  We review scenarios of baryogenesis through leptogenesis at early epochs of
the universe, in string-inspired minimal extensions of the Standard Model (SM),
involving heavy right-handed Majorana neutrinos. Spontaneous violation of CPT
symmetry is induced by appropriate (in general, temperature-dependent)
backgrounds of the Kalb-Ramond (KR) axion field, which has its origins in the
(bosonic) massless string multiplet. As interesting features of the model, we
also discuss two issues associated with quantum (chiral) anomalies: (i) the
non-contribution of the KR axion background to the (anomalous) chiral magnetic
effect, which arises in the presence of external electromagnetic fields and
non-zero chiral chemical potentials of charged fermions and (ii) the potential
r\^ole of quantum fluctuations of the KR axion on the (anomalous) radiative
generation of a Majorana mass for the right-handed neutrinos themselves.
