Holography as a highly efficient RG flow: Part 1

We investigate how the holographic correspondence can be reconstructed as a special RG flow in a strongly interacting large $N$ field theory. We firstly define a "highly efficient RG flow" as one in which the cut-off in momentum space can be adjusted as a functional of the elementary fields, and of the external sources and of the background metric in order to be compatible with the following requirement: the Ward identities for single-trace operators involving conservation of energy, momentum and global charges must preserve the same form at every scale. In order to absorb the contributions of the multi-trace operators to these effective Ward identities, the external sources and the background metric need to be redefined at each scale, and thus they become dynamical as in the dual gravity equations. We give a schematic construction of such highly efficient RG flows using appropriate collective variables, leaving a more explicit construction in certain limits to the second part of this work. We find that all highly efficient RG flows that can be mapped to classical gravity equations have an additional "lifted Weyl symmetry", which is related to the ultraviolet Weyl symmetry, and which also has complete information about the gauge fixing of the diffeomorphism symmetry of the equivalent classical gravity equations. We present strong arguments for our claim that the presence of the lifted Weyl symmetry along with the requirement that the infrared end point can be characterised by a finite number of parameters, are sufficient conditions for a highly efficient RG flow to have a precise dual classical gravity description.