A central charge for sub-AdS holography

We synthesize and sharpen various observations about sub-AdS holography in the literature to associate a central charge to sub-AdS scales of AdS×X geometries. A key ingredient in our proposal is the idea that the AdS length is the length of the long string in a stack of N backreacted D-branes. This allows us to make statements about sub-AdS scales by considering long strings in sub-stacks of M < N branes. Our proposal applies in general dimensions, connects with and refines previous ideas about sub-matrix deconfinement & long strings, makes crucial use of the compact space, and is consistent with the expected thermodynamics of small black holes localized on X. Some of our arguments draw intuition from separating branes, and have natural connections to (heating up) the Coulomb branch of the gauge theory. We apply related ideas to non-conformal D-branes at zero and finite temperature, discuss the holographic bound and IR/UV duality in theories with 16 supercharges, and find broad consistency.