Symmetry, Duality and Anholonomy of Point Interactions in One Dimension

We analyze the spectral structure of the one dimensional quantum mechanical system with point interaction, which is known to be parametrized by the group U(2). Based on the classification of the interactions in terms of symmetries, we show, on a general ground, how the fermion-boson duality and the spectral anholonomy recently discovered can arise. A vital role is played by a hidden su(2) formed by a certain set of discrete transformations, which becomes a symmetry if the point interaction belongs to a distinguished U(1) subfamily in which all states are doubly degenerate. Within the U(1), there is a particular interaction which admits the interpretation of the system as a supersymmetric Witten model.