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This matches eqn 1.17 of Mike's notes under the identification $\phi_I \equiv \phi^R_I,\phi_{N+I} \equiv \phi^L_I$. Moreover, by using the above we can convince ourselves that:  $$ [\rho_{R}, e^{-i \phi_R^g] \left[ \rho_R, e^{-i\phi_R^G} \right]  = [ \frac{1}{2\pi} \partial_x \phi_R^g}, \left[ \frac{1}{2\pi}\partial_x \phi_R^G,  e^{-i \phi_R^g}} ] \phi_R^G} \right]  = \delta(x-y) e^{-i\phi_R^g}(y)} delta(x-y) e^{- i \phi_R^G}  $$