Abstract
We consider holographic CFTs and study their large N expansion. We use Polyakov-Mellin bootstrap to extract the CFT data of all operators, including scalars, till O(1/N4). We add a contact term in Mellin space, which corresponds to an effective ϕ4 theory in AdS and leads to anomalous dimensions for scalars at O(1/N2). Using this we fix O(1/N4) anomalous dimensions for double trace operators finding perfect agreement with [1] (for ∆ϕ = 2). Our approach generalizes this to any dimensions and any value of conformal dimensions of external scalar field. In the second part of the paper, we compute the loop amplitude in AdS which corresponds to non-planar correlators of in CFT. More precisely, using CFT data at O(1/N4) we fix the AdS bubble diagram and the triangle diagram for the general case.
Highlights
In this paper we will focus on holographic CFTs aka CFTs with large number of degrees of freedom (N )
We add a contact term in Mellin space, which corresponds to an effective φ4 theory in AdS and leads to anomalous dimensions for scalars at O(1/N 2)
Only finite number of operators with non-zero anomalous dimension present depending on the number of contact diagrams are present in the bulk, at O(1/N 4) we will see that γn(2,) is non-zero for all spins
Summary
We give a brief overview of working rules of Polyakov Mellin bootstrap. Since we are expanding the same four point function the two conformal block decomposition should be same but this is not manifest. Demanding this equivalence give us the usual bootstrap condition (crossing equation), u ∆φ. It is a legitimate thing to do to add all three channel (s,t and u) and expand four point function in that basis which would be consistent with OPE. One will find contributions from double trace operators (with exact dimension 2∆φ + 2n) which are absent in the OPE Cancellation of this contribution constraint the spectrum of CFT.
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