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Consequences of symmetry-breaking on conformal defect data

Bastien Girault, Miguel F. Paulos, Philine van Vliet

TL;DR

This work develops a general, elementary framework for defect-induced symmetry breaking in conformal field theories. By deriving defect soft theorems from broken Ward identities, the authors obtain integrated constraints on defect data (tilts and displacements) and on bulk–defect correlations, and then recast these into dispersive sum rules that yield sharp bounds on defect OPE data, especially for line defects. They instantiate the formalism with the pinning line defect in the O(N) model, performing perturbative checks and generating new predictions for defect three-point functions, bulk–defect two-point functions, and current-related data to order ε. The results offer a powerful, nonperturbative toolkit for constraining and bootstrapping defect CFTs, with direct implications for a wide class of line and higher-codimension defects across dimensions. The approach also produces concrete perturbative predictions that agree with existing calculations and provide new data for future checks and bootstrap applications.

Abstract

Conformal defects spontaneously break part of the symmetry algebra of a bulk CFT. We show that the broken Ward identities imply very general sum rules on the defect CFT data as well as on the DOE data of bulk operators, which we call defect soft theorems. Our derivation is elementary, allowing us to easily reproduce and generalize constraints on displacement and tilt operators previously obtained in the literature as well as a plethora of new ones, including constraints on bulk-defect correlation functions. For line defects we rewrite constraints in dispersive sum rule form, showing they lead to exact, optimal bounds on the OPE data of the defect. We test these sum rules in concrete perturbative examples, finding perfect agreement with existing calculations and making new predictions for various dCFT data.

Consequences of symmetry-breaking on conformal defect data

TL;DR

This work develops a general, elementary framework for defect-induced symmetry breaking in conformal field theories. By deriving defect soft theorems from broken Ward identities, the authors obtain integrated constraints on defect data (tilts and displacements) and on bulk–defect correlations, and then recast these into dispersive sum rules that yield sharp bounds on defect OPE data, especially for line defects. They instantiate the formalism with the pinning line defect in the O(N) model, performing perturbative checks and generating new predictions for defect three-point functions, bulk–defect two-point functions, and current-related data to order ε. The results offer a powerful, nonperturbative toolkit for constraining and bootstrapping defect CFTs, with direct implications for a wide class of line and higher-codimension defects across dimensions. The approach also produces concrete perturbative predictions that agree with existing calculations and provide new data for future checks and bootstrap applications.

Abstract

Conformal defects spontaneously break part of the symmetry algebra of a bulk CFT. We show that the broken Ward identities imply very general sum rules on the defect CFT data as well as on the DOE data of bulk operators, which we call defect soft theorems. Our derivation is elementary, allowing us to easily reproduce and generalize constraints on displacement and tilt operators previously obtained in the literature as well as a plethora of new ones, including constraints on bulk-defect correlation functions. For line defects we rewrite constraints in dispersive sum rule form, showing they lead to exact, optimal bounds on the OPE data of the defect. We test these sum rules in concrete perturbative examples, finding perfect agreement with existing calculations and making new predictions for various dCFT data.

Paper Structure

This paper contains 68 sections, 269 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Defects and broken symmetries
  3. Kinematics and correlators
  4. Broken Ward identities
  5. Integrated constraints: tilts
  6. Setup
  7. Form factor identities
  8. Defect correlator identities
  9. First identity
  10. Second identity
  11. Integrated constraints: displacements
  12. Setup
  13. Form factor identities
  14. Defect correlator identities
  15. First identity
  16. ...and 53 more sections

Figures (2)

  • Figure 1: Insertion of charge operator along the codimension-1 hypersurface $x^0=T$.
  • Figure 2: Soft sum rules and bounds. In this figure we show the S-channel action of soft sum rules on tilts and displacements, which have simple poles at $\Delta=1$ for the first and $\Delta=1,3$ for the second. Negative contributions to the sum rule from the negative regions in the vicinity of the poles are required to cancel positive contributions from the blue regions.