In this paper http://www.hcs.harvard.edu/~jus/0302/song.pdf when Song was explaining dirac string. He said "In the presence of a magnetic monopole, the vector potential cannot be defined everywhere. ...
I have asked this on this link http://math.stackexchange.com/q/1058307 But I will try to ask it here again. In this paper http://www.hcs.harvard.edu/~jus/0302/song.pdf Song explains the following ...
To introduce magnetic monopoles in Maxwell equations, Dirac uses special strings, that are singularities in space, allowing potentials to be gauge potentials. A consequence of this is the quantization ...
For a non-compact space, the Dirac string can be defined as a line joining the Dirac monopole to infinity (or another Dirac monopole). The region where the gauge connection is ill-defined. (as can be ...