Abstract
We study the operator product expansion (OPE) for scalar conformal defects of any codimension in CFT. The OPE for defects is decomposed into “defect OPE blocks”, the irreducible representations of the conformal group, each of which packages the contribution from a primary operator and its descendants. We use the shadow formalism to deduce an integral representation of the defect OPE blocks. They are shown to obey a set of constraint equations that can be regarded as equations of motion for a scalar field propagating on the moduli space of the defects. By employing the Radon transform between the AdS space and the moduli space, we obtain a formula of constructing an AdS scalar field from the defect OPE block for a conformal defect of any codimension in a scalar representation of the conformal group, which turns out to be the Euclidean version of the HKLL formula. We also introduce a duality between conformal defects of different codimensions and prove the equivalence between the defect OPE block for codimension-two defects and the OPE block for a pair of local operators.
Highlights
Them a important class is conformal defects preserving a part of the conformal symmetry in CFT and allowing better control of determining correlation functions by the residual symmetry
We study the operator product expansion (OPE) for scalar conformal defects of any codimension in CFT
By employing the Radon transform between the AdS space and the moduli space, we obtain a formula of constructing an AdS scalar field from the defect OPE block for a conformal defect of any codimension in a scalar representation of the conformal group, which turns out to be the Euclidean version of the HKLL formula
Summary
The conformal symmetry SO(d + 1, 1), acting non-linearly on fields in d dimensions, can be realized linearly in the embedding space Rd+1,1 [13, 14, 46], which makes it easy to determine the correlators of higher spin fields [15, 16]. This formalism has been employed and expanded to describing conformal defects of any codimension and their correlators with local operators in [11, 12].
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