  We study the dissipative Hofstadter model on a triangular lattice, making use
of the $O(2,2;R)$ T-dual transformation of string theory. The $O(2,2;R)$ dual
transformation transcribes the model in a commutative basis into the model in a
non-commutative basis. In the zero temperature limit, the model exhibits an
exact duality, which identifies equivalent points on the two dimensional
parameter space of the model. The exact duality also defines magic circles on
the parameter space, where the model can be mapped onto the boundary
sine-Gordon on a triangular lattice. The model describes the junction of three
quantum wires in a uniform magnetic field background. An explicit expression of
the equivalence condition, which identifies the points on the two dimensional
parameter space of the model by the exact duality, is obtained. It may help us
to understand the structure of the phase diagram of the model.
