Holographic Correlators and Effective Actions for Quantum Gravity

In this thesis we study holographic correlators in three different examples of the AdS/CFT correspondence. In particular, we consider the low-energy effective actions of quantum gravity, where the leading term describes supergravity and the corrections correspond to higher-derivative interactions. Firstly, we consider the duality between string theory in AdS_5xS^5 and 4d maximally supersymmetric Yang-Mills theory (N=4 SYM). We propose a systematic procedure for obtaining all single-trace half-BPS correlators in N=4 SYM corresponding to the four-point tree-level amplitude for type IIB string theory in AdS_5xS^5. The underlying idea is to compute generalised ten-dimensional contact Witten diagrams, treating AdS and S on equal footing, which are obtained from a 10d scalar effective field theory in AdS_5xS^5. Next, we study holographic correlators in AdS_7xS^4. M-theory in this background is dual to the 6d (2,0) theory. In particular, we derive recursion relations for the anomalous dimensions of double-trace operators occurring in the conformal block expansion of four-point stress tensor correlators in the 6d (2,0) theory. These anomalous dimensions encode higher-derivative corrections to supergravity in AdS_7xS^4 arising from M-theory. Finally, we consider quantum gravity in AdS_2xS^2 whose dual CFT has superconformal group SU(1,1|2). Firstly, we propose an AdS_2xS^2 effective action which describes both supergravity and higher-derivative corrections and compute the four-point half-BPS correlators using generalised 4d Witten diagrams, analogous to AdS_5xS^5 above. Moreover, it was recently shown that IIB supergravity in AdS_5xS^5 enjoys 10d conformal symmetry. We adapt this approach, which is complementary to the effective action approach, to quantum gravity in AdS_2xS^2. We show that the 1d supergravity and free theory correlators exhibit 4d conformal symmetry and discuss implications for higher-derivative corrections where the symmetry is generically broken, except for specific cases.