# Sketching a Proof of the Maldacena Conjecture at Small Radius

## TL;DR

This paper provides a sketch of how superstring theory on small-radius AdS space can be related to super-Yang-Mills theory through a detailed analysis of the pure spinor worldsheet action and its components, supporting the Maldacena conjecture.

## Contribution

It demonstrates that the pure spinor superstring action can be decomposed to reproduce super-Yang-Mills Feynman diagrams at small AdS radius, offering a new proof approach.

## Key findings

- Superstring amplitudes match super-Yang-Mills Feynman diagrams at small radius.
- The BRST-trivial term describes propagators in the boundary region.
- The antisymmetric B-term generates super-Yang-Mills vertices.

## Abstract

At small AdS radius, the superstring on $AdS_5\times S^5$ was conjectured by Maldacena to be equivalent to ${\cal N}=4$ super-Yang-Mills at small `t Hooft coupling where thickened Feynman diagrams can be used to compute scattering amplitudes. It was previously shown that the pure spinor worldsheet action of the $AdS_5\times S^5$ superstring can be expressed as the sum of a BRST-trivial term and a `B-term' which is antisymmetric in worldsheet derivatives. Using the explicit form of the pure spinor vertex operators, it will be argued here that the free super-Yang-Mills Feynman diagrams are described by the BRST-trivial term where the thickened propagators are the regions of the string worldsheet near the AdS boundary and the holes are the regions near the AdS horizon. Evidence will then be presented that the antisymmetric B-term generates the super-Yang-Mills vertex so that, at small radius and arbitrary genus, the superstring amplitudes correctly reproduce the super-Yang-Mills Feynman diagram expansion.