  We discuss here a specific field-theory model, inspired from string theory,
in which the generation of a matter-antimatter asymmetry in the Cosmos is due
to the propagation of fermions in a non-trivial, spherically asymmetric (and
hence Lorentz violating) gravitational background that may characterise the
epochs of the early universe. The background induces different dispersion
relations, hence populations, between fermions and antifermions, and thus CPT
Violation (CPTV) already in thermal equilibrium. Species populations may freeze
out leading to leptogenesis and baryogenesis. More specifically, after
reviewing some generic models of background-induced CPTV in early epochs of the
Universe, we consider a string-inspired scenario, in which the CPTV is
associated with a cosmological background with torsion provided by the
Kalb-Ramond (KR) antisymemtric tensor field of the string gravitational
multiplet. In a four-dimensional space time this field is dual to a
pseudoscalar ``axion-like'' field. The thermalising processes in this model are
(right-handed) Majorana neutrino-antineutrino oscillations, which are induced
in the presence of the KR axion background. These processes freeze out at a
(high) temperature $T_c >> m$, where $m$ is the Majorana neutrino mass, at
which the KR background goes to zero or is diminished significantly, through
appropriate phase transitions of the (string) universe. An additional, but
equally important, r\^ole, of the KR field is that its quantum fluctuations and
mixing with an ordinary axion, which couples to the Majorana neutrinos via
appropriate Yukawa couplings, can also lead to the generation of a Majorana
neutrino mass through quantum anomalies. This provides a novel way for
generating neutrino masses, independent of the traditional seesaw mechanism.
