Asymptotic symmetries in Carrollian theories of gravity with a negative cosmological constant

Asymptotic symmetries of electric and magnetic Carrollian gravitational theories with a negative cosmological constant $\Lambda$ are analyzed in 3+1 space-time dimensions. In the magnetic theory, the asymptotic symmetry algebra is given by the conformal Carroll algebra in three dimensions, which is infinite-dimensional and isomorphic to the BMS$_{4}$ algebra. These results are in full agreement with holographic expectations, providing a new framework for the study of "Carrollian holography." On the contrary, in the case of the electric theory, the presence of a negative $\Lambda$ turns out to be incompatible with a consistent set of asymptotic conditions, that can be traced back to the absence of a sensible ground state configuration. This can be improved if the Carrollian theory obtained from an electric contraction of Euclidean General Relativity is considered. In this case, asymptotic conditions can be constructed with an asymptotic symmetry algebra given by $so\left(1,4\right)$. However, it is shown that the space of spherically symmetric solutions of this theory is degenerate.