Free Boson CFT — Operator Product Expansion

The free boson CFT with central charge c=1. Visualize correlation functions of primary fields V_α(z)=e^{iαφ}, the stress tensor OPE, and conformal dimensions on the cylinder.

Parameters

V_α(z) = :e^{iαφ(z)}: h_α = ᾱ·α/2 = α²/2 ⟨V_α(z)V_{-α}(w)⟩ = |z-w|^{-2α²} T(z)T(w) ~ c/2·(z-w)^{-4} + 2T(w)/(z-w)² + ∂T(w)/(z-w) c = 1
Primary fields have conformal dimension h=α²/2. The OPE V_α·V_β contains V_{α+β} with structure constant C(α,β). On the cylinder z=e^{2π(τ+iσ)/L}, states have energy E=2π(h+h̄-c/12)/L.