Dirac Strings and the Nonperturbative Photon Propagator in Compact QED

A ``3D’’ Mathematica@-generated picture of monopoles and Dirac strings in $D=2+1$ compact QED is given. In the Villain approximation, the monopole part of the partition function factorizes from the Dirac string part, which generates the photon propagator. Numerical experiments in exact compact QED confirm this result: photon mass pole $M_{\gamma }$, originally nonzero, is insensitive to monopole prohibition but almost vanishes if Dirac strings are prohibited.