String-inspired running-vacuum cosmology, quantum corrections and the current cosmological tensions

In the context of a string-inspired running vacuum model (RVM) of cosmology with anomalies and torsion-induced axion-like fields, we discuss quantum corrections to the corresponding energy density, in approximately de Sitter eras, during which the Hubble parameter $H(t)$ varies very slowly with the cosmic time $t$. Such corrections arise either from graviton loops in the corresponding gravitational theory, or from path integration of massive quantum matter fields. They depend logarithmically on $H(t)$, in the form $H^n\, {\rm ln}(H^2)$, $n \in 2Z^+$. In the modern eras, for which the $n=2$ terns are dominant, such corrections may contribute to an alleviation of the currently observed cosmological $H_0$ and structure-growth tensions. In particular, we argue that such an effect is accomplished for a (dynamically-broken) supergravity-based RVM cosmological model. In the current de-Sitter era, for this case, rather surprisingly, the quantum graviton corrections dominate those due to matter fields, provided the scale of the primordial (pre-RVM inflation) dynamical breaking of local supersymmetry lies near the reduced Planck scale, which is a natural assumption in the context of the model.