the original spacetime is 3-dimensional, call the space m. now we are considering a circle in configuration space, which is the same thing as a single configuration on the space formula_3. to find out the number of times that the sign of the square root vanishes as one goes around the circle, it suffices to count the number of zeroes of the determinant on formula_3, because each time that a pair of eigenvalues changes sign there will be a zero. notice that the eigenvalues come in pairs, as discussed for example in supersymmetric index of three-dimensional gauge theory, and so whenever one eigenvalue crosses zero, two will cross.