Constraints for warped branes

We investigate singular geometries which can be associated with warped branes in arbitrary dimensions. If the brane tension is allowed to be variable, the extremum condition for the action requires additional constraints beyond the solution of the field equations. In a higher dimensional world, such constraints arise from variations of the metric which are local in the usual four-dimensional spacetime, without changing the geometry of internal space. As a consequence, continuous families of singular solutions of the field equations, with arbitrary integration constants, are generically reduced to a discrete subset of extrema of the action, similar to regular spaces. As an example, no static extrema of the action with effective four-dimensional gravity exist for six-dimensional gravity with a cosmological constant. These findings explain why the field equations of the reduced four-dimensional theory are not consistent with arbitrary solutions of the higher dimensional field equations - consistency requires the additional constraints. The characteristic solutions for variable tension branes are non-static runaway solutions where the effective four-dimensional cosmological constant vanishes as time goes to infinity.Comment: 25 page

On discrete integrable equations with convex variational principles

We investigate the variational structure of discrete Laplace-type equations that are motivated by discrete integrable quad-equations. In particular, we explain why the reality conditions we consider should be all that are reasonable, and we derive sufficient conditions (that are often necessary) on the labeling of the edges under which the corresponding generalized discrete action functional is convex. Convexity is an essential tool to discuss existence and uniqueness of solutions to Dirichlet boundary value problems. Furthermore, we study which combinatorial data allow convex action functionals of discrete Laplace-type equations that are actually induced by discrete integrable quad-equations, and we present how the equations and functionals corresponding to (Q3) are related to circle patterns.Comment: 39 pages, 8 figures. Revision of the whole manuscript, reorder of sections. Major changes due to additional reality conditions for (Q3) and (Q4): new Section 2.3; Theorem 1 and Sections 3.5, 3.6, 3.7 update

Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions

We address the generalized thermodynamics and the collapse of a system of self-gravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. The equilibrium states correspond to polytropic distributions. The index n of the polytrope is related to the exponent of anomalous diffusion. We consider a high-friction limit and reduce the problem to the study of the nonlinear Smoluchowski-Poisson system. We show that the associated Lyapunov functional is the Tsallis free energy. We discuss in detail the equilibrium phase diagram of self-gravitating polytropes as a function of D and n and determine their stability by using turning points arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions describing the collapse. These results can be relevant for astrophysical systems, two-dimensional vortices and for the chemotaxis of bacterial populations. Above all, this model constitutes a prototypical dynamical model of systems with long-range interactions which possesses a rich structure and which can be studied in great detail.Comment: Submitted to Phys. Rev.

Gauging CSO groups in N=4 Supergravity

We investigate a class of CSO-gaugings of N=4 supergravity coupled to six vector multiplets. Using the CSO-gaugings we do not find a vacuum that is stable against all scalar perturbations at the point where the matter fields are turned off. However, at this point we do find a stable cosmological scaling solution.Comment: 21 page

Multifractality and Freezing Phenomena in Random Energy Landscapes: an Introduction

The Boltzmann-Gibbs probability distributions generated by logarithmically correlated random potentials provide a simple yet nontrivial example of disorder-induced multifractal measures. We introduce and discuss two analytically tractable models for such potentials. The first model uses multiplicative cascades and is equivalent to statistical mechanics of directed polymers on disordered trees studied long ago by B. Derrida and H. Spohn. Second model is the infinite-dimensional version of the problem in Euclidean space which can be solved by employing the replica trick. In particular, the latter version allows one to identify the freezing of multifractality exponents with a mechanism of the replica symmetry breaking (RSB) and to elucidate its physical meaning. The corresponding 1-step RSB solution turns out to be {\it marginally stable} everywhere in the low-temperature phase. In the end we put the model in a more general context by relating to the Gaussian Free Field and briefly discussing recent developments and extensions. The appendices provide a detailed exposition of the replica analysis of the model discussed in the lectures.Comment: Extended version of lecture notes for the International Summer School "Fundamental Problems in Statistical Physics XII" at Leuven, Belgium on Aug 31 - Sept 11, 2009 ; 34 page

Hamiltonian approach to GR - Part 2: covariant theory of quantum gravity

A non-perturbative quantum field theory of General Relativity is presented which leads to a new realization of the theory of Covariant Quantum-Gravity (CQG-theory). The treatment is founded on the recently-identified Hamiltonian structure associated with the classical space-time, i.e., the corresponding manifestly-covariant Hamilton equations and the related Hamilton-Jacobi theory. The quantum Hamiltonian operator and the CQG-wave equation for the corresponding CQG-state and wave-function are realized in $$scalar form. The new quantum wave equation is shown to be equivalent to a set of quantum hydrodynamic equations which warrant the consistency with the classical GR Hamilton-Jacobi equation in the semiclassical limit. A perturbative approximation scheme is developed, which permits the adoption of the harmonic oscillator approximation for the treatment of the Hamiltonian potential. As an application of the theory, the stationary vacuum CQG-wave equation is studied, yielding a stationary equation for the CQG-state in terms of the $4-$scalar invariant-energy eigenvalue associated with the corresponding approximate quantum Hamiltonian operator. The conditions for the existence of a discrete invariant-energy spectrum are pointed out. This yields a possible estimate for the graviton mass together with a new interpretation about the quantum origin of the cosmological constant