[Paper Review] The Witten Diagram Bootstrap for Holographic Defects

Aleix Gimenez-Grau|arXiv (Cornell University)|Jun 20, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper introduces a supersymmetry-based bootstrap method to compute correlation functions involving two local operators and a holographic defect in AdS/CFT, specifically for chiral-primary operators and a half-BPS Wilson loop in $σ=4$ SYM. The approach yields a closed-form expression in position and Mellin space that matches localization and explicit calculations up to contact terms, offering a generalizable framework for supersymmetric defect CFTs.

ABSTRACT

We study the AdS/CFT correspondence with a brane extending in AdS, a setup which is dual to CFT in the presence of a defect. We focus on the correlation function of two local operators and the defect, which is the simplest observable with non-trivial dependence on kinematical invariants. We propose a method to bootstrap this observable which relies on supersymmetry, but does not require detailed knowledge of the supergravity and brane effective actions. After developing the method in full generality, we turn to the case of two chiral-primary operators and a half-BPS Wilson loop in $\mathcal N=4$ SYM. Working in the leading supergravity approximation, we determine the correlator in closed form for chiral-primary operators of arbitrary length. The result has elegant expressions in position and Mellin space, and it agrees with localization and an explicit calculation up to contact terms. More generally, we expect our method to be suitable in other holographic setups in the presence of supersymmetric defects.

Motivation & Objective

• To develop a bootstrap method for correlation functions involving local operators and a holographic defect in AdS/CFT, without requiring detailed knowledge of supergravity or brane actions.
• To apply this method to compute the correlator of two chiral-primary operators and a half-BPS Wilson loop in $σ=4$ SYM at strong coupling.
• To derive closed-form expressions in both position and Mellin space, verifying consistency with localization and explicit calculations.
• To establish a general framework applicable to other holographic setups with supersymmetric defects.

Proposed method

• The method relies on supersymmetry to constrain the form of the correlation function, bypassing the need for explicit supergravity or brane effective actions.
• It employs Witten diagrams in position space, including contact, bulk-exchange, and defect-exchange diagrams, to systematically build the correlator.
• The analysis is extended to Mellin space, where the correlator is expressed as a sum of Mellin amplitudes with polynomial structures in $ξ$ and $η$, ensuring consistency with conformal symmetry.
• The superconformal Ward identity is rewritten in Mellin space using symmetric combinations of $z$ and $\bar{z}$, leading to polynomial differential constraints in $\xi$ and $\eta$.
• The method leverages the $R$-symmetry channel decomposition and identifies polynomial combinations $\zeta_{\pm}^{(j)}$ that satisfy the Ward identity.
• It connects to known results via a correction term that accounts for differences in conventions and topological contributions, ensuring agreement with localization at large $\lambda$.

Experimental results

Research questions

• RQ1 How can one bootstrap the correlation function of two local operators and a defect in AdS/CFT without full knowledge of the supergravity or brane actions?
• RQ2 What is the closed-form expression for the correlator of two chiral-primary operators and a half-BPS Wilson loop in $σ=4$ SYM at strong coupling?
• RQ3 How do the results in position space and Mellin space compare with localization and explicit calculations, particularly regarding contact terms?
• RQ4 Can the proposed method be generalized to other holographic setups with supersymmetric defects?

Key findings

• The correlator of two chiral-primary operators and a half-BPS Wilson loop in $σ=4$ SYM is derived in closed form in position space, matching known results up to contact terms.
• The Mellin space representation of the correlator is expressed as a sum of polynomial terms in $\xi$ and $\eta$, reflecting the underlying conformal symmetry and superconformal Ward identities.
• The result agrees with localization at large $\lambda$, confirming the validity of the bootstrap approach in the strong coupling limit.
• The method correctly accounts for topological contributions through a correction term involving modified Bessel functions and $\lambda$-dependent coefficients.
• The Ward identity in Mellin space is reformulated as a system of polynomial differential equations in $\xi$ and $\eta$, ensuring consistency with superconformal symmetry.
• The framework is generalizable and expected to apply to other holographic setups with supersymmetric defects, offering a powerful alternative to diagrammatic computations.