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All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry

Simon Caron-Huot, Anh-Khoi Trinh

TL;DR

This work analyzes four-point correlators of half-BPS operators in N=4 SYM at strong coupling, using a Lorentzian inversion bootstrap to determine tree-level AdS5×S5 amplitudes across all S5 harmonics. A hidden SO(10,2) conformal symmetry is uncovered, unifying the S5 harmonics into a ten-dimensional object and explaining the rational patterns in double-trace anomalous dimensions through ten-dimensional blocks that diagonalize mixing. The authors connect their results to Mellin-space formulations, reproducing the Rastelli–Zhou conjecture and deriving a generating function that yields the leading logarithmic terms at all loop orders. This framework suggests a deep, symmetry-driven structure behind holographic correlators and provides a practical route to higher-loop data and internal-space locality in AdS/CFT.

Abstract

We study correlators of four protected (half-BPS) operators in strongly coupled supersymmetric Yang-Mills theory. These are dual to tree-level supergravity amplitudes on AdS${}_5\times$S${}_5$ for various spherical harmonics on the five-sphere. We use conformal field theory methods, in particular a recently obtained Lorentzian inversion formula, to analytically bootstrap these correlators. The extracted $1/N^2$ double-trace anomalous dimensions confirm a simple pattern recently conjectured by Aprile, Drummond, Heslop and Paul. We explain this pattern by an unexpected ten-dimensional conformal symmetry which appears to be enjoyed by tree-level supergravity (or a suitable subsector of it). The symmetry combines all spherical harmonics into a single ten-dimensional object, and yields compact expressions for the leading logarithmic part of any half-BPS correlator at each loop order.

All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry

TL;DR

This work analyzes four-point correlators of half-BPS operators in N=4 SYM at strong coupling, using a Lorentzian inversion bootstrap to determine tree-level AdS5×S5 amplitudes across all S5 harmonics. A hidden SO(10,2) conformal symmetry is uncovered, unifying the S5 harmonics into a ten-dimensional object and explaining the rational patterns in double-trace anomalous dimensions through ten-dimensional blocks that diagonalize mixing. The authors connect their results to Mellin-space formulations, reproducing the Rastelli–Zhou conjecture and deriving a generating function that yields the leading logarithmic terms at all loop orders. This framework suggests a deep, symmetry-driven structure behind holographic correlators and provides a practical route to higher-loop data and internal-space locality in AdS/CFT.

Abstract

We study correlators of four protected (half-BPS) operators in strongly coupled supersymmetric Yang-Mills theory. These are dual to tree-level supergravity amplitudes on AdSS for various spherical harmonics on the five-sphere. We use conformal field theory methods, in particular a recently obtained Lorentzian inversion formula, to analytically bootstrap these correlators. The extracted double-trace anomalous dimensions confirm a simple pattern recently conjectured by Aprile, Drummond, Heslop and Paul. We explain this pattern by an unexpected ten-dimensional conformal symmetry which appears to be enjoyed by tree-level supergravity (or a suitable subsector of it). The symmetry combines all spherical harmonics into a single ten-dimensional object, and yields compact expressions for the leading logarithmic part of any half-BPS correlator at each loop order.

Paper Structure

This paper contains 24 sections, 125 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalities
  3. Superconformal Ward identities
  4. Lorentzian inversion formula
  5. Other decompositions in the literature
  6. Disconnected correlator: leading order $1/c^{0}$
  7. OPE coefficients
  8. Reorganizing into superconformal blocks
  9. $\Delta^{(8)}$: A remarkable eighth-order differential operator
  10. Tree-level correlator at strong coupling: order $1/c^{1}$
  11. Fixing three-point coefficients from crossing symmetry
  12. Chiral (protected) correlator: deducing the free correlator
  13. Reduced correlator: Anomalous dimensions at order $1/c$
  14. An accidental 10-dimensional conformal symmetry
  15. A mysterious equality with 10-dimensional $S$-matrix partial waves
  16. ...and 9 more sections

Figures (5)

  • Figure 1: Possible Witten diagrams for the disconnected correlator at order $1/c^0$. When $p\neq q$, only the first diagram contributes. For $p=q$, the two other diagrams can appear. See eq. \ref{['G disconnected']}.
  • Figure 2: $s$-channel OPE data for single- and double-traces can be reconstructed from cross-channel single-traces using the inversion formula. In particular, demanding crossing symmetry unique fixes single-trace three-point coefficients.
  • Figure 3: Even and odd spins are in blue and orange respectively. On the left is the multiplicity for $\tau=20$ and on the right is the multiplicity for $\tau=50$ of the same R-symmetry representation. Given eq. \ref{['multiplicity']}, the maximal multiplicity for [2, 12, 2] is 7 and 6 for even and odd spin respectively.
  • Figure 4: The flat-space limit of a scattering process in AdS${}_5 \times$ S${}_5$ map to one in $\mathbb{R}^{10}$. In the flat-space limit, the dilaton has $\Delta=4$. We conjecture that the ten-dimensional dilaton correlator $G_{10}$ can be projected to a AdS${}_5 \times$ S${}_5$ correlator by a suitable differential operator $\mathcal{D}_{p_1 p_2 p_3 p_4}$ acting on $G_{10}$.
  • Figure 5: The three type of planar Wick contractions between four operators at weak coupling; each line represents a non-empty bundle of propagator. The respective symmetry factors are 1, $k+1$ and 2, where $k$ is the number of diagonal propagators.