Kinematical superspaces

Abstract

A. We classify N=1 d = 4 kinematical and aristotelian Lie superalgebras with spatial isotropy, but not necessarily parity nor time-reversal invariance. Employing a quaternionic formalism which makes rotational covariance manifest and simplifies many of the calculations, we find a list of 43 isomorphism classes of Lie superalgebras, some with parameters, whose (nontrivial) central extensions are also determined. We then classify their corresponding simply-connected homogeneous (4|4)-dimensional superspaces, resulting in a list of 27 homogeneous superspaces, some with parameters, all of which are reductive. We determine the invariants of low rank and explore how these superspaces are related via geometric limits.

Unification of Gravity and Internal Interactions

“…[49] for a review, or ref. [50] for a more concise description) that the type of spinors one obtains for $$SO_{(p, q)}$$ in the real case is governed by the signature $$(p-q)\ {\rm mod}\ 8$$. Among even signatures, signature 0 gives a real representation, signature 4 a quaternionic representation, while signatures 2 and 6 give complex representations.…”

Carrollian manifolds and null infinity: a view from Cartan geometry

“…They typically result of a limiting process or appear as the geometry induced on a null hypersurface. Applications include Hamiltonian analysis [4], Carrollian particles [5,6], ultra-relativistic fluid mechanics [7][8][9][10][11][12][13], cosmology [14], Carrollian [15][16][17][18][19] and conformally Carrollian [20][21][22][23] field theories, higher-spin [24][25][26], super-symmetric [27][28][29][30][31][32] and non-commutative [33] extensions, null hypersurfaces and isolated horizons [34][35][36][37][38][39][40][41][42][43][44][45], geometry of space/time-like infinities [46][47][48] and, finally, geometry of null infinity which is the main concern of this article.…”

Carrollian manifolds and null infinity: A view from Cartan geometry

“…They typically result of a limiting process or appear as the geometry induced on a null hypersurface. Applications include Hamiltonian analysis [4], Carrollian particles [5,6], ultra-relativistic fluid mechanics [7][8][9][10][11][12][13], cosmology [14], Carrollian [15][16][17][18][19] and conformally Carrollian [20][21][22][23] field theories, higher-spin [24,25], super-symmetric [26][27][28][29][30][31] and non-commutative [32] extensions, null hypersurfaces and isolated horizons [33][34][35][36][37][38][39][40][41][42][43][44], geometry of space/time-like infinities [45][46][47] and, finally, geometry of null infinity which is the main concern of this article. This is a well-known fact that the conformal boundary of an asymptotically flat spacetime, which we will generically refer to as "null infinity", is a (conformal) Carrollian manifold.…”