Proof of the Witten-Yau Conjecture

TL;DR

This paper rigorously proves the Witten-Yau conjecture, establishing that the conformal boundary of AdS space must have a non-negative scalar curvature to ensure a well-defined conformal field theory, using advanced conformal geometry tools.

Contribution

The paper provides a rigorous proof of the Witten-Yau conjecture employing conformal geometry techniques, clarifying conditions for the boundary metric in AdS/CFT correspondence.

Findings

Confirmed the necessity of non-negative scalar curvature on the boundary.

Applied conformal geometry tools like tractor bundles in the proof.

Established a rigorous mathematical foundation for the conjecture.

Abstract

The Witten-Yau theorem in the AdS/CFT correspondence conjectures that the conformal boundary to AdS space must possess a metric of non-negative scalar curvature for the conformal field theory defined thereon to be free of pathologies. By employing various tools from conformal geometry - such as almost Einstein structures, collapsing sphere products and tractor bundles - we rigorously prove this conjecture.