Holographic renormalization for z=2 Lifshitz spacetimes from AdS

Lifshitz spacetimes with the critical exponent z = 2 can be obtained by the dimensional reduction of Schrodinger 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.