  In this paper we shall study the inflationary aspects of a logarithmic
corrected $R^2$ Starobinsky inflation model, in the presence of a Kalb-Ramond
field in the gravitational action of $F(R)$ gravity. Our main interest is to
pin down the effect of this rank two antisymmetric tensor field on the
inflationary phenomenology of the $F(R)$ gravity theory at hand. The effects of
the Kalb-Ramond field are expected to be strong during the inflationary era,
however as the Universe expands, the energy density of the Kalb-Ramond field
scales as $\sim a^{-6}$ so dark matter and radiation dominate over the
Kalb-Ramond field effects. In general, antisymmetric fields constitute the
field content of superstring theories, and thus their effect at the low-energy
limit of the theory is expected to be significant. As we will show, for a flat
Friedmann-Robertson-Walker metric, the Kalb-Ramond field actually reduces to a
scalar field, so it is feasible to calculate the observational indices of
inflation. We shall calculate the spectral index and the tensor-to-scalar ratio
for the model at hand, by assuming two conditions for the resulting Kalb-Ramond
scalar field, the slow-roll and the constant-roll condition. As we shall
demonstrate, in both the slow-roll and constant-roll cases, compatibility with
the latest observational data can be achieved. Also the effect of the
Kalb-Ramond field on the inflationary phenomenology is to increase the amount
of the predicted primordial gravitational radiation, in comparison to the
corresponding $f(R)$ gravities, however the results are still compatible with
the observational data.
