Lavinia Heisenberg

A bstract: We establish a non-renormalization theorem for a class of ghost-free local functionals describing massive vector field theories coupled to dynamical geometry. Under the assumptions of locality, Lorentz invariance, and validity of the effective field theory expansion below a fixed cutoff, we show that quantum corrections do not generate local operators that renormalize the classical derivative self-interactions responsible for the constraint structure of the theory. The proof combines an operator-level analysis of the space of allowed local counterterms with a systematic decoupling-limit argument, which isolates the leading contributions to the effective action at each order in the derivative expansion. As a consequence, all radiatively induced local functionals necessarily involve additional derivatives per field and are suppressed by the intrinsic strong-coupling scales of the theory. In particular, the classical interactions defining ghost-free vector field theories are stable under renormalization, and any additional degrees of freedom arising from quantum corrections appear only above the effective field theory cutoff. This result extends known non-renormalization properties of flat-space vector theories to the case of dynamical geometry and provides a structural explanation for their perturbative stability to all loop orders.

We prove that autoparallel curves associated with a torsion‐free but not necessarily metric‐compatible affine connection can be derived from an action principle. We explicitly construct the action functional and show by standard variational techniques that it produces the desired equations. Our analysis is based on systematically solving the inverse problem of the calculus of variation and the associated Helmholtz conditions. This demonstrates that the dynamics of autoparallels admit a consistent variational formulation even in the presence of non‐metricity. Our results provide a variational framework for particle motion in metric‐affine geometries and thereby contribute to the mathematical foundations of the geodesic principle in relativistic gravity.

Infinite towers of higher-order corrections to General Relativity have been proposed as a mechanism to resolve singularities in early-universe cosmology and black holes, in a variety of settings. In this work, we consider an infinite tower of higher-order Proca corrections inspired by dimensional regularizations of Lovelock invariants. We find that the Big Bang singularity present in General Relativity is replaced by an inflationary epoch. Furthermore, the Lovelock-Proca tower allows for regular planar black hole solutions and spherically symmetric black holes with primary hair.

This paper undertakes a conceptual re‐examination of several foundational elements of cosmology through the lens of spacetime symmetries. A new derivation of the Friedmann–Lemaître–Robertson–Walker metric is obtained by a careful conceptual examination of rotations and translations on generic manifolds, followed by solving the rotational and translational Killing equations, yielding both the metric and its translational generators for without any further assumptions. We then analyze how continuous symmetries are inherited by the Einstein tensor and the Hilbert energy–momentum tensor, proving two general propositions. Furthermore, we use the Maxwell and Kalb–Ramond fields to show that a homogeneous and isotropic energy–momentum tensor, in general, does not give rise to field configurations which share these symmetries. In particular, the Kalb–Ramond field we derive is significantly more general than what is usually encountered in the cosmological context. Finally, we provide a rigorous but accessible, elementary, and transparent derivation of the scalar–vector–tensor decomposition from the linearized Einstein equations. Together, these results highlight the value of multiple complementary formulations of the same cosmological physics.

Generalized Proca Theories are the most general higher-derivative extensions of a massive vector field that retain second-order equations of motion. They are phenomenologically interesting as models of dynamical dark energy that, unlike scalar-tensor theories, can naturally accommodate cosmological anisotropies. A key open question is whether such theories can be fundamental. As a first step in this direction, we investigate whether they admit an ultraviolet completion within a quantum field theory framework, working with a truncation comprising up to four powers of the Proca field and up to two derivatives. We find a triplet of non-Gaussian ultraviolet fixed points, that lie very close to one another. Only one of them features a non-tachyonic Proca mass and could thus serve as a consistent ultraviolet completion for Generalized Proca Theories. We name it the Proca fixed point. We discuss its stability and contrast its features with those of the standard Reuter fixed point of the asymptotic safety scenario for quantum gravity and matter. In particular, we show that the Gaussian and Reuter fixed points lie on singular hypersurfaces of the flow of Generalized Proca Theories, yet can act as quasi-fixed points in certain regimes.

We establish a non-renormalization theorem for a class of ghost-free local functionals describing massive vector field theories coupled to dynamical geometry. Under the assumptions of locality, Lorentz invariance, and validity of the effective field theory expansion below a fixed cutoff, we show that quantum corrections do not generate local operators that renormalize the classical derivative self-interactions responsible for the constraint structure of the theory. The proof combines an operator-level analysis of the space of allowed local counterterms with a systematic decoupling-limit argument, which isolates the leading contributions to the effective action at each order in the derivative expansion. As a consequence, all radiatively induced local functionals necessarily involve additional derivatives per field and are suppressed by the intrinsic strong-coupling scales of the theory. In particular, the classical interactions defining ghost-free vector field theories are stable under renormalization, and any additional degrees of freedom arising from quantum corrections appear only above the effective field theory cutoff. This result extends known non-renormalization properties of flat-space vector theories to the case of dynamical geometry and provides a structural explanation for their perturbative stability to all loop orders.

Infinite towers of higher-order corrections to General Relativity have been proposed as a mechanism to resolve singularities in early-universe cosmology and black holes, in a variety of settings. In this work, we consider an infinite tower of higher-order Proca corrections inspired by dimensional regularizations of Lovelock invariants. We find that the Big Bang singularity present in General Relativity is replaced by an inflationary epoch. Furthermore, the Lovelock-Proca tower allows for regular planar black hole solutions and spherically symmetric black holes with primary hair.

We consider evolving, spatially flat isotropic and homogeneous (FLRW) cosmologies in ghost-free (dRGT) massive gravity. In this theory, no dynamical flat FLRW background exists if the reference metric is chosen to be Minkowski and the Stueckelberg fields are homogeneous. Relaxing the assumptions on the Stueckelberg profiles gives access to dynamical backgrounds. We propose a classification of the viable flat FLRW cosmological solutions of dRGT massive gravity. Instead of specifying an initial ansatz for the Stueckelberg fields ϕ ^a and the reference metric f _ab , we show that imposing homogeneity and isotropy on the square root tensor X ^μ _v = (√(g ^-1∂ϕ^a∂ϕ^b f _ab ))^μ _v leads to dynamical cosmological solutions, and we characterize their properties. These solutions become dynamical only when the Stueckelberg fields acquire a sufficiently inhomogeneous and/or anisotropic profile. We explore the consequences for the minimal model and the complete dRGT theory, and show that perturbations are strongly coupled, at the quadratic level, on these backgrounds.