Lorentz anomaly in the semi-light-cone gauge superstrings

We study the Lorentz invariance of D=4 and 6 superstrings in the double-spinor formalism, which are equivalent to the D=4 and 6 superstrings in the pure-spinor formalism in the sense of the BRST cohomology. We first re-examine how the conformal and Lorentz anomalies appear in the D=4 and 6 Green-Schwarz superstrings in the semi-light-cone gauge in the framework of BRST quantization. We construct a set of BRST invariant Lorentz generators and show that they do not form a closed algebra, even cohomologically. We then turn to the construction of Lorentz generators in the D=4 and 6 double-spinor superstrings, and show that the Lorentz invariance is again anomalous. We also discuss the relation between the anomaly-free Lorentz generators in the lower-dimensional pure-spinor formalisms and that obtained in this paper.