Quantum theory of dual relativistic parastring models

We paraquantize the classical massless relativistic-string action and find that the resulting theory is Poincaré-invariant in four space-time dimensions if we use para-Bose commutation relations of order 12. More generally, we find that if the dimension $D$ of the space-time and the order $q$ of parabosons are related by the expression $D=2+\frac{24}{q}$, then the quantized theory is Poincaré-invariant. We also construct a fermionic parastring model which is the analog of the Ramond-Neveu-Schwarz model and find that it is invariant in $D$ dimensions if $D=2+\frac{8}{q}$, both the fermions and the bosons being of order $q$. We show by explicit Klein transformations that these theories are equivalent to "color"-endowed canonically quantized strings with $\mathrm{SO}\mathrm{}(q-1\mathrm{})\mathrm{}$ "color" symmetry. We obtain dual tree amplitudes by suitable choice of vertices. Finally, we consider second-quantized parastring theories and show, by an explicit example, that they can be Poincaré-invariant in four space-time dimensions.