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Elements of Vasiliev theory

V. E. Didenko, E. D. Skvortsov

TL;DR

The notes present a self-contained overview of nonlinear Vasiliev higher-spin theory, emphasizing the unfolded framework that treats dynamics with first-order differential forms and background independence in AdS-like settings. They connect gravity to a gauge-theory viewpoint via vielbein and spin connection, derive the frame-like–unfolded language, and show how Fronsdal-free HS fields on Minkowski and (A)dS backgrounds lay the groundwork for the full Vasiliev system. The work highlights the role of higher-spin algebras, the unfolded approach, and the necessity of AdS backgrounds to achieve consistent interactions, while outlining the path from metric-like formulations to fully nonlinear, background-independent equations. Overall, it provides the methodological tools and notational foundations for understanding Vasiliev equations and their place in the landscape of higher-spin gauge theories and AdS/CFT-inspired contexts.

Abstract

We propose a self-contained description of Vasiliev higher-spin theories with the emphasis on nonlinear equations. The main sections are supplemented with some additional material, including introduction to gravity as a gauge theory; the review of the Fronsdal formulation of free higher-spin fields; Young diagrams and tensors as well as sections with advanced topics. The shortest route to Vasiliev equations covers 40 pages. The general discussion is dimension independent, while the essence of the Vasiliev formulation is discussed on the base of the four-dimensional higher-spin theory. Three-dimensional and $d$-dimensional higher-spin theories follow the same logic.

Elements of Vasiliev theory

TL;DR

The notes present a self-contained overview of nonlinear Vasiliev higher-spin theory, emphasizing the unfolded framework that treats dynamics with first-order differential forms and background independence in AdS-like settings. They connect gravity to a gauge-theory viewpoint via vielbein and spin connection, derive the frame-like–unfolded language, and show how Fronsdal-free HS fields on Minkowski and (A)dS backgrounds lay the groundwork for the full Vasiliev system. The work highlights the role of higher-spin algebras, the unfolded approach, and the necessity of AdS backgrounds to achieve consistent interactions, while outlining the path from metric-like formulations to fully nonlinear, background-independent equations. Overall, it provides the methodological tools and notational foundations for understanding Vasiliev equations and their place in the landscape of higher-spin gauge theories and AdS/CFT-inspired contexts.

Abstract

We propose a self-contained description of Vasiliev higher-spin theories with the emphasis on nonlinear equations. The main sections are supplemented with some additional material, including introduction to gravity as a gauge theory; the review of the Fronsdal formulation of free higher-spin fields; Young diagrams and tensors as well as sections with advanced topics. The shortest route to Vasiliev equations covers 40 pages. The general discussion is dimension independent, while the essence of the Vasiliev formulation is discussed on the base of the four-dimensional higher-spin theory. Three-dimensional and -dimensional higher-spin theories follow the same logic.

Paper Structure

This paper contains 21 sections, 81 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Metric-like formulation for free HS fields
  3. Massless fields on Minkowski background
  4. Going on-shell.
  5. Counting degrees of freedom.
  6. Lagrangian.
  7. The long and winding road from representations to Lagrangians.
  8. Massless fields on (anti)-de Sitter background
  9. Summary.
  10. Gravity as gauge theory
  11. Tetrad, Vielbein, Frame, Vierbein,....
  12. Covariant derivative.
  13. Gravity as a gauge theory
  14. Short summary on Yang-Mills.
  15. Back to gravity.
  16. ...and 6 more sections

Figures (1)

  • Figure 1: The basis vectors $\vec{e}_i$in the tangent plane $TM_O$ at some point $O$ are by definition the vectors that are tangent to the coordinate lines. Let $p(x_1,...,x_n)$ be the point on the manifold parameterized by Cartesian coordinates $(x_1,...,x_n)$ in some chart, we can think of it as a point in a bigger Euclidian space where the given manifold is embedded. Then, $\vec{e}_i=\frac{d}{dt}p(x_1,...,x_i+t,...,x_n)|_{t=0}$, which are shown below.