Holographic renormalization for z = 2 Lifshitz spacetimes from AdS

Chemissany, Wissam; Geissbühler, David; Hartong, Jelle; Rollier, Blaise (2012). Holographic renormalization for z = 2 Lifshitz spacetimes from AdS. Classical and quantum gravity, 29(23), p. 235017. Bristol, UK: Institute of Physics Publishing IOP 10.1088/0264-9381/29/23/235017

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Lifshitz spacetimes with the critical exponent z = 2 can be obtained by the dimensional reduction of Schrödinger spacetimes with the critical exponent z = 0. The latter spacetimes 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 four-dimensional asymptotically z = 2 locally Lifshitz
spacetimes by the Scherk–Schwarz dimensional reduction of the corresponding problem of holographic renormalization for five-dimensional asymptotically locally AdS spacetimes coupled to an axion–dilaton system. We can thus define and characterize a four-dimensional asymptotically locally z = 2 Lifshitz spacetime in terms of five-dimensional AdS boundary data. In this
setup the four-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 spacetimes
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 five-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 a nonzero potential term for z = 2 conformal gravity.

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