# Growing Extra Dimensions in AdS/CFT

## TL;DR

This paper develops a method to determine the number of large bulk dimensions in AdS/CFT from strongly-coupled CFT data, revealing universal properties and consistency conditions for emergent extra dimensions in holography.

## Contribution

It provides a prescription to compute large bulk dimensions from CFT data and establishes a positive sum rule for 1-loop amplitudes, advancing understanding of emergent dimensions in holography.

## Key findings

- Derived a positive-definite sum rule for 1-loop double-discontinuity.
- Proved AdS/CFT folklore relating unitarity to bulk dimensions.
- Discovered OPE universality at the string scale for heavy-heavy-light correlators.

## Abstract

What is the dimension of spacetime? We address this question in the context of the AdS/CFT Correspondence. We give a prescription for computing the number of large bulk dimensions, $D$, from strongly-coupled CFT$_d$ data, where "large" means parametrically of order the AdS scale. The idea is that unitarity of 1-loop AdS amplitudes, dual to non-planar CFT correlators, fixes $D$ in terms of tree-level data. We make this observation rigorous by deriving a positive-definite sum rule for the 1-loop double-discontinuity in the flat space/bulk-point limit. This enables us to prove an array of AdS/CFT folklore, and to infer new properties of large $N$ CFTs at strong coupling that ensure consistency of emergent large extra dimensions with string/M-theory. We discover an OPE universality at the string scale: to leading order in large $N$, heavy-heavy-light three-point functions, with heavy operators that are parametrically lighter than a power of $N$, are linear in the heavy conformal dimension. We explore its consequences for supersymmetric CFTs and explain how emergent large extra dimensions relate to a Sublattice Weak Gravity Conjecture for CFTs. Lastly, we conjecture, building on a claim of arXiv:0908.0756, that any CFT with large higher-spin gap and no global symmetries has a holographic hierarchy: $D=d+1$