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Bootstrapping M-theory Orbifolds

Shai M. Chester, Silviu S. Pufu, Yifan Wang, Xi Yin

Abstract

We analyze correlation functions of $SU(k) \times SU(2)_F$ flavor currents in a family of three-dimensional ${\cal N}=4$ superconformal field theories, combining analytic bootstrap methods with input from supersymmetric localization. Via holographic duality, we extract gluon and graviton scattering amplitudes of M-theory on ${\rm AdS}_4\times S^7/\mathbb{Z}_k$ which contains a $\mathbb{C}^2/\mathbb{Z}_{k}$ orbifold singularity. From these results, we derive aspects of the effective description of M-theory on the orbifold singularity beyond its leading low energy limit. We also determine a threshold correction to the holographic correlator, which vanishes due to equal and opposite contributions from the two-loop gluon and the tree-level bulk graviton exchange.

Bootstrapping M-theory Orbifolds

Abstract

We analyze correlation functions of flavor currents in a family of three-dimensional superconformal field theories, combining analytic bootstrap methods with input from supersymmetric localization. Via holographic duality, we extract gluon and graviton scattering amplitudes of M-theory on which contains a orbifold singularity. From these results, we derive aspects of the effective description of M-theory on the orbifold singularity beyond its leading low energy limit. We also determine a threshold correction to the holographic correlator, which vanishes due to equal and opposite contributions from the two-loop gluon and the tree-level bulk graviton exchange.
Paper Structure (36 sections, 153 equations, 6 figures, 2 tables)

This paper contains 36 sections, 153 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The M-theory orbifold
  3. Effective theory and scattering amplitudes
  4. Constraints on the Coulomb branch
  5. AdS deformation and the holographic dual
  6. Relation between 7d fields and flavor current multiplets in the SCFT
  7. Superconformal kinematics
  8. Correlators and multiplet structure
  9. Central charges
  10. Kinematics of the $SU(k)$ current 4-point function ${\mathcal{G}}^{abcd}$
  11. Kinematics of the $SU(k)$-$SU(2)_F$ mixed 4-point function ${\mathcal{G}}^{abCD}$
  12. Mellin amplitudes in the large $N$ expansion
  13. Analytic bootstrap conditions and the large radius limit
  14. Relation to 7d amplitudes
  15. Structure of the four-gluon Mellin amplitude $M^{abcd}$
  16. ...and 21 more sections

Figures (6)

  • Figure 1: A poor man's portrait of the ${\mathbb C}^2/{\mathbb Z}_k$ orbifold singularity in M-theory.
  • Figure 2: Propagators and minimal coupling vertices for the gluons localized at the singularity (represented as the blue-shaded subspace) and the bulk graviton. In a Wilsonian effective theory, there are infinitely many more higher derivative vertices, e.g. those of Figures \ref{['fig:higherderAAAA']} and \ref{['fig:higherderAAgg']}.
  • Figure 3: Diagrams that exhibit unitarity cuts of the four-gluon amplitude into minimal coupling vertices, of momentum scalings $s^0,s^{3\over 2},s^3,s^3$ respectively. Note that the graviton propagator in the third diagram is not restricted to the orbifold locus, leading to an enhanced momentum scaling.
  • Figure 4: Some higher-derivative supervertices (represented by the red dots) that potentially contribute to the four-gluon amplitude. The distinct color structures have been suppressed in the diagrams and the corresponding momentum scalings are $s^2$ and $s^3$ respectively.
  • Figure 5: Diagrams that exhibit unitarity cuts of two-gluon-two-graviton amplitudes into minimal coupling vertices, of momentum scalings $s,s,s^{5\over 2}, s^{9\over 2}$ respectively. Note that the last diagram is beyond the derivative orders consider in this paper.
  • ...and 1 more figures