Comment on "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, 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.