  In order to have a new perspective on the long-standing problem of the mass
gap in Yang-Mills theory, we study the quantum Yang-Mills theory in the
presence of topologically nontrivial backgrounds in this paper. The
topologically stable gauge fields are constrained by the form invariance
condition and the topological properties. Obeying these constraints, the known
classical solutions to the Yang-Mills equation in the 3- and 4-dimensional
Euclidean spaces are recovered, and the other allowed configurations form the
nontrivial topological fluctuations at quantum level. Together, they constitute
the background configurations, upon which the quantum Yang-Mills theory can be
constructed. We demonstrate that the theory mimics the Higgs mechanism in a
certain limit and develops a mass gap at semi-classical level on a flat space
with finite size or on a sphere.
