A Critical String Theory in 3+1 Dimensions

Redefining the vacuum state of a free twofold N=1 covariant supersymmetric string action as the one with all the world sheet fermionic excited states occupied, makes the theory anomaly free in D=4 with Minkowski signature. The theory thus describes a critical string in 3+1 dimensions as opposed to earlier N=2 supersymmetric theories describing a 2+2 dimensional target space. While in the NS sector the spectrum basically resembles the same for the standard N=1 superstring theory with one of the $N$ species in the background, in the R sector both the species of fermions and superconformal ghosts are required to describe the relevant spin operators to describe the fermion spectra. A crucial difference from D=10 case is that the fermion states are Dirac particles instead of Majorana-Weyl. Even though the full spectrum of the theory contains both bosons and fermions of various spin, there is no space-time supersymmetry due to obvious lack of triality.