black holes. However it is also quite poorly understood. In this thesis aspects of a conformal quantum mechanics model are studied, using an infinite-dimensional representation which provides a natural candidate for the CFT1 side of the correspondence. A similar model using a finite-dimensional representation, which has possible applications in Hodge theory, is also constructed. After reviewing some basic aspects of AdS/CFT and AdS2/CFT1 specifically, the conformal quantum mechanics model is introduced. This model captures some of the aspects of conformal field theories and forms a discrete series representation of SL(2,R). We then propose a completeness relation inspired by the shadow transform in conformal field theories, which provides a method with which all higher correlation functions can be expressed as integrals. Afterwards we propose an interpretation of the conformal quantum mechanics as the boundary model of quantum mechanics on the Poincar´e upper half plane, which is the canonical way the discrete series representation of SL(2,R) is realized. Finally we propose a similar construction for a finite dimensional representation of SL(2,R) which has possible applications in Hodge theory. For this representation we also calculate correlation functions and also use the shadow transform again to define a completeness relation. Finally we define an integral transformation in order to calculate this shadow transform explicitly.