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 doublediscontinuity 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 , heavyheavy-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 + 1.2) String/M-theory landscape beyond supergravity: What is the landscape of consistent AdS vacua? This question is old because it is challenging. The scales in the problem are the AdS scale L, the KK scale L M (where M is some internal manifold), the Planck scale ℓ p , and (in string theory) the string scale ℓ s . Reliable bulk construction of scale-separated AdS vacua (L M ≪ L) in string theory requires control at finite α ′ and, perhaps, at finite g s . This is not currently possible without resorting to parametric effective field theory arguments, and/or assumptions about the structure of α ′ perturbation theory and/or backreaction of sources, which existing works all employ in some way. 13) Bulk reconstruction from large N bootstrap: There has been recent progress in building up AdS amplitudes from large N bootstrap, or bootstrap-inspired, methods. This is true both for "bottom-up" ingredients such as Witten diagrams [2,[20][21][22][23][24][25], and topdown, complete amplitudes in string/M-theory at both genus zero [26][27][28][29][30][31][32][33][34] and genus one [35][36][37][38][39][40][41][42][43]. It is natural to apply these insights to more abstract investigations of the AdS landscape.These topics invite many questions. We will answer the following one:Define D as the number of "large" bulk dimensions, of order the AdS scale. Given the local operator data of a large ∆ gap CFT to leading order in 1/c, what is D?Unlike other questions in the realm of holographic spacetime, this is not possible to answer classically using a finite number of fields: consistent truncations exist.On the other hand, quantum effects in AdS can tell the difference between D dimensions and d + 1 dimensions. Our key idea is that AdS loop amplitudes are se...

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