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In previous works we have embedded the Running Vacuum Model (RVM) ofCosmology in the framework of string theory. Specifically, we considered astring-inspired Cosmology with primordial gravitational waves (GW) andgravitational anomalies, which were argued to lead, via appropriatecondensation during the very-early-universe era, to dynamical inflation, of RVMtype, without the need for extra inflaton fields. A crucial role for theassociated slow-roll nature of the inflationary era was played by thefundamental axion field that exists in the gravitational multiplet of strings,viz. the Kalb-Ramond (KR) axion. In this paper, we study further this model anddemonstrate several novel facts, completing our previous studies. We clarifythe different roles played by the background KR axions, which constitute a formof stiff matter that dominates a pre-RVM-inflationary epoch of the Universe. Weshow that the KR axion when combined with the gravitational Chern-Simonscontribution obey together a peculiar equation of state $p=-\rho$, withnegative energy density $\rho < 0$, which we call `phantom vacuum'. Eventually,this state transmutes into the standard vacuum state thanks to the contributionfrom the gravitational Chern-Simons condensate, which makes the total$\rho_{\rm total} =-p_{\rm total} >0$. At this point the RVM vacuum pictureemerges naturally, with an overall equation of state of true vacuum type. Wefind that our scenario is consistent with the trans-Planckian censorshiphypothesis, given that the rate of change of the KR axion takes onlysub-Planckian values. We also discuss some explicit scenarios on the creationof GWs within the context of supergravity models that could be embedded in ourstring theory framework. {Finally, we argue how the RVM in our context can helpto alleviate the current tensions of the $\Lambda$CDM.}

Inflationary physics and transplanckian conjecture in the Stringy Running-Vacuum-Model: from the phantom vacuum to the true vacuum