Conformal dispersion relations for defects and boundaries

Abstract

We derive a dispersion relation for two-point correlation functions in defect conformal field theories. The correlator is expressed as an integral over a (single) discontinuity that is controlled by the bulk channel operator product expansion (OPE). This very simple relation is particularly useful in perturbative settings where the discontinuity is determined by a subset of bulk operators. In particular, we apply it to holographic correlators of two chiral primary operators in $\mathcal{N}=4$ Super Yang-Mills theory in the presence of a supersymmetric Wilson line. With a very simple computation, we are able to reproduce and extend existing results. We also propose a second relation, which reconstructs the correlator from a double discontinuity, and is controlled by the defect channel OPE. Finally, for the case of codimension-one defects (boundaries and interfaces) we derive a dispersion relation which receives contributions from both OPE channels and we apply it to the boundary correlator in the $O(N)$ critical model. We reproduce the order ${\epsilon }^2$ result in the $\epsilon$-expansion using as input a finite number of boundary CFT data.

arXiv comment36 pages, 3 figures

References in corpus (52)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Analyticity in Spin in Conformal Theories
• The Bootstrap Program for Boundary CFT_d
• On Short and Semi-Short Representations for Four Dimensional Superconformal Symmetry
• Boundary and Interface CFTs from the Conformal Bootstrap
• One-point Functions in Defect CFT and Integrability
• Half-BPS Wilson loop and AdS${}_2$/CFT${}_1$
• Defects in conformal field theory
• Bootstrap equations for $\mathcal{N}=4$ SYM with defects
• Bootstrapping the half-BPS line defect
• Dispersive CFT Sum Rules
• Universality at large transverse spin in defect CFT
• Taming the $\epsilon$-expansion with Large Spin Perturbation Theory
• The extraordinary boundary transition in the 3d O(N) model via conformal bootstrap
• Line and surface defects for the free scalar field
• The Mellin Formalism for Boundary CFT${}_d$
• An Analytic Approach to BCFT${}_d$
• On the OPE of 1/2 BPS Short Operators in N=4 SCFT${}_4$
• Analytic Bootstrap for Boundary CFT
• Line defects and radiation in $\mathcal{N}=2$ theories
• Integrability and Conformal Bootstrap: One Dimensional Defect CFT
• CFT in AdS and boundary RG flows
• Superconformal surfaces in four dimensions
• Wilson lines as superconformal defects in ABJM theory: a formula for the emitted radiation
• Bootstrapping the half-BPS line defect CFT in $\mathcal{N}=4$ SYM at strong coupling
• 3d Abelian Gauge Theories at the Boundary
• Defect CFT in the 6d (2,0) theory from M2 brane dynamics in AdS${}_7\times$S${}^4$
• Analytic bootstrap and Witten diagrams for the ABJM Wilson line as defect CFT${}_1$
• Bootstrapping boundary-localized interactions
• Two-point functions in AdS/dCFT and the boundary conformal bootstrap equations
• Bootstrapping holographic defect correlators in $\mathcal{N}=4$ super Yang-Mills
• $O(N)$ Models with Boundary Interactions and their Long Range Generalizations
• Calogero-Sutherland Approach to Defect Blocks
• Taming Defects in $\mathcal{N}=4$ Super-Yang-Mills
• Exact Bremsstrahlung functions in ABJM theory
• Monodromy Defects in Free Field Theories
• Bootstrapping boundary-localized interactions II: Minimal models at the boundary
• Exact Correlators on the Wilson Loop in $\mathcal{N}=4$ SYM: Localization, Defect CFT, and Integrability
• Dispersion Relation for CFT Four-Point Functions
• On Analytic Bootstrap for Interface and Boundary CFT
• Bootstrapping line defects in $\mathcal{N}=2$ theories
• A dispersion relation for defect CFT
• Conformal Surface Defects in Maxwell Theory are Trivial
• Bootstrapping Monodromy Defects in the Wess-Zumino Model
• Free energy and defect $C$-theorem in free scalar theory
• Classifying Superconformal Defects in Diverse Dimensions Part I: Superconformal Lines
• Free energy and defect $C$-theorem in free fermion
• Superconformal Boundaries in $4-\epsilon$ dimensions
• Mellin amplitudes for 1$d$ CFT
• Dispersive Sum Rules in AdS${}_2$
• Emitted Radiation and Geometry
• Surface Defects from Fractional Branes -- I

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• Perturbative bootstrap of the Wilson-line defect CFT: Bulk-defect-defect correlators
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• Long-Range Order in a Strictly Short-Range Quasi-2D XY Model: When Critical Fluctuations Matter