  The geometric properties of critical fluctuations in the 3D XY model are
analyzed. The 3D XY model is a lattice model describing superfluids. We present
a direct evaluation of the Hausdorff dimension D_H of the vortex loops which
are the critical fluctuations of the 3D XY model. We also present analytical
arguments for why \vartheta in the scaling relation \eta_{\phi} + D_H = 2 +
\vartheta between D_H and the anomalous scaling dimension of the corresponding
field theory, must be zero.
