  The subject of this work is a three-dimensional topological field theory with
a non-semisimple group of gauge symmetry with observables consisting in the
holonomies of connections around three closed loops. The connections are a
linear combination of gauge potentials with coefficients containing a set of
one-dimensional scalar fields. It is checked that these observables are both
metric independent and gauge invariant. The gauge invariance is achieved by
requiring non-trivial gauge transformations in the scalar field sector. This
topological field theory is solvable and has only a relevant amplitude which
has been computed exactly. From this amplitude it is possible to isolate a
topological invariant which is Milnor's triple linking invariant. The
topological invariant obtained in this way is in the form of a sum of multiple
contour integrals. The contours coincide with the trajectories of the three
loops mentioned before. The introduction of the one-dimensional scalar field is
necessary in order to reproduce correctly the particular path ordering of the
integration over the contours which is present in the triple Milnor linking
coefficient. This is the first example of a local topological gauge field
theory that is solvable and can be associated to a topological invariant of the
complexity of the triple Milnor linking coefficient.
