Abstract
We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.
Highlights
The most natural observables in a conformal field theory (CFT) are correlation functions
We study the kinematics of correlation functions of local and extended operators in a conformal field theory
As we shall point out, it is clear from the defect channel that the coupling to the defect is proportional to λλ, and it is easy to see that only the one-point functions of operators of with twist two are compatible with the requirement
Summary
The most natural observables in a conformal field theory (CFT) are correlation functions. The two-point function of the stress tensor is again a natural observable to focus on in this context Another example is provided by the class of line defects, e.g. Wilson and ’t Hooft lines, which correspond to massive probes. The tensor structures which appear in the correlation function of mixed symmetry bulk operators were recently studied in [21]. The bulk OPE was considered from a different point of view in [22]: this paper studies the expansion of a spherical defect in a sum over local operators, and describes the OPE-blocks for this kind of fusion. The minimal correlator which admits an expansion both in the bulk and the defect channels is the two-point function of local operators in the presence of the defect.
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