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On Superconformal Four-Point Mellin Amplitudes in Dimension $d>2$

Xinan Zhou

TL;DR

This work develops a universal Mellin-space framework to impose the superconformal Ward identities for SCFTs with $3\le d\le 6$, translating position-space constraints into Mellin-space difference equations that govern holographic four-point amplitudes. It implements a two-pronged holographic bootstrap in Mellin space, enabling both finite-pole and infinite-pole analyses, and applies the method to compute new results in $AdS_7\times S^4$ and $AdS_4\times S^7$, including next-next-to-extremal four-point functions, the first one-half BPS four-point function in $AdS_4\times S^7$, and the anomalous dimension of the lowest-dimension $R$-singlet double-trace operator. The results reproduce known AdS$_5\times S^5$ and AdS$_7\times S^4$ results, and yield predictions compatible with large-$C_T$ bootstrap bounds in 3d $\mathcal{N}=8$, providing a robust, scalable route to holographic correlators beyond traditional Witten-diagram methods. Overall, the paper showcases a universal, dimension-agnostic approach to holographic four-point functions that aligns with bootstrap constraints and enhances the computational toolkit for maximally supersymmetric AdS/CFT setups.

Abstract

We present a universal treatment for imposing superconformal constraints on Mellin amplitudes for $\mathrm{SCFT_d}$ with $3\leq d\leq 6$. This leads to a new technique to compute holographic correlators, which is similar but complementary to the ones introduced in [1,2]. We apply this technique to theories in various spacetime dimensions. In addition to reproducing known results, we obtain a simple expression for next-next-to-extremal four-point functions in $AdS_7\times S^4$. We also use this machinery on $AdS_4\times S^7$ and compute the first holographic one-half BPS four-point function. We extract the anomalous dimension of the R-symmetry singlet double-trace operator with the lowest conformal dimension and find agreement with the 3d $\mathcal{N}=8$ numerical bootstrap bound at large central charge.

On Superconformal Four-Point Mellin Amplitudes in Dimension $d>2$

TL;DR

This work develops a universal Mellin-space framework to impose the superconformal Ward identities for SCFTs with , translating position-space constraints into Mellin-space difference equations that govern holographic four-point amplitudes. It implements a two-pronged holographic bootstrap in Mellin space, enabling both finite-pole and infinite-pole analyses, and applies the method to compute new results in and , including next-next-to-extremal four-point functions, the first one-half BPS four-point function in , and the anomalous dimension of the lowest-dimension -singlet double-trace operator. The results reproduce known AdS and AdS results, and yield predictions compatible with large- bootstrap bounds in 3d , providing a robust, scalable route to holographic correlators beyond traditional Witten-diagram methods. Overall, the paper showcases a universal, dimension-agnostic approach to holographic four-point functions that aligns with bootstrap constraints and enhances the computational toolkit for maximally supersymmetric AdS/CFT setups.

Abstract

We present a universal treatment for imposing superconformal constraints on Mellin amplitudes for with . This leads to a new technique to compute holographic correlators, which is similar but complementary to the ones introduced in [1,2]. We apply this technique to theories in various spacetime dimensions. In addition to reproducing known results, we obtain a simple expression for next-next-to-extremal four-point functions in . We also use this machinery on and compute the first holographic one-half BPS four-point function. We extract the anomalous dimension of the R-symmetry singlet double-trace operator with the lowest conformal dimension and find agreement with the 3d numerical bootstrap bound at large central charge.

Paper Structure

This paper contains 16 sections, 93 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Part I: Superconformal Ward Identity for Mellin Amplitudes
  3. Superconformal Ward Identity in Position Space for $\mathrm{SCFT_{d>2}}$
  4. Mellin Representation
  5. Superconformal Ward Identity in Mellin Space
  6. Bootstrapping Holographic Mellin Amplitudes
  7. An example: stress-tensor multiplet four-point function for $AdS_5\times S^5$
  8. Part II: Applications
  9. 6D (2,0): 11D Supergravity on $AdS_7\times S^4$
  10. Some Superconformal Kinematics
  11. Selection Rules
  12. Next-Next-to-Extremal Four-Point Functions
  13. 3D $\mathcal{N}=8$: 11D Supergravity on $AdS_4\times S^7$
  14. The Stress-Tensor Multiplet Four-Point Function
  15. Anomalous Dimension of $[\mathcal{O}_2\mathcal{O}_2]_{\Delta=2,\ell=0,\text{singlet}}$
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Solution to the $\gamma_{ij}$ constraints.