  We develop a systematic approach to bosonization and vertex algebras on
quantum wires of the form of star graphs. The related bosonic fields propagate
freely in the bulk of the graph, but interact at its vertex. Our framework
covers all possible interactions preserving unitarity. Special attention is
devoted to the scale invariant interactions, which determine the critical
properties of the system. Using the associated scattering matrices, we give a
complete classification of the critical points on a star graph with any number
of edges. Critical points where the system is not invariant under wire
permutations are discovered. By means of an appropriate vertex algebra we
perform the bosonization of fermions and solve the massless Thirring model. In
this context we derive an explicit expression for the conductance and
investigate its behavior at the critical points. A simple relation between the
conductance and the Casimir energy density is pointed out.
