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We present a model for the Universe in which quantum anomalies are argued toplay an important dual role: they are responsible for generatingmatter-antimatter asymmetry, but also provide time-dependent contributions tothe vacuum energy density of "running-vacuum" type, which drive the Universeevolution. During the inflationary phase of a string-inspired Universe, and itssubsequent exit, the existence of primordial gravitational waves inducegravitational anomalies, which couple to the Kalb-Ramond (KR)axion field of themassless gravitational string multiplet. Such anomalous CP violatinginteractions have two important effects: first, they lead to contributions tothe vacuum energy density of the form appearing in the "running vacuum model"(RVM) framework, proportional to both the square and the fourth power of theeffective Hubble parameter $H$, the $H^4$ terms being associated withanomaly-induced inflation within the RVM framework. Second, there is anundiluted KR axion at the end of inflation, which plays an important role ingenerating matter-antimatter asymmetry through baryogenesis via leptogenesis inmodels with heavy right handed neutrinos. As the Universe exits inflation andenters a radiation dominated era, the generation of chiral fermionic matter isresponsible for the cancellation of gravitational anomalies, thus restoringdiffeomorphism invariance for the matter/radiation (quantum) theory, asrequired for consistency. Chiral U(1) anomalies may remain uncompensated,though, during matter/radiation dominance, providing RVM-like $H^2$ and $H^4$contributions to the Universe energy density. Finally, in the current era,gravitational anomalies resurface, leading to much weaker RVM-like $H_0^2$contributions to the vacuum energy density.Our model favours axionic DarkMatter,the source of which can be the KR axion.

Gravitational and Chiral Anomalies in the Running Vacuum Universe and Matter-Antimatter Asymmetry