Coupling of c  =  −2 and c$=\frac{1}{2}$ and c  =  0 conformal field theories: the geometrical point of view

The coupling of the c  =  −2, $c=\frac{1}{2}$ and c  =  0 conformal field theories are numerically considered in this paper. As the prototypes of the couplings, $(c_1=-2)\oplus (c_2=0)$ and $(c_1=-2)\oplus (c_2=\frac{1}{2})$, we consider the Bak–Tang–Weisenfeld (BTW) model on the 2D square critical site-percolation and the BTW model on Ising-correlated percolation lattices respectively. Some geometrical techniques are used to characterize the presumable conformal symmetry of the resultant systems. Based on the numerical analysis of the diffusivity parameter (κ) in the Schramm–Loewner evolution (SLE) theory we propose that the algebra of the central charges of the coupled models is closed. This result is based on the analysis of the conformal loop ensemble (CLE) analysis. The diffusivity parameter in each case is obtained by calculating the fractal dimension of loops (and the corresponding exponent of mean-square root distance), the direct SLE mapping method, the left passage probability and the winding angle analysis. More precisely we numerically show that the coupling $(c_1=-2)\oplus (c_2=\frac{1}{2})$ results to 2D self-avoiding walk (SAW) fixed point corresponding to c  =  0 conformal field theory, whereas the coupling $(c_1=-2)\oplus (c_2=0)$ results to the 2D critical Ising fixed point corresponding to the $c=\frac{1}{2}$ conformal field theory. (paper)