Conformal dispersion relations for defects and boundaries

TL;DR

This paper develops dispersion relations for defect conformal field theories, enabling the reconstruction of two-point correlators from discontinuities, with applications to holographic correlators and boundary CFT data in the epsilon expansion.

Contribution

It introduces simple dispersion relations for defect CFT correlators, connecting bulk and defect OPEs, and applies them to holographic and boundary models.

Findings

Reproduced known results in $ ext{AdS}_5$/CFT$_4$ holography.

Derived boundary correlator results in the $O(N)$ model.

Proposed a double discontinuity relation controlled by defect OPE.

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.