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 $e^{-2\varphi }$ 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 ${{\tilde{h}} =3-h}$oftheuniversalclass$h$ of fractons defined in the interval 1< h <2 is satisfied.