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 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? In other words, I would like a more explicit derivation of the above equation. Thank you.
Best Answer
The integral $$ I_\gamma=\oint_\gamma d\theta= \oint (\partial_x\theta dx +\partial_y \theta dy)= \oint\nabla \theta \cdot d{\bf r} $$ is $2\pi$ if the loop $\gamma$ encloses the origin and zero if it does not. Now use Stokes' theorem $$ I_\gamma = \int_\Omega \nabla\times (\nabla\theta) d^2r , \quad \partial \Omega=\gamma. $$ Interpreting the fact that $I_\gamma$ is $2\pi$ or zero depending on whether the origin lies within $\Omega$ as a statement about distributions, we read off that $\nabla\times (\nabla \theta)= 2\pi \delta^2(x,y)$. In components this is $\partial_x \partial_y \theta - \partial_y\partial_x \theta = 2\pi \delta^2(x,y)$.