Kummer variety, geometry ofN-representability, and phase transitions

For a system of N indistinguishable fermions or bosons, we give a parametrization of the surface of a convex set of N-representable second-order density matrices, a polynomial equation for the exact energy levels, and an explanation of quantum phase transitions. We display the crucial role played by the Kummer variety as a graph of the 2-matrices for pure discrete states of an N-body system in quantum mechanics.