More gravity solutions of AdS/CMT

TL;DR

The paper generalizes gravity solutions of AdS/CMT to arbitrary but even spacetime dimensions with scaling symmetry $r → r/λ$, $x_i → λ^b x_i$, $t → λ^a t$, and shows that for $b=0$ the solution exists in any dimension with only temporal scale invariance. It employs a bulk action with a 2-form flux $F_2$ and a (d−1)-form flux $F_{d-1}$ with a topological coupling to satisfy the equations of motion, obtaining relations such as $c^2 L^2 = a b (d-2)$ and consistent flux configurations that realize the scaling. It identifies the dual scalar operator dimensions via $Δ_± = \frac{a+b(d-2)}{2} \pm \sqrt{\frac{[a+b(d-2)]^2}{4} + m^2 L^2}$ and derives the BF bound $(mL)^2 \ge -\left(\frac{a+b(d-2)}{2}\right)^2$ together with $a/b$-dependent unitarity bounds. The work also discusses boundary actions compatible with the scaling, highlighting possibilities with first-order time and second-order space derivatives or quadratic actions, and notes a solvable massless case at $a=2b$ with a known Green's function.

Abstract

We have generalized the gravity solutions presented in arXiv:0808.1725 and arXiv:0808.3232 to arbitrary but even space time dimensions with the scaling symmetry $r \to \f{r}λ, x_i \to λ^b x_i, t \to λ^a t$. However, for $b=0$, we have the solution in arbitrary space time dimension, not restricted to even dimensional. The physical meaning of this particular choice of $b$ is that we can have a solution with only temporal scale invariance. From the dual field theory point of view, the BF bound and the unitarity bound for operators dual to scalar field is discussed.