Hausdorff dimension of critical fluctuations in abelian gauge theories

The geometric properties of the critical fluctuations in abelian gauge theories such as the Ginzburg-Landau model are analyzed in zero background field. Using a dual description, we obtain scaling relations between exponents of geometric and thermodynamic nature. In particular we connect the anomalous scaling dimension $\eta$ of the dual matter field to the Hausdorff dimension $D_H$ of the critical fluctuations, {\it which are fractal objects}. The connection between the values of $\eta$ and $D_H$, and the possibility of having a thermodynamic transition in finite background field, is discussed.