  We demonstrate the equivalence between two distinct Luttinger liquid impurity
problems. The first concerns a one-dimensional topological superconductor
coupled at one end to the ends of two single channel Luttinger liquids. The
second concerns a point contact in the quantum spin Hall effect, where four
helical Luttinger liquids meet at a point. Both problems have been studied
previously and exhibit several stable phases depending on the Luttinger
parameter K, that can be characterized in terms of simple conformally invariant
boundary conditions describing perfect normal (or Andreev) transmission or
reflection. In addition, both problems exhibit critical points that are
described by "intermediate" fixed points similar to those found in earlier
studies of an impurity in a Luttinger liquid with spin. Though these two models
have different symmetries and numbers of modes, we show they are equivalent and
are related by a duality transformation, and we show that the non-trivial
intermediate critical points are the same. In the non-interacting limit, K=1,
the duality involves two distinct free fermion representations that are related
by a non-local transformation that derives from the triality of SO(8). Using
the explicit translation between the two theories, we translate results from
one problem to the other and vice versa. This allows us to make new predictions
about the topological superconductor-Luttinger liquid junction, including
predictions about the global behavior of the critical conductance G*(K), as
well predictions for the critical exponents and universal crossover scaling
functions. In this paper we introduce both models from scratch, using a common
notation that facilitates their comparison, and we discuss in detail the
dualities that relate them, along with their free fermion limits. We close with
a discussion of open problems and future directions.
