On Lifshitz scaling and hyperscaling violation in string theory

We explore string/M-theory constructions of holographic theories with Lifshitz scaling exponent $z$ and hyperscaling violation exponent $\theta$, finding a range of $z,\theta$-values. Some of these arise as effective metrics from dimensional reduction of certain kinds of null deformations of $AdS$ spacetimes appearing in the near horizon geometries of extremal D3-, M2- and M5-brane theories. The $AdS_5$ solution in particular gives rise to $\theta=1$ in $d=2$ (boundary) space dimensions. Other solutions arise as the IIA D2- and D4-brane solutions with appropriate null deformations, and we discuss the phase structure of these systems.