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(in particular, we are assuming the Fermi momenta of the right and left movers in channel I coincide)  And similarly, for the bosons, but with no minus sign:  $$  \delta \phi_I = \epsilon( k_{F,I} + \partial_x \phi_I ) $$  $$  \delta \phi_{N+I} = \epsilon( k_{F,I} + \partial_x \phi_{N+I}) $$