We give a basic introduction to Unimodular Gravity both at the classical and quantum level, discussing the role it might play in the interpretation of the Cosmological Constant problem. The aim of this work is at the basic level but techniques used at the research level are presented. The goal of the paper is to enable the reader to grasp the interest of UG while making the recent literature on the topic accessible.

We have studied a lagrangian in which the Einstein-Hilbert term is deformed by the Weyl cube operator, which is the lowest-dimension operator that is non-vanishing on shell and appears as a two-loop counterterm. There is a tension between the Schwarzschild de Sitter (SdS) spacetime and this operator, which we study in some detail.

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For unimodular gravity, we work out, by using dimensional regularization, the complete one-loop correction to the graviton propagator in any space-time dimension. The computation is carried out within the framework where unimodular gravity has Weyl invariance in addition to the transverse diffeomorphism gauge symmetry. Thus, no Lagrange multiplier is introduced to enforce the unimodularity condition. The quantization of the theory is carried out by using the BRST framework and there considering a large continuous family of gauge-fixing terms. The BRST formalism is developed in such a way that the set of ghost, anti-ghost and auxiliary fields and their BRST changes do not depend on the space-time dimension, as befits dimensional regularization. As an application of our general result, and at D = 4, we obtain the renormalized one-loop graviton propagator in the dimensional regularization minimal subtraction scheme. We do so by considering two simplifying gauge-fixing choices.