Hausdorff dimension of critical fluctuations in Abelian gauge theories

The geometric properties of the critical fluctuations in Abelian gauge theo ries such as the Ginzburg-Landau model are analyzed in zero background fiel d. Using a dual description, we obtain scaling relations between exponents of geometric and thermodynamic nature. In particular, we connect the anomal ous scaling dimension eta of the dual matter field to the Hausdorff dimensi on D-H Of the critical fluctuations, which are fractal objects. The connect ion between the values of eta and D-H, and the possibility of having a ther modynamic transition in finite background field, is discussed.