Recently, a class of gravitational backgrounds in 3 + 1 dimensions have been proposed as holographic duals to a Lifshitz theory describing critical phenomena in 2 + 1 dimensions with critical exponent z ≥ 1. We numerically explore black holes in these backgrounds for a range of values of z. We find drastically different behavior for z > 2 and z < 2. We find that for z > 2 (z < 2) the Lifshitz fixed point is repulsive (attractive) when going to larger radial parameter r. For the repulsive z > 2 backgrounds, we find a continuous family of black holes satisfying a finite energy condition. However, for z < 2 we find that the finite energy condition is more restrictive, and we expect only a discrete set of black hole solutions, unless some unexpected cancellations occur. For all black holes, we plot temperature T as a function of horizon radius r 0 . For z 1.761 we find that this curve develops a negative slope for certain values of r 0 possibly indicating a thermodynamic instability.

Recently, a class of gravitational backgrounds in 3+1 dimensions have been proposed as holographic duals to a Lifshitz theory describing critical phenomena in 2+1 dimensions with critical exponent $z\geq 1$. We continue our earlier work \cite{Bertoldi:2009vn}, exploring the thermodynamic properties of the "black brane" solutions with horizon topology $\mathbb{R}^2$. We find that the black branes satisfy the relation $\mathcal{E}=\frac{2}{2+z}Ts$ where $\mathcal{E}$ is the energy density, $T$ is the temperature, and $s$ is the entropy density. This matches the expected behavior for a 2+1 dimensional theory with a scaling symmetry $(x_1,x_2)\to \lambda (x_1,x_2)$, $t\to \lambda^z t$.Comment: 8 pages, references added and regroupe

We show that the recently formulated Equivalence Principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one-dimension is sufficient to fix the Schwarzian equation [6], implies a fundamental higher dimensional Möbius invariance which in turn univocally fixes the quantum version of the Hamilton-Jacobi equation. This holds also in the relativistic case, so that we obtain both the time-dependent Schrödinger equation and the Klein-Gordon equation in any dimension.We then show that the EP implies that masses are related by maps induced by the coordinate transformations connecting different physical systems. Furthermore, we show that the minimal coupling prescription, and therefore gauge invariance, arises quite naturally in implementing the EP. Finally, we show that there is an antisymmetric two-tensor which underlies Quantum Mechanics and sheds new light on the nature of the Quantum Hamilton-Jacobi equation.

Several classes of gravitational backgrounds in 3 + 1 dimensions have been proposed as holographic duals to Lifshitz-like theories describing critical phenomena in 2 + 1 dimensions with critical exponent z ≥ 1. We numerically explore one such model, characterized by a temperature T and chemical potential µ, and find how to embed these solutions into AdS for a range of values of z. We find no phase transition going from the T ≪ µ to the T ≫ µ regimes, and find that the solutions smoothly interpolate between the Lifshitz-like behavior and the relativistic AdS-like behavior. Finally, we exploit some conserved quantities to find a relationship between the energy density E, entropy density s, and number density n, E = 2 3 (T s + µn). We show that this result is expected from general scaling arguments, and generalizes to E = d d+1 (T s + µn) for a theory dual to AdS d+2 (Poincaré patch) asymptotics with a local U (1) gauge invariance.

We study large-N double-scaling limits of U(N) gauge theories in four dimensions. We focus on theories in a partially confining phase where an abelian subgroupĜ of the gauge group remains unconfined. Double-scaling is defined near critical points in the parameter/moduli space where states charged underĜ become massless. In specific cases, we present evidence that the double-scaled theory is dual to a non-critical superstring background. Models studied include the β-deformation of N = 4 SUSY Yang-Mills which leads to a non-critical string theory with sixteen supercharges. We also study N = 1 SUSY Yang-Mills theory coupled to a single chiral superfield with a polynomial superpotential which leads to a related string theory with eight supercharges. In both cases the string coupling is small and the background is free from Ramond-Ramond flux.