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

Logarithmic-corrected $R^2$ Gravity Inflation in the Presence of Kalb-Ramond Fields