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Lectures on Higher Spin Holography

Charlotte Sleight

TL;DR

These notes situate higher-spin gauge theories within AdS/CFT, explaining how the ambient-space formalism renders higher-spin structures transparent and how holography can be used to reconstruct cubic interactions. They outline the GKP/W dictionary, the field/operator map for HS fields, and provide concrete computations of Witten diagrams with external HS legs, culminating in the holographic reconstruction of 0-0-$s$ cubic vertices from free CFT correlators. The approach leverages a scalar-dominated baseline controlled by differential operators to obtain spinning results, and it connects the HS bulk couplings to boundary three-point coefficients, supporting the HS/vector-model duality and offering a framework extendable to more complex HS interactions and mixed-symmetry cases. Overall, the work demonstrates a practical, holography-driven pathway to determining consistent HS interactions in AdS$_{d+1}$ from simple CFT data with explicit ambient-space techniques.

Abstract

These notes were prepared for the introductory lectures on Higher Spin Holography presented at the Twelfth Modave Summer School in Mathematical Physics 2016, aimed at Ph.D. students and researchers new to this topic. Contents: 1. Higher spin particles in AdS 2. The AdS/CFT correspondence and higher spins 3. The ambient space formalism 4. Witten diagrams for higher spin fields / higher spin interactions from CFT

Lectures on Higher Spin Holography

TL;DR

These notes situate higher-spin gauge theories within AdS/CFT, explaining how the ambient-space formalism renders higher-spin structures transparent and how holography can be used to reconstruct cubic interactions. They outline the GKP/W dictionary, the field/operator map for HS fields, and provide concrete computations of Witten diagrams with external HS legs, culminating in the holographic reconstruction of 0-0- cubic vertices from free CFT correlators. The approach leverages a scalar-dominated baseline controlled by differential operators to obtain spinning results, and it connects the HS bulk couplings to boundary three-point coefficients, supporting the HS/vector-model duality and offering a framework extendable to more complex HS interactions and mixed-symmetry cases. Overall, the work demonstrates a practical, holography-driven pathway to determining consistent HS interactions in AdS from simple CFT data with explicit ambient-space techniques.

Abstract

These notes were prepared for the introductory lectures on Higher Spin Holography presented at the Twelfth Modave Summer School in Mathematical Physics 2016, aimed at Ph.D. students and researchers new to this topic. Contents: 1. Higher spin particles in AdS 2. The AdS/CFT correspondence and higher spins 3. The ambient space formalism 4. Witten diagrams for higher spin fields / higher spin interactions from CFT

Paper Structure

This paper contains 20 sections, 187 equations, 3 figures.

Table of Contents

  1. Higher Spin Particles in AdS
  2. The AdS Geometry and Isometry Group
  3. The Conformal Boundary
  4. Unitary Irreducible Representations: Particles in AdS
  5. Lagrangian Formulation
  6. The AdS/CFT Correspondence and Higher Spins
  7. The GKP/W Formula
  8. Higher Spin Holography
  9. The Ambient Space Formalism
  10. Bulk Fields
  11. Boundary Fields
  12. Generating Functions
  13. Higher Spin Interactions from CFT
  14. Witten Diagrams in Higher Spin Theories
  15. Warm-up: Scalar Fields in AdS
  16. ...and 5 more sections

Figures (3)

  • Figure 1:
  • Figure 2: For a given dual pair \ref{['adscft0']}, we can place the CFT$_{d}$ on the boundary (solid black boundary of disc) asymptotically AdS boundary of the dual gravity theory (entire disc). The grey curves are geodesics. This perspective can be obtained by taking Euclidean AdS, adding a point at infinity to the $\mathbb{R}^d$ boundary and compactifying it to $S^d$.
  • Figure 3: