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Operator product expansion for conformal defects

Masayuki Fukuda, Nozomu Kobayashi, Tatsuma Nishioka

TL;DR

The paper analyzes the operator product expansion for conformal defects of any codimension by organizing defect contributions into defect OPE blocks, each corresponding to a conformal multiplet of a primary operator and its descendants. It uses the shadow formalism to obtain an integral representation of these blocks and imposes a monodromy condition to project out shadow contributions, deriving Casimir and range-type constraint equations that the blocks satisfy on the defect moduli space. A Radon transform framework links the defect OPE blocks to local AdS bulk fields, yielding an inversion formula that reproduces the Euclidean HKLL reconstruction of bulk operators from boundary defect data. A duality between defects of complementary codimensions and a codimension-two defect–local-operator equivalence are established, extending the holographic dictionary for non-local objects in AdS/CFT and providing a unified OPE/defect-block perspective across codimensions.

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.

Operator product expansion for conformal defects

TL;DR

The paper analyzes the operator product expansion for conformal defects of any codimension by organizing defect contributions into defect OPE blocks, each corresponding to a conformal multiplet of a primary operator and its descendants. It uses the shadow formalism to obtain an integral representation of these blocks and imposes a monodromy condition to project out shadow contributions, deriving Casimir and range-type constraint equations that the blocks satisfy on the defect moduli space. A Radon transform framework links the defect OPE blocks to local AdS bulk fields, yielding an inversion formula that reproduces the Euclidean HKLL reconstruction of bulk operators from boundary defect data. A duality between defects of complementary codimensions and a codimension-two defect–local-operator equivalence are established, extending the holographic dictionary for non-local objects in AdS/CFT and providing a unified OPE/defect-block perspective across codimensions.

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.

Paper Structure

This paper contains 29 sections, 176 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Conformal defects
  3. Embedding space formalism
  4. Correlators in embedding space
  5. Conformal defects in embedding space
  6. Correlators in defect CFT
  7. Dual frame and dual defect
  8. Codimension-two defects
  9. Defect OPE blocks
  10. Integral representation and monodromy prescription
  11. Constraint equations
  12. Dual description
  13. Codimension-two defect
  14. Comparison to the OPE block
  15. Moduli space of conformal defects
  16. ...and 14 more sections

Figures (8)

  • Figure 1: The Radon transform between the AdS space and the moduli space ${\cal M}^{(d,m)}$ of conformal defects
  • Figure 2: A codimension-$m$ spherical defect and its dual defect
  • Figure 3: The defect duality between a codimension-two defect and a codimension-$d$ defect when $d=3$
  • Figure 4: A schematic picture of the OPE for conformal defects
  • Figure 5: A totally geodesic submanifold in $\mathbb{H}^{d+1}$ that is anchored on a conformal defect in $\mathbb{R}^d$
  • ...and 3 more figures