Applied foliated conformal Carroll symmetries

TL;DR

This paper explores the limitations and extensions of conformal Carroll symmetries on foliated manifolds, revealing that different algebraic structures are needed for various p-brane geometries and connecting to foliated Galilean and Aristotelian geometries.

Contribution

It identifies the constraints of existing conformal Carroll algebras for p-branes and proposes new conformal extensions suitable for higher-dimensional objects, also relating to foliated Galilean geometries.

Findings

Standard conformal Carroll algebra is unsuitable for p-branes with p > 0.

String-like geometries require a different conformal extension due to invariance issues.

New conformal extensions are proposed for various foliated geometries.

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.