Visualising conformal transformations and primary operator scaling dimensions
In a 2D CFT, primary operators O have scaling dimension Δ and spin s; the full dimensions are (h,h̄)=((Δ+s)/2,(Δ−s)/2). The two-point function is fixed: ⟨O(z)O(w)⟩=1/|z−w|^{2Δ}. Conformal maps w=f(z) transform coordinates while preserving angles. A primary operator transforms as O'(w)=(dw/dz)^{−h}(dw̄/dz̄)^{−h̄}O(z). The left panel shows the grid before mapping; the right shows after — angles are preserved everywhere (conformality).