Fermions on a Riemann surface and the Kadomtsev–Petviashvili equation

It is shown that the scattering matrix for free massless fermions on a Riemann surface of finite genus generates the quasiperiodic solutions of the Kadomtsev–Petviashvili equation. The operator changing the genus of the solution is constructed and the composition law of such operators is discussed. The construction extends the well-known operator approach in the case of soliton solutions to the general case of the quasiperiodic $\tau$-functions.