Galilean conformal algebras and AdS/CFT

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 different nonrelativistic 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 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 .

Extended Galilean symmetries of non-relativistic strings

“…These generators are similar to those appearing in the Galilean conformal algebra [19,40]. Notice, however, that the relative sign inD andK between time and space directions is negative, while in the standard Galilean conformal algebra one hasD = t∂ t + x i ∂ i , K = t 2 ∂ t + 2tx i ∂ i .…”

“…Other algebras of the same type have been considered in the literature; see for instance [25,40] and references therein. The standard Galilean algebra is generated bŷ…”

Flat-space energy-momentum tensor from BMS/GCA correspondence

“…If one considers a two dimensional CFT and takes its non-relativistic limit by contracting one of the coordinates, the contracted algebra is exactly (2.17) [10][11][12][13]. In this context the algebra (2.17) is called Galilean conformal algebra due to appearance of Galilean subalgebra.…”

“…The observation of [1,2] was that BMS algebra is isomorphic Galilean conformal algebra (GCA) which is a result of contraction of conformal algebra [10][11][12][13]. This connection dubbed the BMS/GCA correspondence was studied carefully in [3] where it was shown that in order to have a well-defined correspondence at the level of centrally extended algebras, at the level of spacetime, the time direction must be contracted in the CFT and the resultant theory is an ultra-relativistic field theory.…”

Flat-Space Chiral Gravity

“…So, if we believe that quantum gravity on AdS is dual to a CFT, the structure of the field theory dual for flat-space would be given by a contraction of a CFT. Interestingly, these contracted CFTs were studied earlier in the context of non-relativistic limits of CFTs and are called Galilean conformal algebras (GCA) [13]. This intriguing connection was dubbed the BMS/GCA correspondence [9].…”