Polyakov-Mellin Bootstrap for AdS loops
Kausik Ghosh
TL;DR
The paper develops the Polyakov-Mellin bootstrap for holographic CFTs in the large-$N$ expansion and extracts the CFT data of all operators up to $O(1/N^4)$. It introduces a Mellin-space contact term corresponding to a bulk $\phi^4$ interaction, producing $O(1/N^2)$ scalar anomalous dimensions and enabling closed-form $O(1/N^4)$ anomalous dimensions for double-trace operators, agreeing with LoopAl for $\Delta_{\phi}=2$ in $d=4$ and generalizing to arbitrary $\Delta_{\phi}$ and dimensions. It then computes AdS loop amplitudes, fixing the AdS bubble and triangle diagrams from the PM data, thus connecting CFT data to bulk loop computations. The work demonstrates the power of the PM bootstrap in odd dimensions and provides explicit large-spin expansions for $\gamma^{(2)}_{n,\ell}$, offering a practical route to AdS loop reconstruction from boundary data.
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 $O(1/N^4)$. We add a contact term in Mellin space, which corresponds to an effective $φ^4$ theory in AdS and leads to anomalous dimensions for scalars at $O(1/N^2)$. Using this we fix $O(1/N^4)$ anomalous dimensions for double trace operators finding perfect agreement with \cite{loopal} (for $Δ_φ=2$). 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 $O(1/N^4)$ we fix the AdS bubble diagram and the triangle diagram for the general case.