Holographic Renormalization for z=2 Lifshitz Space-Times from AdS

TL;DR

This work embeds 4D $z=2$ Lifshitz holography into a 5D AdS framework by Scherk–Schwarz reduction of an axion–dilaton system, allowing holographic renormalization to proceed via familiar AlAdS techniques. The FG structure, counterterms, and anomalies are computed explicitly, revealing two independent scale anomalies whose on-shell reduction yields a Horava–Lifshitz type action with a nonzero potential. The analysis shows how Lifshitz data arise from AdS boundary data and identifies two central charges tied to the 5D conformal anomaly, now appearing as anisotropic (Horava-like) anomalies in 3D. The results provide a concrete string-theoretic route to Lifshitz holography, with implications for dual DLCQ field theories and the structure of asymptotic symmetries.

Abstract

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.