Toy superstrings

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Abstract
Superstrings with critical dimension two and two real bosonic spacetime coordinates may serve as useful toy models for the study of string properties. We present details of several such toy superstrings including open, closed, and heterotic models. Conformal methods are used to establish the spectrum and dimensionality. The spin fields are provided and Becchi-Rouet-Stora-Tyutin- (BRST-) invariant vertex operators are constructed. Four-point tree-level and four-point one-loop amplitudes in these models are obtained. The closed and heterotic toy superstrings are shown to be modular invariant to this order.