5 research outputs found
Holographic Renormalization for z=2 Lifshitz Space-Times from AdS
Lifshitz space-times with critical exponent z=2 can be obtained by
dimensional reduction of Schroedinger space-times with critical exponent z=0.
The latter space-times are asymptotically AdS solutions of AdS gravity coupled
to an axion-dilaton system and can be uplifted to solutions of type IIB
supergravity. This basic observation is used to perform holographic
renormalization for 4-dimensional asymptotically z=2 locally Lifshitz
space-times by Scherk-Schwarz dimensional reduction of the corresponding
problem of holographic renormalization for 5-dimensional asymptotically locally
AdS space-times coupled to an axion-dilaton system. We can thus define and
characterize a 4-dimensional asymptotically locally z=2 Lifshitz space-time in
terms of 5-dimensional AdS boundary data. In this setup the 4-dimensional
structure of the Fefferman-Graham expansion and the structure of the
counterterm action, including the scale anomaly, will be discussed. We find
that for asymptotically locally z=2 Lifshitz space-times obtained in this way
there are two anomalies each with their own associated nonzero central charge.
Both anomalies follow from the Scherk--Schwarz dimensional reduction of the
5-dimensional conformal anomaly of AdS gravity coupled to an axion-dilaton
system. Together they make up an action that is of the Horava-Lifshitz type
with nonzero potential term for z=2 conformal gravity.Comment: 32 pages, v2: modified discussion of the central charge
Schr\"odinger Manifolds
This article propounds, in the wake of influential work of Fefferman and
Graham about Poincar\'e extensions of conformal structures, a definition of a
(Poincar\'e-)Schr\"odinger manifold whose boundary is endowed with a conformal
Bargmann structure above a non-relativistic Newton-Cartan spacetime. Examples
of such manifolds are worked out in terms of homogeneous spaces of the
Schr\"odinger group in any spatial dimension, and their global topology is
carefully analyzed. These archetypes of Schr\"odinger manifolds carry a Lorentz
structure together with a preferred null Killing vector field; they are shown
to admit the Schr\"odinger group as their maximal group of isometries. The
relationship to similar objects arising in the non-relativisitc AdS/CFT
correspondence is discussed and clarified.Comment: 42 pages, 1 figure, published version: J. Phys. A: Math. Theor. 45
(2012) 395203 (24pp
Boundary stress-energy tensor and Newton-Cartan geometry in Lifshitz holography
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 Hořava-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