Applied foliated conformal Carroll symmetries

TL;DR

The work addresses how to systematically couple matter to foliated Carroll gravity across $p$-branes. It shows that the conventional $p$-brane conformal Carroll algebra fails to yield consistent magnetic gravity couplings for $p>0$, prompting the introduction of a new conformal extension with longitudinal orchestration and an anisotropic dilatation; this unveils viable magnetic Carroll gravity actions for $p>1$ and clarifies when electric variants arise. By exploiting brane duality with Galilei gravity and constructing a domain-wall analog, the authors map the conformal framework to Aristotelian foliations and delineate special cases ($p=0$, $p=1$, domain walls) where the approach behaves differently. The results provide a systematic path to magnetic Carroll gravity for generic $p$ and deepen the understanding of conformal couplings in foliated nonrelativistic geometries, with potential extensions to NS-NS gravity and supersymmetry. Overall, the paper clarifies when conformal compensating techniques are viable for $p$-brane Carroll theories and outlines a clear, algebraic route to consistent matter couplings in these foliated spacetimes.

Abstract

We apply the conformal compensating technique for constructing matter couplings to conformal scalars on a $D$-dimensional foliated conformal Carroll manifold dividing the tangent space into $(p+1)$-dimensional longitudinal and $(D-p-1)$-dimensional transversal directions corresponding to $p$-branes. We show that the conformal Carroll algebra that was used for particle-like foliated geometries with $p=0$ cannot be used for higher-dimensional objects, called $p$-branes, with $0 < p \le D-2$. Furthermore, string-like foliated geometries are not suitable for the conformal compensating technique due to the conformal invariance in the longitudinal directions that is present for $p=1$. All other cases can be dealt with provided one uses a different conformal extension of the Carroll algebra that amounts to a conformal extension in the longitudinal directions only supplemented with an additional an-isotropic dilatation. By brane-duality similar results hold for foliated Galilean geometries which we present as well. Our results nicely fit in with recent work on foliated Aristotelian geometries.