Assume a tau-function of the KP integrable hierarchy is fixed by the point of the Sato grassmannian (that is by a semi-infinite set of Laurant series $\varphi\_k(z)=\sum\_{m>0}a\_{km}z^{-k+m}$). Can one restore a spectral curve that corresponds to this solution?



From Sato grassmannian to spectral curve
This is explained in Segal-Wilson. Essentially, realize your point in the Grassmannian as as a space W of functions w(z), then look for all functions g(z) such that g(z)W is included in W. These functions form a commutative algebra, and Spec of this algebra is your spectral curve. Of course for most points of the Grassmannian the commutative algebra is just C.


But this is all beautifully explained in Segal-Wilson.


