Operator product expansion for conformal defects

TL;DR

This paper develops a comprehensive framework for the operator product expansion of scalar conformal defects in conformal field theory, introducing defect OPE blocks, integral representations, and a duality between defects of different codimensions.

Contribution

It introduces defect OPE blocks, integral representations via shadow formalism, and a duality between conformal defects of different codimensions, extending the understanding of defect structures in CFT.

Findings

Derived integral representation of defect OPE blocks.

Established equations of motion for defect OPE blocks as scalar fields.

Proved the equivalence between defect OPE blocks and local operator OPE blocks.

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.