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Absence of $D^4 R^4$ in M-Theory From ABJM

Damon J. Binder, Shai M. Chester, Silviu S. Pufu

TL;DR

The work establishes the absence of the protected $D^4R^4$ interaction in the 11d M-theory S-matrix by connecting ABJM ${ m N}=8$ SCFT data to the flat-space limit of the four-graviton amplitude. It uses a two-pronged approach: (i) relate the fourth mass derivatives of the ${S^3}$ free energy, computed exactly via localization, to integrated four-point functions of the stress-tensor multiplet, and (ii) fix Mellin amplitudes for $raket{SSSS}$, $raket{PPPP}$, and $raket{SSPP}$ through supersymmetry and Ward identities, then translate to the ${11d}$ S-matrix. Localization data fixes $B_4^4$ (reproducing the known ${R^4}$ coefficient) and, crucially, forces $B_6^6$ and $B_4^6$ to vanish, implying $f_{D^4R^4}(s,t)=0$. This provides a direct CFT-based proof of the absence of the ${D^4R^4}$ term and highlights how protected CFT data can determine parts of the quantum gravity S-matrix with potential extensions to higher-derivative terms and other dimensions.

Abstract

Supersymmetry allows a $D^4 R^4$ interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the $D^4 R^4$ M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the $R^4$ interaction. The new ingredient in this extension is the interpretation of the fourth derivative of the mass deformed $S^3$ partition function of ABJM theory, which can be determined using supersymmetric localization, as a constraint on the Mellin amplitude associated with the stress tensor multiplet 4-point function. As part of this computation, we relate the 4-point function of the superconformal primary of the stress tensor multiplet of any 3d ${\cal N} = 8$ SCFT to some of the 4-point functions of its superconformal descendants. We also provide a concise formula for a general integrated 4-point function on $S^d$ for any $d$.

Absence of $D^4 R^4$ in M-Theory From ABJM

TL;DR

The work establishes the absence of the protected interaction in the 11d M-theory S-matrix by connecting ABJM SCFT data to the flat-space limit of the four-graviton amplitude. It uses a two-pronged approach: (i) relate the fourth mass derivatives of the free energy, computed exactly via localization, to integrated four-point functions of the stress-tensor multiplet, and (ii) fix Mellin amplitudes for , , and through supersymmetry and Ward identities, then translate to the S-matrix. Localization data fixes (reproducing the known coefficient) and, crucially, forces and to vanish, implying . This provides a direct CFT-based proof of the absence of the term and highlights how protected CFT data can determine parts of the quantum gravity S-matrix with potential extensions to higher-derivative terms and other dimensions.

Abstract

Supersymmetry allows a interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the interaction. The new ingredient in this extension is the interpretation of the fourth derivative of the mass deformed partition function of ABJM theory, which can be determined using supersymmetric localization, as a constraint on the Mellin amplitude associated with the stress tensor multiplet 4-point function. As part of this computation, we relate the 4-point function of the superconformal primary of the stress tensor multiplet of any 3d SCFT to some of the 4-point functions of its superconformal descendants. We also provide a concise formula for a general integrated 4-point function on for any .

Paper Structure

This paper contains 26 sections, 125 equations, 2 tables.

Table of Contents

  1. Introduction and Summary
  2. Stress tensor multiplet 4-point functions
  3. The ${\cal N} = 8$ stress tensor multiplet
  4. Ward identities
  5. A check: superconformal blocks
  6. Integrated correlators on $S^3$
  7. Three-parameter family of real mass deformations
  8. Integrated four point function
  9. Order $m^2$ terms
  10. Order $m^2$ terms in the free theory
  11. Order $m^2$ terms at strong coupling
  12. From ABJM to the 11d S-matrix
  13. Mellin amplitudes for $\langle SSSS\rangle$, $\langle PPPP\rangle$, and $\langle SSPP\rangle$
  14. From Mellin to position space
  15. Integrals of $\langle SSSS\rangle$, $\langle PPPP\rangle$, and $\langle SSPP\rangle$
  16. ...and 11 more sections