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Simplicity of AdS Supergravity at One Loop

Luis F. Alday, Xinan Zhou

TL;DR

This work shows that $AdS_5\times S^5$ IIB supergravity becomes surprisingly simple at one loop when described in Mellin space, focusing on non-planar four-point holographic correlators. The authors develop a systematic algorithm to build one-loop Mellin amplitudes from tree-level data and prove a closed-form solution for the family $\langle \mathcal{O}_2^{SG}\mathcal{O}_2^{SG}\mathcal{O}_p^{SG}\mathcal{O}_p^{SG}\rangle$ that features only simultaneous poles in Mellin variables. The flat-space limit consistently reproduces the ten-dimensional IIB one-loop amplitude, providing nontrivial evidence that the hidden ten-dimensional conformal symmetry persists at one loop. Overall, the paper highlights a symmetry-driven, highly constrained structure for holographic one-loop corrections and proposes a framework—the pre-amplitudes—that may extend to broader classes of correlators and backgrounds.

Abstract

We demonstrate the simplicity of $AdS_5\times S^5$ IIB supergravity at one loop level, by studying non-planar holographic four-point correlators in Mellin space. We develop a systematic algorithm for constructing one-loop Mellin amplitudes from tree-level data, and obtain a simple closed form answer for the $\langle \mathcal{O}_2^{SG}\mathcal{O}_2^{SG}\mathcal{O}_p^{SG}\mathcal{O}_p^{SG} \rangle$ correlators. The structure of this expression is remarkably simple, containing only simultaneous poles in the Mellin variables. We also study the flat space limit of the Mellin amplitudes, which reproduces precisely the IIB supergravity one-loop amplitude in ten dimensions. Our results provide nontrivial evidence for the persistence of the hidden conformal symmetry at one loop.

Simplicity of AdS Supergravity at One Loop

TL;DR

This work shows that IIB supergravity becomes surprisingly simple at one loop when described in Mellin space, focusing on non-planar four-point holographic correlators. The authors develop a systematic algorithm to build one-loop Mellin amplitudes from tree-level data and prove a closed-form solution for the family that features only simultaneous poles in Mellin variables. The flat-space limit consistently reproduces the ten-dimensional IIB one-loop amplitude, providing nontrivial evidence that the hidden ten-dimensional conformal symmetry persists at one loop. Overall, the paper highlights a symmetry-driven, highly constrained structure for holographic one-loop corrections and proposes a framework—the pre-amplitudes—that may extend to broader classes of correlators and backgrounds.

Abstract

We demonstrate the simplicity of IIB supergravity at one loop level, by studying non-planar holographic four-point correlators in Mellin space. We develop a systematic algorithm for constructing one-loop Mellin amplitudes from tree-level data, and obtain a simple closed form answer for the correlators. The structure of this expression is remarkably simple, containing only simultaneous poles in the Mellin variables. We also study the flat space limit of the Mellin amplitudes, which reproduces precisely the IIB supergravity one-loop amplitude in ten dimensions. Our results provide nontrivial evidence for the persistence of the hidden conformal symmetry at one loop.

Paper Structure

This paper contains 18 sections, 136 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Four-Point Functions at Large $c$ and Strong Coupling
  3. The supergravity basis of one-half BPS operators
  4. Superconformal kinematics of four-point functions
  5. Solution to the superconformal constraints
  6. Correlators as functions of cross ratios
  7. Superconformal block decomposition
  8. Implications of ten dimensional hidden conformal symmetry
  9. Mellin representation
  10. Supergravity One-Loop Correlators in Mellin Space
  11. General structure of $\langle \mathcal{O}_2^{SG}\mathcal{O}_2^{SG}\mathcal{O}_p^{SG}\mathcal{O}_p^{SG}\rangle$ and outline of the algorithm
  12. Explicit examples
  13. Pre-amplitudes and general Mellin amplitudes of $\langle \mathcal{O}_2^{SG}\mathcal{O}_2^{SG}\mathcal{O}_p^{SG}\mathcal{O}_p^{SG}\rangle$
  14. Comments on higher-weight correlators
  15. Flat Space Limit
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Pole structures of the Mellin amplitude $\widetilde{\mathcal{M}}^{(2)}_{22pp}$ (for the example of $p=4$). The poles of the Gamma functions are represented by the straight lines, and the simultaneous poles in the ansatz (\ref{['ansatz22pp']}) are denoted by the dots at the intersections of the lines. The light blue region is where the poles of the tree level supergravity solution (\ref{['treeMellin']}) sit, which is bounded by the Gamma function poles. We have used the green color for the region $2\leq Re(s)< 8$ to emphasize that the two series of Gamma function poles do not overlap in this region, which corresponds to the long operators $[\mathcal{O}_2\mathcal{O}_2]_{n,\ell}$ with $n=2,\ldots p-1$.
  • Figure 2: Different Wick contractions for $\langle O_2^{SG}O_2^{SG}O_p^{SG}O_p^{SG}\rangle_{free}$. The thick grey line stands for Wick contractions of any number of strands indicated by the number beside it.