## Conformal quantum mechanics in holography and Hodge theory

##### Summary

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.