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We discuss the weight of vacuum energy in various contexts. First, we compute the vacuum energy for flat spacetimes of the form T³×R, where T³ stands for a general 3-torus. We discover a quite simple relationship between energy at radius R and energy at radius l2/R. Then we consider quantum gravity effects in the vacuum energy of a scalar field in M 3×S¹ where M 3 is a general curved spacetime, and the circle S 1 refers to a spacelike coordinate. We compute it for general relativity and generic transverse diffeomorphisms theories. In the particular case of unimodular gravity vacuum energy does not gravitate. [ABSTRACT FROM AUTHOR]

A new class of conformal field theories is presented, where the background gravitational field is conformally flat. Conformally flat (CF) spacetimes enjoy conformal properties quite similar to the ones of flat spacetime. The conformal isometry group is of maximal rank and the conformal Killing vectors in conformally flat coordinates are exactly the same as the ones of flat spacetime. In this work, a new concept of distance is introduced, the conformal distance, which transforms covariantly under all conformal isometries of the CF space. It is shown that precisely for CF spacetimes, an adequate power of the said conformal distance is a solution of the nonminimal d'Alembert equation. [ABSTRACT FROM AUTHOR]

The unimodular version of the ghost-free higher derivative gravity is obtained. It is the unimodular reduction of some particular Lagrangians quadratic in curvature. [ABSTRACT FROM AUTHOR]