Abstract
We prove a no-go theorem for the construction of a Galilean boost invariant and $z\neq2$ anisotropic scale invariant field theory with a finite dimensional basis of fields. Two point correlators in such theories, we show, grow unboundedly with spatial separation. Correlators of theories with an infinite dimensional basis of fields, for example, labeled by a continuous parameter, do not necessarily exhibit this bad behavior. Hence, such theories behave effectively as if in one extra dimension. Embedding the symmetry algebra into the conformal algebra of one higher dimension also reveals the existence of an internal continuous parameter. Consideration of isometries shows that the non-relativistic holographic picture assumes a canonical form, where the bulk gravitational theory lives in a space-time with one extra dimension. This can be contrasted with the original proposal by Balasubramanian and McGreevy, and by Son, where the metric of a $d+2$ dimensional space-time is proposed to be dual of a $d$ dimensional field theory. We provide explicit examples of theories living at fixed point with anisotropic scaling exponent $z=\frac{2\ell}{\ell+1}\,,\ell\in \mathbb{Z}$
Highlights
Gravity duals of nonrelativistic field theories have been proposed in [1,2]
While the metric having isometry of this generalized Schrödinger group has been used with the holographic dictionary to construct correlators of a putative field theory [3,4,5,6,7,8,9], there is no explicit field theoretic realization of such a symmetry for z ≠ 2.1 One surprising feature, noted as a “strange aspect” in Ref. [1], is that, unlike in the canonical AdS=CFT correspondence, where the conformal field theory (CFT) in d dimensions is dual to the gravity in (d þ 1) dimensions, in the nonrelativistic case the metric is of a space-time with two additional dimensions
The z 1⁄4 2 case is special in that respect since it is possible to obtain a d-dimensional theory with a finite number of fields such that the symmetries on the field theory side match the isometries of a (d þ 2)-dimensional geometry
Summary
Gravity duals of nonrelativistic field theories have been proposed in [1,2]. It has been observed in Ref. [1] that one can consistently define an algebra with Galilean boost invariance and arbitrary anisotropic scaling exponent z. [1], is that, unlike in the canonical AdS=CFT correspondence, where the conformal field theory (CFT) in d dimensions is dual to the gravity in (d þ 1) dimensions, in the nonrelativistic case the metric is of a space-time with two additional dimensions. A question arises naturally as to whether one can get rid of the extra ξ direction and reduce the correspondence to a canonical correspondence between a d-dimensional quantum field theory on flat space and a (d þ 1)-dimensional gravitational theory. Refs. [15,16]
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