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Three-dimensional Spin-3 Theories Based on General Kinematical Algebras

Eric Bergshoeff, Daniel Grumiller, Stefan Prohazka, Jan Rosseel

TL;DR

This work extends Bacry–Lévy–Leblond’s kinematical classification to three-dimensional theories with massless spin-3 fields by classifying all IW contractions of the spin-3 $(A)dS_3$ algebras. It constructs corresponding Chern–Simons actions for ultra-relativistic Carroll and suitably extended non-relativistic (Extended Bargmann) algebras, providing linearized analyses and explicit invariant metrics. A key result is the Carroll case’s infinite-dimensional asymptotic symmetry via Carroll boundary conditions, suggesting similar extensions for the non-/ultra-relativistic spin-3 theories. The paper also develops spin-3 extensions of Extended Bargmann gravity through Medina–Revoy double extensions, enabling nondegenerate invariant bilinear forms and CS actions, thereby enriching the landscape of three-dimensional higher-spin gravity in non-AdS settings and offering avenues for holography, Hořava–Lifshitz connections, and condensed-matter applications.

Abstract

We initiate the study of non- and ultra-relativistic higher spin theories. For sake of simplicity we focus on the spin-3 case in three dimensions. We classify all kinematical algebras that can be obtained by all possible Inönü--Wigner contraction procedures of the kinematical algebra of spin-3 theory in three dimensional (anti-) de Sitter space-time. We demonstrate how to construct associated actions of Chern--Simons type, directly in the ultra-relativistic case and by suitable algebraic extensions in the non-relativistic case. We show how to give these kinematical algebras an infinite-dimensional lift by imposing suitable boundary conditions in a theory we call "Carroll Gravity", whose asymptotic symmetry algebra turns out to be an infinite-dimensional extension of the Carroll algebra.

Three-dimensional Spin-3 Theories Based on General Kinematical Algebras

TL;DR

This work extends Bacry–Lévy–Leblond’s kinematical classification to three-dimensional theories with massless spin-3 fields by classifying all IW contractions of the spin-3 algebras. It constructs corresponding Chern–Simons actions for ultra-relativistic Carroll and suitably extended non-relativistic (Extended Bargmann) algebras, providing linearized analyses and explicit invariant metrics. A key result is the Carroll case’s infinite-dimensional asymptotic symmetry via Carroll boundary conditions, suggesting similar extensions for the non-/ultra-relativistic spin-3 theories. The paper also develops spin-3 extensions of Extended Bargmann gravity through Medina–Revoy double extensions, enabling nondegenerate invariant bilinear forms and CS actions, thereby enriching the landscape of three-dimensional higher-spin gravity in non-AdS settings and offering avenues for holography, Hořava–Lifshitz connections, and condensed-matter applications.

Abstract

We initiate the study of non- and ultra-relativistic higher spin theories. For sake of simplicity we focus on the spin-3 case in three dimensions. We classify all kinematical algebras that can be obtained by all possible Inönü--Wigner contraction procedures of the kinematical algebra of spin-3 theory in three dimensional (anti-) de Sitter space-time. We demonstrate how to construct associated actions of Chern--Simons type, directly in the ultra-relativistic case and by suitable algebraic extensions in the non-relativistic case. We show how to give these kinematical algebras an infinite-dimensional lift by imposing suitable boundary conditions in a theory we call "Carroll Gravity", whose asymptotic symmetry algebra turns out to be an infinite-dimensional extension of the Carroll algebra.

Paper Structure

This paper contains 16 sections, 2 theorems, 105 equations, 2 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Kinematical Spin-3 Algebras
  3. AdS$_3$ and dS$_3$ Spin-3 Algebras
  4. All Kinematical Spin-3 Algebras by Contracting $\mathfrak{hs}_{3}\mathfrak{(A)dS}$
  5. Spin-3 Carroll, Galilei and Extended Bargmann Chern--Simons Theories
  6. Spin-3 Carroll Gravity
  7. Chern--Simons Theory for $\mathfrak{hs}_3\mathfrak{car1}$
  8. Chern--Simons Theory for $\mathfrak{hs}_3\mathfrak{car2}$
  9. Spin-3 Galilei and Extended Bargmann Gravity
  10. Chern--Simons Theory for $\mathfrak{hs}_3\mathfrak{ebarg1}$
  11. Chern--Simons Theory for $\mathfrak{hs}_3\mathfrak{ebarg2}$
  12. Carroll Gravity as Example
  13. Discussion
  14. Conventions
  15. Proof of Theorem 1
  16. ...and 1 more sections

Key Result

Theorem 1

All possible IW contraction procedures, that reduce to those considered in table tab:spin2contr when restricted to the spin-2 part and that are non-abelian on the subspace spanned by the spin-3 generators $\{{\tt J}_{a},{\tt H}_{a},{\tt G}_{ab},{\tt{P}}_{ab} \}$, are given by 10 'democratic' contrac

Figures (2)

  • Figure 1: This cube summarizes the sequential contractions starting from $\mathfrak{(A)dS}$. The lines represent contraction procedures and the dots represent the resulting contractions. We consider contraction procedures starting from AdS and dS simultaneously. Each dot can therefore represent one contraction, if the contraction procedures from AdS and dS lead to the same algebra, or two contractions, if the contraction procedures from AdS and dS lead to two different results. We have indicated this in the cube by using single lines, for contraction procedures that lead to the same contraction, and double lines otherwise. Dashed lines have no specific meaning except that they should convey the feeling of a three-dimensional cube.
  • Figure 2: This figure summarizes the sequential democratic contractions of table \ref{['tab:contr']}. There are 2 space-time (blue; #1,#2), 2 speed-space (red; #3,#4) and 2 speed-time (black; #5,#6) contractions and combining them leads to the full cube. The commutators of the algebras corresponding to the dots are given in table \ref{['tab:adspoin']}-\ref{['tab:hsstat2']}. In comparison to figure \ref{['fig:cube']}, we have for clarity omitted the double lines and the diagonal lines that indicate the direct IW contraction procedures to the static algebras.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2