Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography

Morten H Christensen,Blaise Rollier,Niels A Obers,Jelle Hartong

doi:10.1007/jhep01(2014)057

Abstract

For a specific action supporting z=2 Lifshitz geometries we identify the Lifshitz UV completion by solving for the most general solution near the Lifshitz boundary. We identify all the sources as leading components of bulk fields which requires a vielbein formalism. This includes two linear combinations of the bulk gauge field and timelike vielbein where one asymptotes to the boundary timelike vielbein and the other to the boundary gauge field. The geometry induced from the bulk onto the boundary is a novel extension of Newton-Cartan geometry that we call torsional Newton-Cartan (TNC) geometry. There is a constraint on the sources but its pairing with a Ward identity allows one to reduce the variation of the on-shell action to unconstrained sources. We compute all the vevs along with their Ward identities and derive conditions for the boundary theory to admit conserved currents obtained by contracting the boundary stress-energy tensor with a TNC analogue of a conformal Killing vector. We also obtain the anisotropic Weyl anomaly that takes the form of a Horava-Lifshitz action defined on a TNC geometry. The Fefferman-Graham expansion contains a free function that does not appear in the variation of the on-shell action. We show that this is related to an irrelevant deformation that selects between two different UV completions.

Highlights

1.1 Scope and motivationEver since the birth of the AdS/CFT correspondence [1,2,3] there has been a continuous effort to find more examples of holographic correspondences
For a specific action supporting z = 2 Lifshitz geometries we identify the Lifshitz UV completion by solving for the most general solution near the Lifshitz boundary
Such theories are of relevance to condensed matter theory (CMT) where one frequently finds effective field theory descriptions of a system near some quantum critical point that is invariant under the Lifshitz symmetry group (z-dependent scale transformations t → λzt, x → λx, space-time translations and spatial rotations)

Summary

Introduction

1.1 Scope and motivationEver since the birth of the AdS/CFT correspondence [1,2,3] there has been a continuous effort to find more examples of holographic correspondences. Very often the dual field theories are not known explicitly and one (quasi) defines them in the appropriate regime of the coupling constant at large N via the holographic duality These developments have led to a tremendous activity in the field of applied holography leading e.g. to many new interesting asymptotically AdS black holes solutions and the construction of new types of holographic dualities involving non-asymptotically AdS space-times such as Schrodinger, Lifshitz and hyperscaling violating geometries. The motivation of the present paper lies in understanding the basic ingredients of holographic dualities for scale invariant field theories with dynamical exponent z > 1 Such theories are of relevance to condensed matter theory (CMT) where one frequently finds effective field theory descriptions of a system near some quantum critical point that is invariant under the Lifshitz symmetry group (z-dependent scale transformations t → λzt, x → λx, space-time translations and spatial rotations).

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