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Unimodularity can be implemented in different ways. In this paper we consider only the formulation of Unimodular Gravity in which the unimodular metric is obtained out of an unrestricted one as $\g_{\m\n}=|g|^{-{1\over n}} g_{\m\n}$. This procedure induces an extra Weyl symmetry. Some physical implications of this symmetry on the conserved currents are discussed. Finally the results are illustrated for the Painlev\'e-Gullstrand extension of Schwarzschild spacetime., Comment: LateX, 19 pages, figure included, revised

In this paper some techniques useful to perform quantum field theory computations in a covariant manner are reviewed. In particular the background field gauge, the zeta function regularization and the heat kernel approach are highlighted. Some detailed calculations of the Schwinger-de Witt coefficients of the small proper time expansion of the heat kernel are also repeated in detail. This work reports lectures given by Enrique \'Alvarez at the IFT-UAM-CSIC in Madrid., Comment: 96 pages, LaTeX, lecture notes given by Enrique \'Alvarez at the IFT-UAM-CSIC in Madrid. New section added. References added. Typos corrected

It is claimed that in the unimodular gravity framework the observational fact of exponential expansion of the universe cannot be taken as evidence for the presence for a cosmological constant or similar quintessence., 17 pages

The generalization of scale invariance when gravitational effects are considered is Weyl invariance, namely, invariance under (global or local) rescalings of the metric. In this work, we discuss in some details the implications of the fact that the value of the anomaly for the global Weyl invariant coupling of scalar fields to gravity is sensitive to the dynamics (or absence thereof) of the gravitational field., Comment: 24 pages, streamlined, some errata fixed

A definition of quasi-local energy in a gravitational field based upon its embedding into flat space is discussed. The outcome is not satisfactory from many points of view., Comment: 12 pages

One-loop counterterms are computed in the first order formalism for the Einstein-Hilbert action with a minimally coupled scalar field using the background field method and the heat kernel technique. The {\em off-shell} divergent piece in the harmonic gauge is {\em exactly} the same as the one first found by 't Hooft and Veltman., Some misprints have been correct

A Weyl invariant extension of Einstein gravity is studied. It simply consists in the group averaging of Einstein's action under Weyl transformations. Contradicting cherished beliefs, a conformal anomaly is found in the trace of the equations of motion if diffeomorphism invariance is to be a symmetry of the quantum theory. This anomaly however vanishes on shell, which signals the existence of another gauge in which the anomaly is absent. It can be made also to vanish off-shell at the price of losing diffeomorphism invariance, 6 pages, to be published in the proceedings of "Windows on the universe" (Rencontres du Vietnam, 2013)

Some cosmological solutions of the unimodular theory of gravity are studied. These solutions can always be mapped to solutions of Einstein's general relativity in an appropiate frame. Constant scalar potentials however do not gravitate. Some physical consequences are pointed out., Comment: 14 pages, LaTeX. Withdrawn. Its contents will be incorporated in a forthcoming preprint

The approximate late conformal time wavefunction for self-interacting scalar quantum fields in de Sitter space is computed. It dominates for conformal times much larger than a certain critical value, $\eta=\eta_{\text{crit}}$ which depends on the dimension and on other characteristics of the matter fields. Its self-consistency (i.e., whether the back reaction is consistent with the de Sitter symmetry) is analyzed., Comment: 22 pages. Withdrawn. The conclusion seems to be in error. Some parts will be incorporated in a forthcoming preprint

We point out a generic inconsistency of the coupling of ordinary gravity as described by general relativity with matter invariant under the subgroup of unimodular diffeomorphisms (TDiffs), and some previously studied exceptions are pointed out. The most general Lagrangian invariant under TDiff up to dimension five operators is determined, and restrictions on parameters coming from existing observations are studied in some cases., 17 pages, 4 figures. References added

The static potential, corresponding to the interaction of two heavy sources is computed for $\mathcal{N}=4$ Super Yang Mills in the strong 't Hooft coupling regime by using the AdS/CFT conjecture and performing a computation of a rectangular Wilson loop at a finite distance of the boundary., 12 pages, 4 figures. to be published in Horizons in Physics Research, Nova Science Publishers, Inc

This paper has been superseded by hep-th/0303164, "The Dirichlet Obstruction in AdS/CFT", Comment: This paper has been superseded by hep-th/0303164, "The Dirichlet Obstruction in AdS/CFT", and is hence withdrawn

The renormalization group approach towards the string representation of non abelian gauge theories translates, in terms of the string sigma model beta function equations, the renormalization group evolution of the gauge coupling constant and Zamolodchikov`s $c$ function. Tachyon stability, glueball mass gap, renormalization group evolution of the $c$ function and the area law for the Wilson loop are studied for a critical bosonic string vacuum corresponding to a non abelian gauge theory in four dimensional space-time. We prove that the same intrinsic geometry for the string vacuum is universal in some sense, reproducing the Yang-Mills beta function to arbitrary loop order in perturbation theory., LaTeX, 26 pages. Minor changes