Quantum mechanics not on manifold
  The free scalar field is studied on the Y-junction of three semi infinite axes which is the simplest example of a non-manifold space. It is shown that under an assumption that the junction point can not gain a macroscopic amount of energy and charge the transmission rules for this system uniquely follow from conservation of energy and charge. This result is also obtained in the discrete version of the model. Some alternative approaches to the problem based on quantum mechanics of Hamiltonian systems with constrains are discussed. 
P. N. Bibikov and L. V. Prokhorov