Gluon Mass from Curvature of Gauge Slice

The masslessness of gluons due to gauge invariance appears to be contradictory with the observed massive structures of hadrons. As a dual process of gauge transformation, we have to fix the gauge to quantize the gluons in order to get propagators. The contradictory picture between gauge invariance and massive gluon propagation with vacuum condensates can be overcome by introducing generalized $\theta$ vacuum composed of curved gauge slices. On the curved gauge slices the gluon propagator turns out to be dependent on the curvature and in maximally symmetric space the curvature plays the role of mass, thus providing a solution to the contradiction between masslessness and massive propagation.