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Growing Extra Dimensions in AdS/CFT

Luis F. Alday, Eric Perlmutter

TL;DR

The paper addresses how many large bulk dimensions $D$ can emerge in AdS/CFT from strongly coupled, large-$N$ CFT data by deriving a positive-definite 1-loop double-discontinuity sum rule and examining its flat-space limit. It shows that polynomial growth in the density of single-trace operators signals extra bulk dimensions, with $D$ determined by the growth exponent and potential global-symmetry charges; it also uncovers an OPE universality at the string scale where heavy-heavy-light couplings scale linearly with the heavy dimension. Additional results link these findings to SCFTs with Higgs/Coulomb branches, constrain holographic spectra, and motivate a Holographic Hierarchy Conjecture tying locality to the absence of global symmetries. The work lays a framework for bootstrapping AdS landscapes from CFT data and suggests criteria for viable vacua with $D<10$ and beyond. Overall, it provides concrete connections between high-energy CFT data, bulk spacetime structure, and consistency conditions from string/M-theory.

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: $D=d+1$

Growing Extra Dimensions in AdS/CFT

TL;DR

The paper addresses how many large bulk dimensions can emerge in AdS/CFT from strongly coupled, large- CFT data by deriving a positive-definite 1-loop double-discontinuity sum rule and examining its flat-space limit. It shows that polynomial growth in the density of single-trace operators signals extra bulk dimensions, with determined by the growth exponent and potential global-symmetry charges; it also uncovers an OPE universality at the string scale where heavy-heavy-light couplings scale linearly with the heavy dimension. Additional results link these findings to SCFTs with Higgs/Coulomb branches, constrain holographic spectra, and motivate a Holographic Hierarchy Conjecture tying locality to the absence of global symmetries. The work lays a framework for bootstrapping AdS landscapes from CFT data and suggests criteria for viable vacua with and beyond. Overall, it provides concrete connections between high-energy CFT data, bulk spacetime structure, and consistency conditions from string/M-theory.

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, , from strongly-coupled CFT 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 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 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 , heavy-heavy-light three-point functions, with heavy operators that are parametrically lighter than a power of , 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:

Paper Structure

This paper contains 20 sections, 95 equations.

Table of Contents

  1. Introduction
  2. Counting dimensions with 1-loop amplitudes
  3. 1-loop sum rule
  4. Extra dimensions at large $\Delta_{\rm gap}$
  5. Adding global charge
  6. SCFTs
  7. $1/\Delta_{\rm gap}$ corrections and AdS quartic vertices
  8. Bounding holographic spectra
  9. OPE universality at the string scale
  10. Asymptotic linearity of planar non-BPS spectra
  11. On holographic hierarchies
  12. Future directions
  13. Supplementary details at 1-loop
  14. $C_{\phi\phi[{\cal O}{\cal O}]_{n,\ell}}$ at large $\Delta_{\rm gap}$
  15. Computing $\langle \gamma_{n,\ell}(p)\rangle$
  16. ...and 5 more sections