Operator product expansion for conformal defects

Abstract

We study the operator product expansion (OPE) for scalar conformal defects of\nany codimension in CFT. The OPE for defects is decomposed into "defect OPE\nblocks", the irreducible representations of the conformal group, each of which\npackages the contribution from a primary operator and its descendants. We use\nthe shadow formalism to deduce an integral representation of the defect OPE\nblocks. They are shown to obey a set of constraint equations that can be\nregarded as equations of motion for a scalar field propagating on the moduli\nspace of the defects. By employing the Radon transform between the AdS space\nand the moduli space, we obtain a formula of constructing an AdS scalar field\nfrom the defect OPE block for a conformal defect of any codimension in a scalar\nrepresentation of the conformal group, which turns out to be the Euclidean\nversion of the HKLL formula. We also introduce a duality between conformal\ndefects of different codimensions and prove the equivalence between the defect\nOPE block for codimension-two defects and the OPE block for a pair of local\noperators.\n