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TASI Lectures on the Higher Spin - CFT duality

Simone Giombi

TL;DR

The notes provide a comprehensive, pedagogical overview of the higher-spin gravity/CFT duality, focusing on AdS4/CFT3. They start from free vector and fermionic CFTs, derive an infinite tower of HS currents, and match their bulk duals in Vasiliev theory, including both parity-invariant (Type A/B) and parity-violating generalizations, with CS vector model extensions. Core results include the precise spectra, the mapping of single-trace operators to bulk fields, the role of double-trace deformations, and detailed one-loop tests that support the dualities and hint at UV completeness. The discussion highlights the framework's potential to shed light on quantum gravity, holography for vector models, and the rich structure arising from HS symmetries, while outlining key open problems in bulk quantization, loop corrections, and higher-dimensional generalizations.

Abstract

In these lectures we give an overview of the duality between gravitational theories of massless higher spin fields in AdS and large N vector models. We first review the original higher spin/vector model duality conjectured by Klebanov and Polyakov, and then discuss its generalizations involving vector models coupled to Chern-Simons gauge fields. We proceed to review some aspects of the theory of massless higher spins, starting with the Fronsdal equations for free fields and moving on to the fully non-linear Vasiliev equations in four dimensions. We end by reviewing some recent tests of the higher spin/vector model duality at the level of correlation functions and one-loop partition functions.

TASI Lectures on the Higher Spin - CFT duality

TL;DR

The notes provide a comprehensive, pedagogical overview of the higher-spin gravity/CFT duality, focusing on AdS4/CFT3. They start from free vector and fermionic CFTs, derive an infinite tower of HS currents, and match their bulk duals in Vasiliev theory, including both parity-invariant (Type A/B) and parity-violating generalizations, with CS vector model extensions. Core results include the precise spectra, the mapping of single-trace operators to bulk fields, the role of double-trace deformations, and detailed one-loop tests that support the dualities and hint at UV completeness. The discussion highlights the framework's potential to shed light on quantum gravity, holography for vector models, and the rich structure arising from HS symmetries, while outlining key open problems in bulk quantization, loop corrections, and higher-dimensional generalizations.

Abstract

In these lectures we give an overview of the duality between gravitational theories of massless higher spin fields in AdS and large N vector models. We first review the original higher spin/vector model duality conjectured by Klebanov and Polyakov, and then discuss its generalizations involving vector models coupled to Chern-Simons gauge fields. We proceed to review some aspects of the theory of massless higher spins, starting with the Fronsdal equations for free fields and moving on to the fully non-linear Vasiliev equations in four dimensions. We end by reviewing some recent tests of the higher spin/vector model duality at the level of correlation functions and one-loop partition functions.

Paper Structure

This paper contains 24 sections, 227 equations, 8 figures, 1 table.

Table of Contents

  1. TASI Lectures on the Higher Spin - CFT duality
  2. Introduction
  3. Higher Spins from free CFT
  4. The free $O(N)$ vector model
  5. From CFT to AdS: higher spin/vector model duality
  6. Complex scalars and $U(N)$ vector model
  7. Interacting $O(N)$ model and its AdS dual
  8. Fermionic CFT
  9. Summary of parity invariant HS/CFT dualities
  10. Chern-Simons vector models
  11. Fronsdal equations for free HS fields
  12. Flat spacetime
  13. (A)dS
  14. Frame-like formulation of HS fields
  15. Vasiliev equations in AdS$_4$
  16. ...and 9 more sections

Figures (8)

  • Figure 1: Three-point function of HS currents in the free scalar CFT. Solid lines represent free scalar propagators.
  • Figure 2: Holographic 3-point function
  • Figure 3: RG flow from free to critical $O(N)$ model.
  • Figure 4: Witten diagram contributing to the one-loop correction to the HS current two-point function. The loop involves all fields in the spectrum, in particular the scalar field with $\Delta=2$ boundary condition.
  • Figure 5: RG flow from the UV fixed point of the Gross-Neveu model to free fermions.
  • ...and 3 more figures