Covariant string field theory

The covariant string field theory is presented in its full detail for the open-bosonic-string case. The previously reported gauge-fixed action and Becchi-Rouet-Stora (BRS) transformation are completed by adding the quartic string-interaction term constructed explicitly here. The properties of the 3-string and 4-string vertices are fully clarified. We thus establish the nilpotency of the full non-linear BRS transformation and the BRS invariance of our gauge-fixed action. This, on the other hand, establishes also the gauge invariance of our gauge-unfixed action and the group law of the gauge transformations, which were also reported previously. The general $N$-string amplitudes are computed explicitly at the tree level and shown to reproduce correctly the usual dual amplitudes for the on-shell physical external states. In particular we prove that the on-shell physical amplitudes at the tree level are completely independent of the string length parameter $\alpha$. The zero-slope limit of our covariant string action as well as of the BRS transformation is calculated completely off the mass shell. The resultant action has the same form as the usual covariant Yang-Mills action $-\frac{1}{4}\mathrm{tr}F_{\mu \nu }^{}{}_{}{}^2$ as for the gauge-invariant part, and the nontrivial $\alpha$ dependence appears only in the gauge-fixing terms.