  The Frenet equation governs the extrinsic geometry of a string in
three-dimensional ambient space in terms of the curvature and torsion, which
are both scalar functions under string reparameterisations. The description
engages a local SO(2) gauge symmetry, which emerges from the invariance of the
extrinsic string geometry under local frame rotations around the tangent
vector. Here we inquire how to construct the most general SO(2) gauge invariant
Hamiltonian of strings, in terms of the curvature and torsion. The construction
instructs us to introduce a long-range (self-) interaction between strings,
which is mediated by a three dimensional bulk gauge field with a Chern-Simons
self-interaction. The results support the proposal that fractional statistics
should be prevalent in the case of three dimensional string-like
configurations.
