  The whole picture of gauge theory is described by manifolds, while the field
equation provides only a part (a section) of the manifold. Just as a
three-dimensional object is reconstructed from two planar images, a monopole is
constructed by combining two solutions. The Schwinger and the Unruh effects and
the Hawking radiation are the production of particles out of the ``vacuum''. If
the '''vacuum'' on the manifold is properly defined, these phenomena should be
described as local phenomena. However, calculations using the field equations
have so far resulted in unnatural extrapolations. We present a method for
properly defining the ``vacuum'' and explain how to resolve the local particle
production on manifolds. By defining the Stokes phenomena on the manifold, the
Schwinger effect is naturally accompanied by the Unruh effect. Also, unlike the
conventional Unruh effect, calculations on manifolds do not suffer from the
entanglement between disconnected wedges.
