Galilean Conformal Algebras and AdS/CFT

TL;DR

The paper identifies a non-relativistic conformal symmetry obtained by contracting the relativistic conformal group $SO(d{+}1,2)$ to form the Galilean Conformal Algebra (GCA), which shares the same generator count as the parent group but differs from the Schrödinger group, notably lacking a mass central extension and featuring an intrinsic infinite-dimensional Virasoro–Kac–Moody extension. It analyzes both boundary and bulk realizations: on the boundary, the GCA and its extension act as symmetries of a boundary theory, while in the bulk, a Newton–Cartan–like AdS geometry emerges in which the infinite algebra appears as asymptotic isometries, with the $AdS_2$ base playing a central role. The authors provide explicit bulk contractions of AdS isometries, define bulk vector fields $L^{(n)}$, $M_i^{(n)}$, and $J_{ij}^{(n)}$, and argue that the resulting Virasoro–Kac–Moody structure persists in the bulk, enabling potential Brown–Henneaux–like central charges and a richer holographic dictionary. The work suggests concrete connections to non-relativistic hydrodynamics and fluid equations (e.g., Navier–Stokes in the gapless limit) and outlines future directions, including supersymmetric extensions and string-theoretic embedding of the Newton–Cartan dual geometry.

Abstract

Non-relativistic versions of the AdS/CFT conjecture have recently been investigated in some detail. These have primarily been in the context of the Schrodinger symmetry group. Here we initiate a study based on a {\it different} non-relativistic conformal symmetry: one obtained by a parametric contraction of the relativistic conformal group. The resulting Galilean conformal symmetry has the same number of generators as the relativistic symmetry group and thus is different from the Schrodinger group (which has fewer). One of the interesting features of the Galilean Conformal Algebra is that it admits an extension to an {\it infinite} dimensional symmetry algebra (which can potentially be dynamically realised). The latter contains a Virasoro-Kac-Moody subalgebra. We comment on realisations of this extended symmetry in a boundary field theory. We also propose a somewhat unusual geometric structure for the bulk gravity dual to any realisation of this symmetry. This involves taking a Newton-Cartan like limit of Einstein's equations in anti de Sitter space which singles out an $AdS_2$ comprising of the time and radial direction. The infinite dimensional Virasoro extension is identified with the asymptotic isometries of this $AdS_2$.