  We present a model for the Universe in which quantum anomalies are argued to
play an important dual role: they are responsible for generating
matter-antimatter asymmetry, but also provide time-dependent contributions to
the vacuum energy density of "running-vacuum" type, which drive the Universe
evolution. During the inflationary phase of a string-inspired Universe, and its
subsequent exit, the existence of primordial gravitational waves induce
gravitational anomalies, which couple to the Kalb-Ramond (KR)axion field of the
massless gravitational string multiplet. Such anomalous CP violating
interactions have two important effects: first, they lead to contributions to
the vacuum energy density of the form appearing in the "running vacuum model"
(RVM) framework, proportional to both the square and the fourth power of the
effective Hubble parameter $H$, the $H^4$ terms being associated with
anomaly-induced inflation within the RVM framework. Second, there is an
undiluted KR axion at the end of inflation, which plays an important role in
generating matter-antimatter asymmetry through baryogenesis via leptogenesis in
models with heavy right handed neutrinos. As the Universe exits inflation and
enters a radiation dominated era, the generation of chiral fermionic matter is
responsible for the cancellation of gravitational anomalies, thus restoring
diffeomorphism invariance for the matter/radiation (quantum) theory, as
required 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 Dark
Matter,the source of which can be the KR axion.
