summarizing, we want to know how many times the sign of the square root of the determinant of a dirac operator changes sign as one circumnavigates the circle. the eigenvalues of the dirac operator come in pairs, and the sign changes each time a pair crosses zero. thus we are counting the zeroes of the dirac operator on the space formula_3. these zeroes are counted by the atiyah-singer index theorem, which gives the answer h times the second chern class of the gauge bundle over formula_3. this second chern class may be any integer. in particular it may be one, in which case the sign changes h times. if the sign changes an odd number of times then the partition function is ill-defined, and so there is an anomaly.