  We discuss the role of torsion in string theory on inducing pseudoscalar
degrees of freedom (axions), which in turn couple to (gravitational)
Chern-Simons (CS) anomalous terms. Such interactions can induce inflation, of
running vacuum type, not requiring external inflaton fields, through
condensation of the anomalous terms as a consequence of primordial chiral
gravitational-wave (GW) tensor perturbations in a weak-quantum gravity setting.
The presence of an UV cutoff for the GW quantum graviton modes opens up the
system, leading to a dissipative behaviour realised via the presence of non
trivial imaginary parts of the gravitational CS terms. The naive estimate of
the life time of inflation based on such imaginary parts, which afflict the
pertinent GW Hamiltonian, is quite consistent with the estimates of the
duration of inflation based on an analysis of the condensate-induced
linear-axion-potential by means of dynamical systems. Such quantum-gravity
effects can also contribute positively to the alleviation of cosmological
tensions if they survive today. In the talk we discuss the conditions under
which such a result may be achieved. We also discuss the potential role of
other axions in string theory, coming from compactification, in inducing
enhanced densities of primordial black holes during RVM inflation, thereby
contributing to significantly increased percentages of these black holes that
can play the role of dark matter components. Moreover, under certain
circumstances, that we shall discuss in some detail, it is also possible that
the initially massless torsion-induced axions can acquire a non-trivial mass
during the radiation era, thereby providing additional dark matter components
in the Universe. With regards to this aspect, we also emphasise the role of
massive right-handed neutrinos, provided that such excitations exist in the
relevant spectra.
