Boundary Terms and Three-Point Functions: An AdS/CFT Puzzle Resolved

Abstract

$\mathcal{N}=8$ superconformal field theories, such as the ABJM theory at Chern-Simons level $k=1$ or $2$, contain 35 scalar operators ${\mathcal{O}}_{IJ}$ with $\Delta =1$ in the ${35}_v$ representation of SO(8). The 3-point correlation function of these operators is non-vanishing, and indeed can be calculated non-perturbatively in the field theory. But its AdS${}_4$ gravity dual, obtained from gauged $\mathcal{N}=8$ supergravity, has no cubic $A^3$ couplings in its Lagrangian, where $A^{IJ}$ is the bulk dual of ${\mathcal{O}}_{IJ}$. So conventional Witten diagrams cannot furnish the field theory result. We show that the extension of bulk supersymmetry to the AdS${}_4$ boundary requires the introduction of a finite $A^3$ counterterm that does provide a perfect match to the 3-point correlator. Boundary supersymmetry also requires infinite counterterms which agree with the method of holographic renormalization. The generating functional of correlation functions of the $\Delta =1$ operators is the Legendre transform of the on-shell action, and the supersymmetry properties of this functional play a significant role in our treatment.

67 pages

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• Quantum ${\varphi }^4$ Theory in AdS${}_4$ and its CFT Dual
• A 2d/1d Holographic Duality
• Supersymmetric solitons in gauged $\mathcal{N}=8$ supergravity
• Entropy functional and the holographic attractor mechanism
• Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization
• Hairy Black Holes and Duality in an Extended Supergravity Model
• Mass Deformations of the ABJM Theory: The Holographic Free Energy
• Test of the local form of higher-spin equations via AdS/CFT
• On the On-Shell: The Action of AdS${}_4$ Black Holes
• Four-dimensional Traversable Wormholes and Bouncing Cosmologies in Vacuum
• On the supersymmetry invariance of flat supergravity with boundary
• Susy Q and spatially modulated deformations of ABJM theory
• Unconventional Supersymmetry at the Boundary of AdS_4 Supergravity
• Supersymmetric D3/D5 for massive defects on curved space
• Double-Trace Flows and the Swampland
• Lectures on Higher Spin Holography
• Revisiting Extremal Couplings in AdS/CFT
• On (spinor)-helicity and bosonization in $Ad{S}_4/CF{T}_3$
• Exact holographic RG flows in extended SUGRA
• D-branes and Orbit Average
• On-shell action and the Bekenstein-Hawking entropy of supersymmetric black holes in $Ad{S}_6$
• Topological AdS/CFT and the $\Omega$ deformation
• Conformal field theories dual to quantum gravity with strongly coupled matter
• Super-Weyl Anomaly from Holography and Rigid Supersymmetry Algebra on Two-Sphere
• Holography for field theory solitons
• "$1k_F$" Singularities and Finite Density ABJM Theory at Strong Coupling
• Boundary Conditions in Topological AdS${}_4$/CFT${}_3$
• The Holographic Contributions to the Sphere Free Energy
• Lecture for the Fortieth Anniversary of Supergravity
• Wrapped $M5$-branes and complex saddle points
• Searching for strongly coupled AdS matter with multi-trace deformations