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 CFTd 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 [1], that any CFT with large higher-spin gap and no global symmetries has a holographic hierarchy: D = d + 1.
Highlights
Spectrum, another oft-invoked avatar of strong coupling
The idea is that unitarity of 1-loop AdS amplitudes, dual to nonplanar 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
Summary
As stated in the introduction, we will present a formula from which one can derive D from planar single-trace data in a dual CFT. (This scaling follows from the coincident limit of the connected fourpoint function φφOO .) On the other hand, those with τO = ∆φ + Z lead to double-trace mixing between [φφ]n,l and the finite subset of [OO]n′,l with ∆φ + n = ∆O + n′. These operators contribute in the same way as [φφ]n,l. The t-channel exchange φp → p → φp implies a contribution to tree-level anomalous dimensions γn(1,l)(p) This exchange is manifestly proportional to the squared OPE coefficient Cφ2pp at leading order in 1/c, and so is γn(1,l)(p).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
