Post-Carroll Algebra, Conformal Extensions, and Field Theories
By incorporating leading $c\,$-dependent corrections to the Carroll transformations, we introduce the ``post-Carroll transformations''. We demonstrate that these transformations are consistent with post-Carrollian mechanics \cite{Najafizadeh:2025ksm}; furthermore, they give rise to the so-called ``post-Carroll algebra''. We show that, unlike the Carroll algebra, this new structure allows for a central charge in higher dimensions; we refer to it as the ``Carroll-Bargmann algebra''. To construct conformal extensions, we first build the conformal extension of the post-Carroll algebra and study field theories invariant under this symmetry. We then construct the conformal extension of the Carroll-Bargmann algebra, referred to as the ``Carroll-Schrödinger algebra'', and demonstrate that it precisely matches the symmetry algebra of the higher-dimensional Carroll-Schrödinger theory \cite{Najafizadeh:2024imn}. Finally, we derive the general form of two-point functions in a post-Carrollian CFT, which in $1+1$ dimensions exhibits both electric and magnetic sectors, while in higher dimensions only the magnetic sector survives.