In this 1985 paper by Callan and Harvey, Eq. $11$ seems to claim that in the presence of an infinitely extended string-like topological defect, partial derivatives do not seem to commute on the string:
$$ [\partial_x, \partial_y]\, \theta = 2\pi \delta(x)\delta(y)\,. \tag{11} $$
This is apparently "because of the topology of the axion string." I do not quite follow. Can someone please explain the reasoning to me in a bit more detail? Thank you.