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AdS$_4$/CFT$_3$ for Unprotected Operators

Shai M. Chester

TL;DR

This work advances AdS$_4$/CFT$_3$ by computing the leading $1/c_T$ corrections to the stress-tensor four-point function in $ ext{ABJ(M)}$ via a tree-level AdS$_4$ Mellin amplitude. It develops a practical lightcone/Mellin-residue algorithm to extract corrected scaling dimensions and OPE coefficients for both protected and unprotected operators, and then validates these results against localization formulas and the $ ext{N}=8$ numerical bootstrap at large $c_T$. The protected data, including the short multiplets $(B,2)$ and $(B,+)$, perfectly agrees with all-orders CFT$_3$ results, while the unprotected/semi-short sectors also show precise agreement with bootstrap predictions; the second-twist unprotected data matches bootstrap averages at large spin. Overall, the paper provides a first precise check of unprotected observables in AdS$_4$/CFT$_3$ and lays groundwork for higher-loop and more general operator analyses in this holographic setting.

Abstract

We consider the four-point function of the lowest scalar in the stress-energy tensor multiplet in $\mathcal{N}=8$ ABJ(M) theory \cite{Aharony:2008ug, Aharony:2008gk}. At large central charge $c_T\sim N^{3/2}$, this correlator is given by the corresponding holographic correlation function in 11d supergravity on $AdS_4\times S^7$. We use Mellin space techniques to compute the leading $1/c_T$ correction to anomalous dimensions and OPE coefficients of operators that appear in this holographic correlator. For half and quarter-BPS operators, we find exact agreement with previously computed localization results. For the other BPS and non-BPS operators, our results match the $\mathcal{N}=8$ numerical bootstrap for ABJ(M) at large $c_T$, which provides a precise check of unprotected observables in AdS/CFT.

AdS$_4$/CFT$_3$ for Unprotected Operators

TL;DR

This work advances AdS/CFT by computing the leading corrections to the stress-tensor four-point function in via a tree-level AdS Mellin amplitude. It develops a practical lightcone/Mellin-residue algorithm to extract corrected scaling dimensions and OPE coefficients for both protected and unprotected operators, and then validates these results against localization formulas and the numerical bootstrap at large . The protected data, including the short multiplets and , perfectly agrees with all-orders CFT results, while the unprotected/semi-short sectors also show precise agreement with bootstrap predictions; the second-twist unprotected data matches bootstrap averages at large spin. Overall, the paper provides a first precise check of unprotected observables in AdS/CFT and lays groundwork for higher-loop and more general operator analyses in this holographic setting.

Abstract

We consider the four-point function of the lowest scalar in the stress-energy tensor multiplet in ABJ(M) theory \cite{Aharony:2008ug, Aharony:2008gk}. At large central charge , this correlator is given by the corresponding holographic correlation function in 11d supergravity on . We use Mellin space techniques to compute the leading correction to anomalous dimensions and OPE coefficients of operators that appear in this holographic correlator. For half and quarter-BPS operators, we find exact agreement with previously computed localization results. For the other BPS and non-BPS operators, our results match the numerical bootstrap for ABJ(M) at large , which provides a precise check of unprotected observables in AdS/CFT.

Paper Structure

This paper contains 12 sections, 48 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Four-point function of stress-tensor
  3. $c_T\to\infty$ theory
  4. $1/c_T$ corrections
  5. Mellin space amplitude
  6. Extracting CFT data
  7. $a_{(B,2)}^{(1)}$ and $a_{(B,+)}^{(1)}$
  8. $a_{(A,+)_j}^{(1)}$ for $j=0,2,4,6$ and $a_{(A,2)_j}^{(1)}$ for $j=1,3,5$
  9. $\Delta^{(1)}_{(A,0)_{j,0,0}}$ for $j=0,2,\dots,12$ and $\Delta^{(1)}_{(A,0)_{j,1,\overline{q}}}$ for $j=0,2,\dots,10$
  10. Comparison to exact results and numerical bootstrap
  11. Conclusion
  12. Lightcone blocks

Figures (1)

  • Figure 1: The $\lambda_{(A,2)_j}^2$ and $\lambda_{(A,+)_j}^2$ OPE coefficients with spins $j=1,3,5$ and $j=0,2,4$, respectively, in terms of the stress-tensor coefficient $c_T$, where the plot ranges from the generalized free field theory limit $c_T\to\infty$ to the numerical point $\frac{16}{c_T}\approx.71$ where $\lambda_{(B,2)}^2=0$, which is near the lowest interacting theory ABJ$_1$ with $c_T=.75$. The orange dotted lines show the analytic $1/c_T$ corrections \ref{['A2Final']} and \ref{['AplusFinal']}.