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In the context of a string-inspired running vacuum model (RVM) of cosmologywith anomalies and torsion-induced axion-like fields, we discuss quantumcorrections to the corresponding energy density, in approximately de Sittereras, during which the Hubble parameter $H(t)$ varies very slowly with thecosmic time $t$. Such corrections arise either from graviton loops in thecorresponding gravitational theory, or from path integration of massive quantummatter fields. They depend logarithmically on $H(t)$, in the form $H^n\, {\rmln}(H^2)$, $n \in 2Z^+$. In the modern eras, for which the $n=2$ terns aredominant, such corrections may contribute to an alleviation of the currentlyobserved cosmological $H_0$ and structure-growth tensions. In particular, weargue that such an effect is accomplished for a (dynamically-broken)supergravity-based RVM cosmological model. In the current de-Sitter era, forthis case, rather surprisingly, the quantum graviton corrections dominate thosedue 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.

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