deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 9b8b216..e2397e6 100644  --- a/untitled.tex  +++ b/untitled.tex 
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$$ [\rho_{R,I}, e^{i \phi_I}] = [ \frac{-1}{2\pi} \partial_x \phi^R_I, e^{i \phi^R_I} ] = \delta(x-y) e^{i\phi^R_I(y)} $$  $$ [\rho_L, e^{i \phi_L} e^{-i \phi_{N+I}}  ] = i \delta(x-y)e^{i\phi_L} [ \frac{-1}{2\pi} \partial_x \phi^L_I, e^{-i \phi^L_I} ] = \delta(x-y) e^{-i\phi^L_I(y)}  $$