A. Restuccia

On the consistency of the Horava Theory

With the goal of giving evidence for the theoretical consistency of the Hořava Theory, we perform a Hamiltonian analysis on a classical model suitable for analyzing its effective dynamics at large distances. The model is the lowest-order truncation of the Hořava Theory with the detailed-balance condition. We consider the pure gravitational theory without matter sources. The model has the same potential term of general relativity, but the kinetic term is modified by the inclusion of an arbitrary coupling constant λ. Since this constant breaks the general covariance under space-time diffeomorphisms, it is believed that arbitrary values of λ deviate the model from general relativity. We show that this model is not a deviation at all, instead it is completely equivalent to general relativity in a particular partial gauge fixing for it. In doing this, we clarify the role of a second-class constraint of the model. There have been a lot of interest about Hořava’s proposal of a new theory of gravity which in principle has a renormalizable quantum version [1] (an important part of the conceptual and technical basis was previously developed in Ref. [2]). To build such a theory, Hořava has proposed to abandon the principle of space-time relativity as a fundamental symmetry of nature, reducing the freedom to perform coordinate transformations to those transformations that preserve some preferred universal time-like foliation. The advantage of this scheme is that one can include higher spatial-derivative terms in the Lagrangian that render the theory renormalizable. According to Hořava’s point of view, jorgebellorin@usb.ve arestu@usb.ve

Discreteness of the spectrum of the compactified D=11 supermembrane with nontrivial winding

Abstract We analyze the Hamiltonian of the compactified D=11 supermembrane with nontrivial central charge in terms of the matrix model constructed in [Phys. Rev. D 66 (2002) 045023, hep-th/0103261 ]. Our main result provides a rigorous proof that the quantum Hamiltonian of the supersymmetric model has compact resolvent and thus its spectrum consists of a discrete set of eigenvalues with finite multiplicity.

On the spectrum of a matrix model for the D = 11 supermembrane compactified on a torus with non-trivial winding

The spectrum of the Hamiltonian of the double compactified D = 11 supermembrane with non-trivial central charge or, equivalently, the non-commutative symplectic super Maxwell theory, is analysed. In distinction to what occurs for the D = 11 supermembrane in Minkowski target space where the bosonic potential presents string-like spikes which render the spectrum of the supersymmetric model continuous, we prove that the potential of the bosonic compactified membrane with non-trivial central charge is strictly positive definite and becomes infinity in all directions when the norm of the configuration space goes to infinity. This ensures that the resolvent of the bosonic Hamiltonian is compact. We find an upper bound for the asymptotic distribution of the eigenvalues.

Duality symmetric actions and canonical quantization

We obtain an extension of the Schwarz and Sen formulation of dual symmetric actions, avoiding the second class constraints present in their approach. The new actions are also symmetric under dual transformations and present an infinite set of gauge symmetries.

On the stability of compactified D = 11 supermembranes: Global aspects of the bosonic sector

We prove that the Hamiltonian for the bosonic sector of D = 11 supermembrane theories, wrapped in an irreducible way around S1 × S1 × M9 on the target manifold, only has strict minima without infinite-dimensional valleys. The minima occur at monopole connections of an associated U(1) bundle over topologically non-trivial Riemann surfaces of arbitrary genus. Explicit expressions for the minimal connections in terms of membrane maps are presented.

A global quadratic algorithm for solving a system of mixed equalities and inequalities

A new algorithm is proposed which, under mild assumptions, generates a sequence{xi} that starting at any point inRn will converge to a setX defined by a mixed system of equations and inequalities. Any iteration of the algorithm requires the solution of a linear programming problem with relatively few constraints. By only assuming that the functions involved are continuously differentiable a superlinear rate of convergence is achieved. No convexity whatsoever is required by the algorithm.

The heat kernel of the compactified D=11 supermembrane with non-trivial winding

We study the quantization of the regularized Hamiltonian, H, of the compactified D=11 supermembrane with non-trivial winding. By showing that H is a relatively small perturbation of the bosonic Hamiltonian, we construct a Dyson series for the heat kernel of H and prove its convergence in the topology of the von Neumann–Schatten classes so that e−Ht is ensured to be of finite trace. The results provided have a natural interpretation in terms of the quantum mechanical model associated to regularizations of compactified supermembranes. In this direction, we discuss the validity of the Feynman path integral description of the heat kernel for D=11 supermembranes and obtain rigorously a matrix Feynman–Kac formula.

D=11 supermembrane wrapped on calibrated submanifolds

We construct the Hamiltonian of the D=11 supermembrane with topological conditions on configuration space. It may be interpreted as a supermembrane theory where all configurations are wrapped in an irreducible way on a calibrated submanifold of a compact sector of the target space. We prove that the spectrum of its Hamiltonian is discrete with finite multiplicity. The construction is explicitly perfomed for a compact sector of the target space being a 2g-dimensional flat torus and the base manifold of the supermembrane a genus g compact Riemann surface. The topological conditions on configuration space work in such a way that the g=2 case may be interpreted as the intersection of two D=11 supermembranes over g=1 surfaces, with their corresponding topological conditions. The discreteness of the spectrum is preserved by the intersection procedure. Between the configurations satisfying the topological conditions there are minimal configurations which describe minimal immersions from the base manifold to the compact sector of the target space. They allow to map the D=11 supermembrane with topological conditions to a symplectic noncommutative Yang–Mills theory. We analyze geometrical properties of these configurations in the context of supermembranes and D-branes theories. We show that this class of configurations also minimizes the Hamiltonian of D-branes theories.

Consistent Hořava gravity without extra modes and equivalent to general relativity at the linearized level

We consider a Horava theory that has a consistent structure of constraints and propagates two physical degrees of freedom. The Lagrangian includes the terms of Blas, Pujolas and Sibiryakov. The theory can be obtained from the general Horavas formulation by setting lambda = 1/3. This value of lambda is protected in the quantum formulation of the theory by the presence of a constraint. The theory has two second-class constraints that are absent for other values of lambda. They remove the extra scalar mode. There is no strong-coupling problem in this theory since there is no extra mode. We perform explicit computations on a model that put together a z = 1 term and the IR effective action. We also show that the lowest-order perturbative version of the IR effective theory has a dynamics identical to the one of linearized general relativity. Therefore, this theory is smoothly recovered at the deepest IR without discontinuities in the physical degrees of freedom.

Closure of the algebra of constraints for a non-projectable Ho\v{r}ava model

We perform the Hamiltonian analysis for a nonprojectable Horava model whose potential is composed of R and R{sup 2} terms. We show that Diracs algorithm for the preservation of the constraints can be done in a closed way, hence the algebra of constraints for this model is consistent. The model has an extra, odd, scalar mode whose decoupling limit can be seen in a linear-order perturbative analysis on weakly varying backgrounds. Although our results for this model point in favor of the consistency of the Horava theory, the validity of the full nonprojectable theory still remains unanswered.