Understanding Conformal Field Theory in Mellin Space

Abstract

In the last decade, there has been tremendous progress in understanding conformal eld theories in more than two space-time dimensions. The numerical bootstrap framework has been pushed to a very advanced level of sophistication. Analytical bootstrap progress is still in its infancy. The Polyakov-Mellin bootstrap can be applied to various perturbative theories to derive analytical results purely from CFT arguments and successfully reproduce important results like operator dimensions and OPE coe cients to the rst few orders in epsilon expansion at the Wilson-Fisher xed point. This tool is not yet capable of handling nonperturbative CFTs. In this thesis, we try to understand certain aspects of Polyakov-Mellin (PM) bootstrap in Mellin space. We also set up conventional bootstrap in Mellin space and showed how many things simplify because of the properties of special functions that appear in Mellin space description of conformal eld theories. In particular, we give closed form formula for any general term in the large spin expansion of correction to OPE coe cients and anomalous dimensions of double eld operators which are part of any unitary CFT. We also show in the leading order of perturbation, the PM bootstrap and conventional bootstrap give rise to identical equations, proving the existence of PM basis in any dimension to the rst order of perturbation. Then we also apply PM bootstrap techniques to holographic CFTs and show this framework can be used to reconstruct certain loop diagrams in the AdS, which are hard to compute otherwise. This is essentially showing the CFT version of unitarity from amplitudes, which states that we can construct loops using the lower point data. We also argue the nonperturbative existence of Polyakov Mellin basis in one dimension, including 1d CFTs with global symmetry. We perform various checks to establish our claims and we also developed a method where one can expand the CFT four point function in the basis of transcendental functions and extract CFT data. This independent method agrees with results derived from PM bootstrap. We put stringent constraints on CFT data by studying the correlator in the Regge limit.