Lessons for Conformal Field Theories from Bootstrap and Holography
The work done in this thesis includes an exploration of both the conformal field theory techniques and holographic techniques of the Gauge/Gravity duality. From the field theory, we have analyzed the analytical aspects of the Conformal Bootstrap program to gain handle on at least a part of the CFT spectrum. The program applies equally to the strongly coupled as well as the weakly coupled theories. We have considered both the regimes of interest in this thesis. In the strongly coupled sector, as we have shown that it is possible to extract information about the anomalous dimensions, of a particular subset of large spin operators in the spectrum, as a function of the spin and twist of these operators. The holographic analog of the anomalous dimensions from CFT are the binding energies of generalized free fields in the bulk, which has also been analyzed in this thesis. On the contrary, in the weakly coupled sector, the same idea can be used to calculate the anomalous dimensions of operators, with any spin and dimension in an expansion. We have considered a simple set of scalar operators, whose anomalous dimensions are reproduced correctly up to O( 2). In another holographic calculation, we have analyzed generic higher derivative theories of gravity, which corresponds to boundary theories with in finite colors but finite `t Hooft coupling. Certain universal aspects of these theories, such as anomalies and correlation functions are also calculated. The three point functions for these higher derivative theories will serve as a building block for considering four point functions for finitely coupled boundary CFTs. In the conclusion, we have pointed out the directions of interest which could be locating the bulk duals of large N finitely coupled theories, or that of an intermediate theory with both finite `t Hooft coupling as well as finite gauge group, with a speculative string theory dual.