AdS$_3$ Carroll gravity: asymptotic symmetries and C-thermal configurations

The asymptotic structure of three-dimensional Carroll gravity with negative cosmological constant is studied. We formulate a consistent set of boundary conditions preserved by an infinite-dimensional extension of the AdS$_3$ Carroll algebra, which turns out to be isomorphic to a precise generalized BMS$_3$ algebra. This is described by four independent functions of the circle at infinity, generating spatial superrotations, Carroll superboosts, spatial supertranslations and time supertranslations. Remarkably, this asymptotic symmetry algebra contains as subalgebras to BMS$_3$ (generated by spatial superrotations and time supertranslations) and the two-dimensional conformal algebra (spanned by spatial superrotations and spatial supertranslations). We also introduce a new solution - endowed with a Carroll extremal surface - that fulfills this set of asymptotic conditions. By taking advantage of the Chern-Simons formulation of the theory, Carroll thermal properties, obtained from regularity conditions, and entropy of the configuration are also addressed.