Notes on $n$-point Witten diagrams in AdS${}_2$

TL;DR

This paper analytically computes scalar n-point contact Witten diagrams in AdS$_2$, explores exchange diagrams, and verifies a multipoint Ward identity using perturbative methods in a simplified one-dimensional holographic setting.

Contribution

It provides explicit analytic results for scalar contact diagrams and exchange diagrams in AdS$_2$, and verifies a recent Ward identity in the context of holography.

Findings

Explicit formulas for scalar n-point contact Witten diagrams in AdS$_2$

Calculation of exchange diagrams and a Polyakov block in one dimension

Perturbative verification of a multipoint Ward identity in $ ext{AdS}_2$

Abstract

Witten diagrams provide a perturbative framework for calculations in Anti-de-Sitter space, and play an essential role in a variety of holographic computations. In the case of this study in AdS$_2$, the one-dimensional boundary allows for a simple setup, in which we obtain perturbative analytic results for correlators with the residue theorem. This elementary method is used to find all scalar $n$-point contact Witten diagrams for external operators of conformal dimension $\Delta=1$ and $\Delta=2$, and to determine topological correlators of Yang-Mills in AdS$_2$. Another established method is applied to explicitly compute exchange diagrams and give an example of a Polyakov block in $d=1$. We also check perturbatively a recently proposed multipoint Ward identity with the strong coupling expansion of the six-point function of operators inserted on the 1/2 BPS Wilson line in $\mathcal{N}$=4 SYM.