3-dimensional chern–simons gauge theories are also anomalous when their level is half-integral. in fact, the derivation is identical to that above. using stokes' theorem and the fact that the exterior derivative of the chern–simons action is equal to the instanton number, the 4-dimensional theory on formula_3 has a theta angle equal to the level of the chern–simons theory, and so the 4-dimensional partition function is equal to -1 precisely when the instanton number is odd. this implies that the 3-dimensional partition function is ill-defined by a factor of -1 when considering deformations over a path with an odd number of instantons.