Growing Extra Dimensions in AdS/CFT

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: $D=d+1$

23+8 pages, +refs. v2: corrected typos, clarified discussion in Section 5. v3: minor clarification, added refs

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• Quantum Log-Corrections to Swampland Conjectures
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• Tipler Naked Singularities in $N$ Dimensions
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• Boundary Conformal Field Theory at Large Charge