Absence of the effects of vortices in the gauge glass

We calculate several correlation functions and distribution functions of dynamical variables for the gauge glass and the Villain model using the spin wave approximation and the gauge transformation. The results show that the spin wave approximation gives the exact solutions on the Nishimori line in the phase diagram. This implies that vortices play no role in the thermodynamic behavior of the system as long as some correlation and distribution functions are concerned. These results apply to any dimensions including the two-dimensional case.