On Witten’s global anomaly for higher SU(2) representations

The integer nj depends on the group representation j, and the theory is ill-defined if nj is an odd integer. Witten showed, invoking an Atiyah-Singer index theorem [2] for a certain five-dimensional Dirac operator (the fifth coordinate is essentially the path parameter t), that nj is odd only for the representations in (1). Besides the fundamental representation with j = 1/2, the representations with j = 5/2, 9/2, etc. also suffer from Witten’s anomaly, while the 3/2 and the remaining half-integer representations, as well as all integervalued representations, are anomaly free. However, the index theorem does not predict nj itself. So we can ask the following question: What is the value of nj as a function of j that causes only half of the half-integer representations to be anomalous?