  In this paper we propose a notion of $s$-fractional mass for $1$-currents in
$\R^d$. Such a notion generalizes the notion of $s$-fractional perimeters for
sets in the plane to higher codimension one-dimensional singularities.
Remarkably, the limit as $s\to 1$ of the $s$-fractional mass gives back the
classical notion of length for regular enough curves in $\R^d$.
  We prove a lower semi-continuity and compactness result for sequences of
$1$-currents with uniformly bounded fractional mass and support. Moreover, we
prove the density of weighted polygonal, closed and compact oriented curves in
the class of divergence-free 1-currents with compact support and finite
fractional mass.
  Finally, we discuss some possible applications of our notion of fractional
mass to build up purely geometrical approaches to the variational modeling of
dislocation lines in crystals and to vortex filaments in superconductivity.
