Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity

TL;DR

This work shows that non-relativistic Newton–Hooke and ultra-relativistic Carrollian symmetries with a cosmological constant can be systematically generated from (A)dS algebras in any dimension via Lie algebra expansions, and that these infinite families embed into affine Kac–Moody algebras. By using resonant semigroup expansions $S_E^{(N)}$ and their infinite limit $S^{( olinebreak ext{∞})}$, the authors construct a hierarchy of NH and Carrollian algebras, including central and non‑central extensions, and realize them in 2+1 dimensions as Chern–Simons gravities with corresponding invariant bilinear forms. The CS actions yield extended NH and post‑Newtonian gravity, and their Carrollian counterparts, with explicit truncations to finite $N$ recovering known theories and their cosmological constant generalizations. The results connect the large‑$c$ and Carroll limits of GR to algebraic expansions and affine algebras, offering a framework to explore non‑relativistic and ultra‑relativistic gravitational dynamics, potential Maxwell/branes generalizations, and supersymmetric extensions in curved backgrounds.

Abstract

We construct finite- and infinite-dimensional non-relativistic extensions of the Newton-Hooke and Carroll (A)dS algebras using the algebra expansion method, starting from the (anti-)de Sitter relativistic algebra in D dimensions. These algebras are also shown to be embedded in different affine Kac-Moody algebras. In the three-dimensional case, we construct Chern-Simons actions invariant under these symmetries. This leads to a sequence of non-relativistic gravity theories, where the simplest examples correspond to extended Newton-Hooke and extended (post-)Newtonian gravity together with their Carrollian counterparts.

Figures (2)

• Figure 1: Dynkin diagrams of $B_{r+1}^{(1)}$ (lower) and $D_{r+1}^{(1)}$ (upper). These are relevant for the infinite Newton--Hooke algebras.
• Figure 2: Dynkin diagrams of $B_{r+1}^{(1)}$ (upper) and $D_{r+1}^{(2)}$ (lower). These are relevant to the (A)dS Carroll algebras.