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Global and local properties of AdS(2) higher spin gravity

K. B. Alkalaev

TL;DR

This work constructs and analyzes a two-dimensional BF theory with an infinite tower of higher spin fields as a consistent $AdS_{2}$ higher spin gravity, clarifying both frame-like and metric-like formulations. Central to the approach is an unfolded, cohomological framework built from two nilpotent operators $\sigma_{\pm}$, which yield dual metric-like theories and reveal how massive-scalar and conserved-current sectors arise in the one-form, and how Weyl tensors organize the zero-form sector. The authors realize the higher spin algebras via an $o(2,1)$–$sp(2)$ Howe duality, presenting oscillator constructions, trace decompositions, and a taxonomy of quotient algebras (horizontal, vertical, and double factorizations) including the well-known hs$[ u]$ family. They then extend to a non-linear BF action valued in these quotient algebras and analyze linearization around $AdS_{2}$, showing how reduced models emerge and how the parent action links to dual metric-like formulations. The results illuminate how 2d higher spin gravity can couple topological gauge fields and dilatons in a controlled, algebraically rich setting, with potential implications for boundary dynamics and higher-dimensional extensions.

Abstract

Two-dimensional BF theory with infinitely many higher spin fields is proposed. It is interpreted as the AdS(2) higher spin gravity model describing a consistent interaction between local fields in AdS(2) space including gravitational field, higher spin partially-massless fields, and dilaton fields. We carry out analysis of the frame-like and the metric-like formulation of the theory. Infinite-dimensional higher spin global algebras and their finite-dimensional truncations are realized in terms of o(2,1) - sp(2) Howe dual auxiliary variables.

Global and local properties of AdS(2) higher spin gravity

TL;DR

This work constructs and analyzes a two-dimensional BF theory with an infinite tower of higher spin fields as a consistent higher spin gravity, clarifying both frame-like and metric-like formulations. Central to the approach is an unfolded, cohomological framework built from two nilpotent operators , which yield dual metric-like theories and reveal how massive-scalar and conserved-current sectors arise in the one-form, and how Weyl tensors organize the zero-form sector. The authors realize the higher spin algebras via an – Howe duality, presenting oscillator constructions, trace decompositions, and a taxonomy of quotient algebras (horizontal, vertical, and double factorizations) including the well-known hs family. They then extend to a non-linear BF action valued in these quotient algebras and analyze linearization around , showing how reduced models emerge and how the parent action links to dual metric-like formulations. The results illuminate how 2d higher spin gravity can couple topological gauge fields and dilatons in a controlled, algebraically rich setting, with potential implications for boundary dynamics and higher-dimensional extensions.

Abstract

Two-dimensional BF theory with infinitely many higher spin fields is proposed. It is interpreted as the AdS(2) higher spin gravity model describing a consistent interaction between local fields in AdS(2) space including gravitational field, higher spin partially-massless fields, and dilaton fields. We carry out analysis of the frame-like and the metric-like formulation of the theory. Infinite-dimensional higher spin global algebras and their finite-dimensional truncations are realized in terms of o(2,1) - sp(2) Howe dual auxiliary variables.

Paper Structure

This paper contains 41 sections, 1 theorem, 166 equations.

Table of Contents

  1. Introduction
  2. Quadratic higher spin BF action
  3. Cohomological view of BF equations
  4. $\sigma_{\pm}$ operators
  5. Exact expressions for the coefficients.
  6. Cohomological analysis
  7. Off-shell unfolded equations for one-form fields
  8. Component form of fields.
  9. Off-shell equations of motion.
  10. Explicit $\sigma_+$ - reduction: one-form sector
  11. Explicit $\sigma_-$ - reduction: one-form sector
  12. Off-shell field spaces
  13. Unfolding Weyl tensors
  14. Going on-shell
  15. Off-shell unfolded equations for zero-form fields
  16. ...and 26 more sections

Key Result

Theorem 1

The cohomology groups of operators $\sigma_\pm$ in $\mathcal{G}_s$ are given by where $T^{a_1 ... a_{m}}$ are totally symmetric and traceless $o(1,1)$ tensors.

Theorems & Definitions (1)

  • Theorem