  We propose a new class of gravity-matter-gauge theories in terms of two
different non-Riemannian volume-forms independent of the Riemannian metric. The
nonlinear gauge field system contains a square-root $\sqrt{-F^2}$ of the
standard Maxwell Lagrangian which is known to describe charge confinement in
flat spacetime. In the physical Einstein frame we obtain an effective
Lagrangian of "k-essence" type with quadratic dependence on the scalar
"dilaton" kinetic term X, with a remarkable effective potential possessing two
infinitely large flat regions as well as with nontrivial effective gauge
coupling constants running with the "dilaton" $\varphi$. Corresponding to the
each of the two flat regions we find "vacuum" configurations of the following
types: (i) $\varphi = const$ and a non-zero gauge field vacuum $\sqrt{-F^2}\neq
0$, which corresponds to a charge confining phase; (ii) X = const ("kinetic
vacuum") and ordinary gauge field vacuum $\sqrt{-F^2}=0$ which supports
confinement-free charge dynamics. In one of the flat regions of the effective
potential we also find: (iii) X = const ("kinetic vacuum") and a non-zero gauge
field vacuum $\sqrt{-F^2}\neq 0$, which again corresponds to a charge confining
phase. In all three cases the spacetime metric is de Sitter or Schwarzschild-de
Sitter. Both "kinetic vacuums" (ii) and (iii) can exist only within a
finite-volume space region below a de Sitter horizon. Extension to the whole
space requires matching the latter with the exterior region with a nonstandard
Reissner-Nordstroem-de Sitter geometry carrying an additional constant radial
background electric field. As a result we obtain two classes of gravitational
bag-like configurations with properties, which on one hand partially parallel
some of the properties of the solitonic "constituent quark" model and, on the
other hand, partially mimic some of the properties of MIT bags in QCD
phenomenology.
