  We present the extension of the Lagrangian $loop$ representation in such a
way to introduce matter fields. The partition function of lattice compact U(1)
Gauge-Higgs model is expressed as a sum over closed as much as open surfaces.
These surfaces correspond to world sheets of loop-like pure electric flux
excitations and open electric flux tubes carrying matter fields at their ends.
There is a duality transformation between this description in terms of loop
world sheets and the $topological$ representation in terms of world sheets of
Nielsen-Olesen strings both closed and open joining pairs of monopoles.
