Abstract
We provide a framework for generic 4D conformal bootstrap computations. It is based on the unification of two independent approaches, the covariant (embedding) formalism and the non-covariant (conformal frame) formalism. We construct their main ingredients (tensor structures and differential operators) and establish a precise connection between them. We supplement the discussion by additional details like classification of tensor structures of n-point functions, normalization of 2-point functions and seed conformal blocks, Casimir differential operators and treatment of conserved operators and permutation symmetries. Finally, we implement our framework in a Mathematica package and make it freely available.
Highlights
In recent years a lot of progress has been made in understanding Conformal Field Theories (CFTs) in d ≥ 3 dimensions using the conformal bootstrap approach [1,2,3,4,5]
In this paper we have described a framework for performing computations in 4D CFTs by unifying two different approaches, the covariant embedding formalism and the noncovariant conformal frame formalism
In the embedding formalism we have explained the recipe for constructing tensor structures of n-point functions in the 6D embedding space
Summary
In recent years a lot of progress has been made in understanding Conformal Field Theories (CFTs) in d ≥ 3 dimensions using the conformal bootstrap approach [1,2,3,4,5] (see [6, 7] for recent introduction). In order to set up the crossing equations for a spinning 4-point function, first, one needs to find a basis of its tensor structures and second, to compute all the relevant conformal blocks. The difficulty of this task increases with the dimension d due to an increasing complexity of the d-dimesnional Lorentz group. In this approach 4- and higher-point tensor structures are hard to analyze due to a growing number of non-linear relations between the basic building blocks This problem is alleviated in the conformal frame ap-.
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