Comments on Galilean conformal field theories and their geometric realization

TL;DR

The paper investigates non-relativistic conformal field theories governed by exotic Galilean conformal symmetry, focusing on a spin-l dependent extension that exists in two spatial dimensions with a central charge Θ.A refined contraction from the relativistic conformal group (involving spin) yields an exotic, acceleration-extended Galilei algebra, along with a concrete vector-field realization and a free-field equation that saturates a unitarity bound.The authors construct both field-theoretic realizations (free and Chern–Simons–matter actions) and a geometric realization via an invariant metric on an extended space, and they compute holographic-like two-point functions that match the CFT expectations, despite the metric's non-Lorentzian signature.The results illuminate how exotic Galilean conformal symmetry constrains operator structure, correlation functions, and dynamics, while leaving open questions about a complete gravity dual and richer extensions (fermions, supersymmetry, Newton–Cartan-type formulations).

Abstract

We discuss non-relativistic conformal algebras generalizing the Schrödinger algebra. One instance of these algebras is a conformal, acceleration-extended, Galilei algebra, which arises also as a contraction of the relativistic conformal algebra. In two dimensions, this admits an "exotic" central extension, whereby boosts do not commute. We study general properties of non-relativistic conformal field theories with such symmetry. We realize geometrically the symmetry in terms of a metric invariant under the exotic conformal Galilei algebra, although its signature is neither Lorentzian nor Euclidean. We comment on holographic-type calculations in this background.