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We show that the problem of impurity tunneling in a Luttinger liquid ofelectrons with spin is solvable in the spin isotropic case ($g_\sigma=2$,$g_\rho$ arbitrary). The resulting integrable model is similar to a two channelanisotropic Kondo model, but with the impurity spin in a "cyclicrepresentation" of the quantum algebra $su(2)_q$ associated with theanisotropy. Using exact, non-perturbative techniques we study the RG flow, andcompute the DC conductance. As expected from the analysis of Kane and Fisher wefind that the IR fixed point corresponds to two separate leads. We also provean exact duality between the UV and IR expansions of the current at vanishingtemperature.Comment: Revtex, epsf, 14pgs, 4 figs. One reference adde

Tunneling in quantum wires: Exact solution of the spin isotropic case