Holographic reconstruction for AdS Wilson line networks and scalar Witten diagrams

TL;DR

This paper derives a holographic reconstruction formula for gravitational Wilson line networks in AdS$_2$, connecting them to scalar Witten diagrams and providing explicit integral and series representations.

Contribution

It introduces a new integral formula for gravitational Wilson line networks in AdS$_2$ and relates them to scalar Witten diagrams, with explicit series solutions.

Findings

Explicit integral form for Wilson line networks.

Series solutions as rational functions of endpoints.

Relations established between Wilson lines and Witten diagrams.

Abstract

We find a holographic reconstruction formula for gravitational Wilson line network operators in AdS$_2$ evaluated between Ishibashi states of the algebra $sl(2,\mathbb{R})$. It is given in integral form where the integrand is the global conformal block multiplied by a smearing function which is the product of the scalar bulk-to-boundary propagators. The integral can be explicitly calculated as multidimensional series of which arguments are rational functions of endpoint coordinates. In the case of two and three endpoints the resulting expressions allow one to establish a number of relations between the gravitational Wilson line networks and Witten diagrams for massive scalar fields in AdS$_2$.