  We analyze a model of resonant point-contact tunneling between multiple
Luttinger liquid leads. The model is a variant of the multi-channel Kondo model
and can be related to the quantum Brownian motion of a particle on lattices
with $\pi$-flux through each plaquette (in the 3-lead case, it is a honeycomb
lattice with $\pi$-flux). By comparing the perturbative and instanton gas
expansions, we find a duality property of the model. At the boundary, this
duality exchanges Neumann and Dirichlet boundary conditions on the
Tomonaga-Luttinger bosons which describe the leads; in the bulk, it exchanges
the `momentum' and `winding' modes of these bosons. Over a certain range of
Luttinger liquid parameter, $g$, a novel non-trivial intermediate coupling
fixed point controls the low-energy physics. The finite conductance at this
fixed point can be exactly computed for two special values of $g$. For larger
values of $g$, there is a stable fixed point at strong coupling which has
enhanced conductance resulting from an analogue of Andreev reflection at the
point contact.
