Riccardo Martini

A bstract: We derive the runnings of the R and R ² operators that stem from integrating out quantum torsion fluctuations on a maximally symmetric Euclidean background, while treating the metric as a classical field. Our analysis is performed in a manifestly covariant way, exploiting both the recently-introduced spin-parity decomposition of torsion perturbations and the heat kernel technique. The Lagrangian we start with is the most general one for 1-loop computations on maximally symmetric backgrounds involving kinetic terms and couplings to the scalar curvature that is compatible with a gauge-like symmetry for the torsion. The latter removes the twice-longitudinal vector mode from the spectrum, and it yields operators of maximum rank four. We also examine the conditions required to avoid ghost instabilities and ensure the validity of our assumption to neglect metric quantum fluctuations, demonstrating the compatibility between these two assumptions. Then, we use our findings in the context of Starobinsky’s inflation to calculate the contributions from the torsion tensor to the β -function of the R ² term. While this result is quantitatively reliable only at the 0-th order in the slow-roll parameters or during the very early stages of inflation — due to the background choice — it qualitatively illustrates how to incorporate quantum effects of torsion in the path integral formalism.

We study substructures of the Weyl group of conformal transformations of the metric of (pseudo)Riemannian manifolds. These substructures are identified by differential constraints on the conformal factors of the transformations which are chosen such that their composition is associative. Mathematically, apart from rare exceptions, they are partial associative groupoids, not groups, so they do not have an algebra of infinitesimal transformations, but this limitation can be partially circumvented using some of their properties cleverly. We classify and discuss the substructures with two-derivatives differential constraints, the most famous of which being known as the harmonic or restricted Weyl group in the physics literature, but we also show the existence of a lightcone constraint which realizes a proper subgroup of the Weyl group. We then show the physical implications that come from invariance under the two most important substructures, concentrating on classical properties of the energy-momentum tensor and a generalization of the quantum trace anomaly. We also elaborate further on the harmonic substructure, which can be interpreted as partial gauge fixing of full Weyl invariance using BRST methods. Finally, we discuss how to construct differential constraints of arbitrary higher-derivative order and present, as examples, generalizations involving scalar constraints with four and six derivatives.

We discuss generalizations of the notions of projective transformations acting on affine model of Riemann-Cartan and Riemann-Cartan-Weyl gravity which preserve the projective structure of the light-cones. We show how the invariance under some projective transformations can be used to recast a Riemann-Cartan-Weyl geometry either as a model in which the role of the Weyl gauge potential is played by the torsion vector, which we call torsion-gauging, or as a model with traditional Weyl (conformal) invariance.

We discuss generalizations of the notions of projective transformations acting on affine model of Riemann–Cartan and Riemann–Cartan–Weyl gravity which preserve the projective structure of the light-cones. We show how the invariance under some projective transformations can be used to recast a Riemann–Cartan–Weyl geometry either as a model in which the role of the Weyl gauge potential is played by the torsion vector, which we call torsion-gauging, or as a model with traditional Weyl (conformal) invariance.

We propose a functional measure over the torsion tensor. We discuss two completely equivalent choices for the Wheeler–DeWitt supermetric for this field, the first one is based on its algebraic decomposition and the other is inspired by teleparallel theories of gravity. The measure is formally defined by requiring the normalization of the Gaußian integral. To achieve such a result we split the torsion tensor into its spin-parity eigenstates by constructing a new, York-like, decomposition. Of course, such a decomposition has a wider range of applicability to any kind of tensor sharing the symmetries of the torsion. As a result of this procedure a functional Jacobian naturally arises, whose formal expression is given exactly in the phenomenologically interesting limit of maximally symmetric spaces. We also discuss the explicit computation of this Jacobian in the case of a four-dimensional sphereS ⁴ with particular emphasis on its logarithmic divergences.