Abstract
We reviewed the recent developments in the study of conformal field theories in generic space time dimensions using the methods of the conformal bootstrap, in its analytic aspect. These techniques are solely based on symmetries, particularly on the analytic structure and in the associativity of the operator product expansion. We focused on two applications of the analytic conformal bootstrap: the study of the ϵ expansion of the Wilson–Fisher model via the introduction of a dispersion relation and the large N expansion of the maximally supersymmetric Super Yang–Mills theory in four dimensions.
Highlights
One efficient method is the numerical one, which is a numerical procedure that allows finding bounds on the Conformal field theories (CFTs) data for the operators appearing in the operator product expansion (OPE) decomposition, by using the relation Equation (16) and other symmetries that the theory may possess, as can be seen for instance in a recent review [4]
CFT, where we have an expansion of the CFT data in a perturbative parameter, the fourpoint function only depends on the spectrum of the theory and the OPE coefficients of certain low lying operators2
The presence of additional symmetry will even further constrain the spectrum of these theories and it will further help the analysis of correlation functions, unveiling new and interesting properties
Summary
The main idea is to use the associativity of the OPE inside four-point functions to be able to put numerical bounds on the conformal dimension and the three-point function coefficient (OPE data) of the lightest operator present in the OPE of the two operators appearing in the four-point function we started with Over the years, these techniques proved to be extremely efficient and achieved impressive results, as can be seen in [4] for a recent review. The crossing relations are very intricate equations, and in generic space–time dimensions, it is extremely complicated to systematically find solutions This has been the focus of some investigations and it has become clear that an analytic approach can be developed to give powerful results.
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