Internal free fermions in a four-dimensional bosonic string ($4+N_f/2-26=0$)

Keeping the ordinary reparameterization bc ghost system, four embedding bosons require 44 additional real chiral fermions in each chirality to cancel the worldsheet Weyl anomaly. These are internal matter fields, not an assumed locally supersymmetric RNS construction. Their bilinear currents generate the chiral $\mathfrak{so}(44)$ current algebra, with $\left(\genfrac{}{}{0ex}{}{44}{2}\right)=946$ currents. A full closed fermionic model also requires consistent spin-structure and modular projections; the central-charge count alone does not determine these choices.

In the unprojected polynomial Neveu–Schwarz vertex sector, zero fermion insertions give the usual closed-string tachyon and the massless graviton, two-form and dilaton. Two insertions split into a left-right pair ${\psi }^i{\bar{\psi }}^j{e}^{ipX}$ of internal weights $(1/2,1/2)$ and a same-chirality current pair of weight one. The first has $M^2=-2/{\alpha }^{\prime }$ and 44-squared scalar components. The second requires a bosonic derivative on the opposite side for level matching and gives the two sets of 946 massless gauge vectors. Four-fermion scalar-current products belong to a different insertion count. A chosen global projection may retain a subset, so these operators do not by themselves specify a projected string spectrum.