Omar Zanusso

A bstract: The multicritical generalizations of the Lee-Yang universality class arise as renormalization-group fixed points of scalar field theories with complex iφ ^{2 n +1} interaction, n ∈ ℕ, just below their upper critical dimension. It has been recently conjectured that their continuation to two dimensions corresponds to the non-unitary conformal minimal models ℳ(2 , 2 n + 3). Motivated by that, we revisit the functional renormalization group approach to complex 𝒫𝒯-symmetric scalar field theories in the Local Potential Approximation, without or with wavefunction renormalization (LPA and LPA’ respectively), aiming to explore the fate of the iφ ^{2 n +1} theories from their upper critical dimension to two dimensions. The iφ ^{2 n +1} fixed points are identified using a perturbative expansion of the functional fixed-point equation near their upper critical dimensions, and they are followed to lower dimensions by numerical integration of the full equation. A peculiar feature of the complex 𝒫𝒯-symmetric potentials is that the fixed points are characterized by real but negative anomalous dimensions η , and in low dimension d , this can lead to a change of sign of the scaling dimensions ∆ = ( d − 2 + η ) / 2, thus requiring a novel analysis of the analytical properties of the functional fixed-point equations. We are able to follow the Lee-Yang universality class ( n = 1) down to two dimensions, and numerically determine the scaling dimension of the fundamental field as a function of d . On the other hand, within the LPA’, multicritical Lee-Yang fixed points with n > 1 cannot be continued to d = 2 due to the existence of unexpected non-perturbative fixed points that annihilate with the iφ ^{2 n +1} fixed points.

General Relativity and Yang-Mills theories are the two pillars of our understanding of the physical world. The fact that the former is a classical field theory whereas the latter are usually presented as quantum field theories, tends to obscure the similarities between them and is often given as a motivation to develop a quantum theory of gravity. In this book we instead treat both as classical theories, putting them on a more even basis. Insisting that the dynamical variables must have a global geometrical meaning exposes their deep common foundations and allows to give precise answers to otherwise vague questions such as whether gravity is a gauge theory. The book contains standard presentations of Yang-Mills theories and General Relativity but covers also several topics that are not very well known, such as the geometry of the Higgs phenomenon, the transformation of spinors under diffeomorphisms, the derivation of General Relativity from the principles of special relativistic field theory.

We compute the physical running of a general higher derivative scalar coupled to a nondynamical metric and of higher derivative Weyl invariant gravity with a dynamical metric in four dimensions. In both cases, we find that the physical running differs from the μ -running of dimensional regularization because of infrared divergences which are present in amplitudes also at large momenta, differently from what happens in standard two derivative theories. We use the higher derivative scalar as a toy-model to elaborate on the properties of the conformal limit in relation to the trace anomaly. The physical running of higher derivative Weyl gravity, while different from the μ -running, remains asymptotically free, suggesting that the model is a viable completion of Einstein’s gravity, at least from the point of view of its renormalization group properties.

We use the ambient space construction, in which spacetime is mapped into a special lightcone of a higher dimensional manifold, to derive the integrable terms of the trace anomaly in even dimensions. We argue that the natural topological anomaly is the so-called Q-curvature, which, when projected from the ambient space, always comes with a Weyl covariant operator that can naturally be adopted for the integration of the anomaly itself in the form of a nonlocal action. The use of the ambient space makes transparent the fact that there are some new ambiguities in the integration of the anomaly, which we now understand geometrically from the ambient point of view. These ambiguities, which manifest themselves as undetermined parameters in the integrated nonlocal action, become more severe in dimensions d ≥ 6 and do not seem to be related to a choice of the renormalization scheme.

It is well-known that the results by Bekenstein, Gibbons and Hawking on the thermodynamics of black holes can be reproduced quite simply in the Euclidean path integral approach to quantum gravity. The corresponding partition function is obtained semiclassically, ultimately requiring only the on-shell Einstein–Hilbert action with opportune asymptotic subtractions. We elaborate on the fact that the same expressions for the thermodynamical quantities can be obtained within teleparallel equivalent theories, based on either torsion or nonmetricity, by employing quasilocal relations. Notably, the bulk integrals of these theories do not vanish on-shell but rather result in boundary terms themselves. Asymptotic subtractions of the latter are able to cancel out the divergences, ultimately leading to Bekenstein–Gibbons–Hawking’s results. As a non-trivial cross-check, we compute the bulk integrals directly without reference to the boundary terms. While the result agrees with the previous method for the torsion-based teleparallel theory, it differs for the nonmetricity theory. Specifically, upon regularizing the bulk integral using a fiducial reference frame, we find that the semiclassical partition function vanishes. To address this problem, we propose a simple prescription for Schwarzschild black holes, which involves keeping the nonmetric connection arbitrary and imposing thermal equilibrium. Generalizations of the results to more general modified gravity theories with antisymmetric degrees of freedom are also discussed.

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.