Semi-local quantum criticality in string/M-theory

Semi-local quantum critical behaviour in $D-1$ spacetime dimensions can be holographically described by metrics that are conformal to $Ad{S}_2\times {\mathbb{R}}^{D-2}$, with the conformal factor characterised by a parameter $\eta$. We analyse such “$\eta$-geometries” in a top-down setting by focussing on the $U(1{)}^4$ truncation of D=4 N=8 gauged supergravity. The model has extremal black hole solutions carrying three non-zero electric or magnetic charges which approach $Ad{S}_4$ in the UV and an $\eta =1$ geometry in the IR. Adding a fourth charge provides a mechanism to resolve the singularity of the $\eta$-geometry, replacing it with an $Ad{S}_2\times {\mathbb{R}}^2$ factor in the IR, while maintaining a large region where the $\eta$-geometry scaling is approximately valid. Some of the magnetically charged black hole solutions preserve supersymmetry while others just preserve it in the IR. Finally, we show that $\eta$-geometries, with various values of $\eta$, can be obtained from the dimensional reduction of geometries consisting of $AdS$ or Lifshitz geometries with flat directions.