Absence of (1,0) supersymmetry anomaly in world-sheet gauge theories: a purely cohomological proof

A purely cohomological proof is given for the absence of the (1,0) supersymmetry anomaly in gauge theories on a world sheet. In particular, it is shown that generalized cohomological approaches to anomalies in supersymmetric gauge theory, either formulated in whole superconnection space or only in the Wess-Zumino-gauge surface, yield results which agree with those obtained by noncohomological field-theoretical methods. We argue that the success of the cohomological arguments implies that there should be a generalization of (family) index theorem in the supersymmetric cases.