Fractons and Luttinger liquids

We consider the concept of fractons as particles or quasiparticles which obey a specific fractal statistics in connection with a one-dimensional Luttinger liquid theory. We obtain a dual statistics parameter ${\tilde{\nu}}=\nu+1$ which is identified with the controlling parameter $e^{-2\phi}$ of the Luttinger model. In this way, a bosonic system characterized by a fractal index $i_{f}[h]=i_{f}[2]=1$ is considered as a conformal field theory with central charge $c[\nu=0]=1=i_{f}[2]$ with a compactified radius $R=\frac{1}{\sqrt{\tilde{\nu}}}=1$. Thus, we have a mapping of a bosonic theory to a fermionic one and vice-versa, i.e., the duality symmetry ${{\tilde{h}} =3-h}$ of the universal class $h$ of fractons defined in the interval 1< h <2 is satisfied.