  Non-relativistic charged particles and strings coupled with abelian gauge
fields are quantized in a geometric representation that generalizes the Loop
Representation. We consider three models: the string in self-interaction
through a Kalb-Ramond field in four dimensions, the topological interaction of
two particles due to a BF term in 2+1 dimensions, and the string-particle
interaction mediated by a BF term in 3+1 dimensions. In the first case one
finds that a consistent "surface-representation" can be built provided that the
coupling constant is quantized. The geometrical setting that arises corresponds
to a generalized version of the Faraday's lines picture: quantum states are
labeled by the shape of the string, from which emanate "Faraday`s surfaces". In
the other models, the topological interaction can also be described by
geometrical means. It is shown that the open-path (or open-surface) dependence
carried by the wave functional in these models can be eliminated through an
unitary transformation, except by a remaining dependence on the boundary of the
path (or surface). These feature is closely related to the presence of
anomalous statistics in the 2+1 model, and to a generalized "anyonic behavior"
of the string in the other case.
