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Higher Spin Interactions in Four Dimensions: Vasiliev vs. Fronsdal

Nicolas Boulanger, Pan Kessel, E. D. Skvortsov, Massimo Taronna

TL;DR

This work derives the four-dimensional Higher-Spin theory at the first nontrivial (cubic) order directly from Vasiliev's equations and builds the complete dictionary between unfolded HS variables and Fronsdal fields. It reveals that beyond the algebraically fixed vertices, additional, highly derivative (pseudo-local) interactions contribute, and that their resummation leads to divergences, complicating straightforward AdS/CFT extrapolations. By computing explicit second-order unfolded equations and the corresponding Fronsdal currents, the authors show that only a limited subset of correlators is reliably captured by HS algebra alone, and they discuss potential resolutions, including gauge/frame choices and regularization schemes. These results challenge the idea of a strictly local HS bulk theory and suggest that careful handling of nonlocal tails is essential for a consistent HS holography and for defining a well-behaved bulk action. The study provides a concrete framework for testing locality, canonical current structures, and the interplay between unfolded and metric-like formulations in 4d HS gravity.

Abstract

We consider four-dimensional Higher-Spin Theory at the first nontrivial order corresponding to the cubic action. All Higher-Spin interaction vertices are explicitly obtained from Vasiliev's equations. In particular, we obtain the vertices that are not determined solely by the Higher-Spin algebra structure constants. The dictionary between the Fronsdal fields and Higher-Spin connections is found and the corrections to the Fronsdal equations are derived. These corrections turn out to involve derivatives of arbitrary order. We observe that the vertices not determined by the Higher-Spin algebra produce naked infinities, when decomposed into the minimal derivative vertices and improvements. Therefore, standard methods can only be used to check a rather limited number of correlation functions within the HS AdS/CFT duality. A possible resolution of the puzzle is discussed.

Higher Spin Interactions in Four Dimensions: Vasiliev vs. Fronsdal

TL;DR

This work derives the four-dimensional Higher-Spin theory at the first nontrivial (cubic) order directly from Vasiliev's equations and builds the complete dictionary between unfolded HS variables and Fronsdal fields. It reveals that beyond the algebraically fixed vertices, additional, highly derivative (pseudo-local) interactions contribute, and that their resummation leads to divergences, complicating straightforward AdS/CFT extrapolations. By computing explicit second-order unfolded equations and the corresponding Fronsdal currents, the authors show that only a limited subset of correlators is reliably captured by HS algebra alone, and they discuss potential resolutions, including gauge/frame choices and regularization schemes. These results challenge the idea of a strictly local HS bulk theory and suggest that careful handling of nonlocal tails is essential for a consistent HS holography and for defining a well-behaved bulk action. The study provides a concrete framework for testing locality, canonical current structures, and the interplay between unfolded and metric-like formulations in 4d HS gravity.

Abstract

We consider four-dimensional Higher-Spin Theory at the first nontrivial order corresponding to the cubic action. All Higher-Spin interaction vertices are explicitly obtained from Vasiliev's equations. In particular, we obtain the vertices that are not determined solely by the Higher-Spin algebra structure constants. The dictionary between the Fronsdal fields and Higher-Spin connections is found and the corrections to the Fronsdal equations are derived. These corrections turn out to involve derivatives of arbitrary order. We observe that the vertices not determined by the Higher-Spin algebra produce naked infinities, when decomposed into the minimal derivative vertices and improvements. Therefore, standard methods can only be used to check a rather limited number of correlation functions within the HS AdS/CFT duality. A possible resolution of the puzzle is discussed.

Paper Structure

This paper contains 44 sections, 229 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General Structure of Higher-Spin Theories
  3. Unfolded equations.
  4. HS algebra.
  5. Unfolded equations of HS theories.
  6. HS algebra vs. Dynamics.
  7. AdS as an HS background.
  8. First order.
  9. Second order.
  10. Higher-Spin Theory at the Second Order
  11. Comments on the Results
  12. Unfolded vs. Fronsdal Dictionary
  13. Fronsdal Currents
  14. Canonical Currents
  15. Corrections to Fronsdal Equations: Examples
  16. ...and 29 more sections

Figures (1)

  • Figure 1: A picture from Didenko:2014dwa showing the position of various fields and their mixing via unfolded equations. The coordinates are the number of undotted/dotted indices carried by a field. The fields connected by links talk to each other via the unfolded equations \ref{['firstordereqs']}.