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General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy

The tau function, introduced by the 'Kyoto School' as a central element in the description of soliton equation hierarchies, is identified with the determinant of a family of linear operators solving linear, constant-coefficient PDE in the hierarchy variables, for the Kadomtsev-Petviashvili (KP) hierarchy. For Gel'fand-Levitan-Marchenko integral operators, the tau function is the Fredholm determinant; the author gives a new proof of this fact. He shows further that the determinant of a suitably chosen family of finite-dimensional matrices is a tau function which gives rise to rational solutions of the KP equation. Finally, he proves the 'vertex operator identity' for tau functions in the Fredholm determinant case.