Gravity duals of cyclic RG flows, with strings attached

In this note, we propose gravity duals for 3+1 dimensional Lorentz invariant theories exhibiting discrete scale invariance. We construct non-singular solutions of a six dimensional gravitational theory that are warped products of $AdS_{5}$ and a circle. The presence of non-trivial warp factor explicitly breaks the symmetries of $AdS_{5}$ to discrete scale invariance and also breaks translation symmetry along the circle. The matter content of the 6D gravitational theory does not violate the null energy condition. In addition, we show that the linearized fluctuations around these 6D backgrounds are stable i.e., the fluctuations do not violate the Breitenlohner-Freedman stability criterion. The dual theories display periodic RG trajectories, but these RG flows do not violate the $a$-theorem. In particular, the dual field theory is not a soft deformation (deformation by marginal or relevant operators) of a 3+1 dimensional conformal field theory and hence the $a$-theorem and similar monotonicity constraints on RG flows do not apply to these examples. The holographic $c$-theorem also does not apply to these solutions. Finally, we present solutions of type II supergravity that exhibit discrete scale invariance to demonstrate that such solutions can be embedded in string theory.