Holographic Description of Curved-Space Quantum Field Theory and Gravity

The celebrated AdS/CFT dualities provide a window to strongly-coupled quantum eld theories (QFTs), which are realized in nature at the most fundamental level on the one hand, but are hardly accessible for the standard mathematical tools on the other hand. The prototype examples of AdS/CFT relate classical supergravity theories on ( d + 1)-dimensional anti-de Sitter space (AdS) to strongly-coupled d -dimensional conformal eld theories (CFTs). The AdS spacetimes are characterized by a constant negative curvature and admit a timelike conformal boundary, on which the dual CFT is dened. In that sense the AdS/CFT dualities are holographic, and this new approach has led to remarkable progress in understanding strongly-coupled QFTs dened on Minkowski space and on the Einstein cylinder. On the other hand, the study of QFT on more generic curved spacetimes is of fundamental interest and non-trivial already for free theories. Moreover, understanding the properties of gravity { the actual description of spacetime dynamics { as a quantum theory remains among the hardest problems to solve in physics. Both of these issues can be studied holographically and we are interested here in generalizations of AdS/CFT involving on the lower-dimensional side QFTs on curved backgrounds and as a further generalization gravity. In the rst part we take the natural rst step from at-space QFT towards gravity and expand on the holographic description of QFT on xed curved backgrounds. The description of a CFT on a specic background involves gravity on an asymptotically-AdS space with that prescribed boundary structure. We discuss geometries with de Sitter and AdS as conformal boundary to holographically describe CFTs on these maximally symmetric spacetimes. After setting up the procedure of holographic renormalization we study the re ection of CFT unitarity properties in the dual bulk description. The geometry with AdS on the boundary exhibits a number of interesting features, mainly due to the fact that the boundary itself has a boundary. We study both cases and resolve potential tensions between the unitarity properties of the bulk and boundary theories, which would be incompatible with a duality. The origin of these tensions is partly in the structure of the geometry with AdS conformal boundary, while another one arises for a particular limiting case where the bulk and boundary descriptions navely disagree. Besides technical challenges, the interesting hierarchy of boundaries for the geometry with AdS conformal boundary oers the possibility of multi-layered AdS/CFT dualities. Namely, having the dual theory on the conformal boundary itself dened on an AdS space of codimension 1 oers the logical possibility of implementing a second instance of AdS/CFT. We discuss an appropriate geometric setting allowing for the notion of the boundary of a boundary and draw conclusions on limitations for such multi-layered dualities from our previous investigation of holographic renormalization. In the second part we consider ve-dimensional supergravities resulting from string theory as low-energy limits and whose solutions can be lifted to actual string-theory backgrounds. We work out the asymptotic structure of the theories on asymptotically-AdS spaces, carry out the holographic renormalization and calculate the Weyl anomaly of the dual CFTs. These holographic calculations conrm the expectations from the eld-theory side and provide a non-trivial test of the AdS/CFT conjecture. Moreover, the holographic renormalization also alters the symplectic structure of the bulk theory, and building on the previous results we show that in addition to Dirichlet also Neumann or mixed boundary conditions can be imposed. That deformation of AdS/CFT by changing the boundary conditions in particular promotes the boundary metric to a dynamical quantity and is expected to yield a holographic relation between a conformal supergravity on the boundary and the bulk theory. The boundary theory obtained this way exhibits pathologies such as perturbative ghosts, which is in fact expected for a conformal gravity. The fate of these ghosts beyond perturbation theory actually is a longstanding question and our setting provides a starting point to study it from the string-theory perspective. That discussion leads to a regime where the holographic description of the boundary theory requires quantization of the bulk supergravity. A necessary ingredient of any supergravity is a number of gravitinos as superpartners of the graviton, for which we thus need an eective-QFT description { to make sense of AdS/CFT beyond the limit where bulk theory becomes classical, but also more generally e.g. for cosmological applications. In particular, quantization should be possible not only on rigid AdS, but also on generic asymptotically-AdS spacetimes which may not be Einstein. In the third part we study the quantization and causality properties of the gravitino on Friedmann-Robertson-Walker spacetimes to explicitly show that a consistent quantization can be carried out also on non-Einstein spaces, in contrast to claims in the recent literature. Furthermore, this reveals interesting non-standard eects for the gravitino propagation, which in certain cases is restricted to regions more narrow than the expected light cones.