Galilean Conformal Algebras and AdS/CFT (0902.1385v3)

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 $AdS_2$.

• The paper constructs the Galilean Conformal Algebra by contracting the relativistic conformal group SO(d+1,2), offering a novel non-relativistic approach.
• It reveals that the finite GCA extends to an infinite-dimensional symmetry including a Virasoro-Kac-Moody subalgebra, deepening our understanding of symmetry structures.
• The study proposes a bulk gravity dual via a Newton-Cartan-like limit in AdS space, which may advance non-relativistic gauge/gravity dualities.

Insights from "Galilean Conformal Algebras and AdS/CFT"

The paper "Galilean Conformal Algebras and AdS/CFT" by Arjun Bagchi and Rajesh Gopakumar explores a novel non-relativistic extension of the AdS/CFT correspondence, emphasizing Galilean conformal symmetry. This research diverges from the well-studied Schrödinger symmetry, introducing an alternative approach by employing group contraction of the relativistic conformal group SO(d+1,2). The resulting Galilean Conformal Algebra (GCA) retains the same number of generators as its relativistic counterpart, distinguishing it significantly from the Schrödinger algebra. Additionally, the paper identifies a natural extension of the GCA to an infinite-dimensional symmetry algebra, which includes a Virasoro-Kac-Moody subalgebra. This work has potential implications for understanding non-relativistic systems, proposing a new geometric structure for their bulk gravity dual.

Key Contributions and Findings

1. Galilean Conformal Algebra (GCA):
   • The paper constructs the GCA by contracting the relativistic conformal group. This contraction yields a fifteen-parameter group akin to the parent SO(d+1,2). Unlike the Schrödinger group, which only shares the ten-parameter Galilean subgroup with the GCA, the GCA does not support a central mass extension, thus aligning more closely with "massless" or "gapless" non-relativistic theories.
2. Infinite Dimensional Extension:
   • The GCA admits an extension analogous to how the finite conformal algebra in two dimensions extends to the Virasoro algebra. This suggests the possibility of dynamic realizations in systems possessing finite-dimensional Galilean conformal symmetry.
   • The extended GCA encompasses "Galilean conformal isometries," hinting at potential applications and a deeper understanding of such symmetries within boundary field theories.
3. Bulk Gravity Dual Proposal:
   • The authors propose a bulk gravity dual using a Newton-Cartan-like limit of Einstein’s equations in anti-de Sitter (AdS) space. This geometry includes an AdS space comprising the time and radial directions, distinguishing it from typical approaches involving the Schrödinger symmetry. The solution involves an underlying AdS base with unforeseen implications for non-relativistic gauge-gravity dualities.
4. Implications and Speculative Considerations:
   • The incorporation of the Galilean Conformal Algebra into the AdS/CFT framework implies potential applications to non-relativistic systems, such as quantum fluids or condensed matter systems.
   • The paper paves the way to explore sectors within relativistic conformal theories where the GCA might be realized. Additionally, speculations about extended symmetries becoming dynamically realized in specific sub-theories are intriguing aspects warranting further exploration.
   • The presence of an infinite-dimensional Virasoro extension as asymptotic isometries in the bulk points to the likelihood of central charge dynamics and implications for quantum gravity theories adapted to non-relativistic symmetries.

Future Directions

The considerations raised in this paper highlight several avenues for further research:

• Identifying and analyzing relativistic CFTs where the GCA could surface as a symmetry in specific sectors.
• Extending the formalism to encompass supersymmetric extensions and examining the role of additional symmetry structures in non-relativistic setups.
• Investigating potential real-world systems that could embody these proposed theoretical constructs, such as non-relativistic fluids or materials under specific physical conditions.
• Developing a concrete bulk-boundary duality framework to thoroughly describe the Galilean limit's implications for string theory.

This paper provides a significant contribution to non-relativistic adaptations of the AdS/CFT correspondence, potentially enriching our understanding of strongly interacting non-relativistic systems through the lens of gauge/gravity duality.