  We study geometric variational problems for a class of effective models in
quantum field theory known as Faddeev-Skyrme models. Mathematically one
considers minimizing an energy functional on homotopy classes of maps from
closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy
minimizers known as Hopfions describe stable configurations of subatomic
particles such as protons and their strong interactions.
  In particular, we introduce novel Sobolev spaces suitable for our variational
problem and develop the notion of homotopy type for maps in such spaces that
generalizes homotopy for smooth and continuous maps. This notion is based on
defining a generalized Hopf invariant for Sobolev maps into homogeneous spaces
that was previously known only for maps into $S^2$. Since the spaces in
question are neither linear nor even convex we represent maps by gauge
potentials that form a linear space and reformulate the problem in terms of
these potentials. However, this representation of maps introduces some gauge
ambiguity into the picture and we work out 'gauge calculus' for the principal
bundles involved to apply the gauge-fixing techniques that eliminate the
ambiguity. These bundles arise as pullbacks of the structure bundles $H\hra
G\to G/H$ of homogeneous spaces and we study their topology and geometry that
are of independent interest.
  Our main results include proving existence of Hopfions as finite energy
Sobolev maps in each (generalized) homotopy class when the target space is a
symmetric space. For more general spaces we obtain a weaker result on existence
of minimizers only in each 2-homotopy class.
