# From Sato grassmannian to spectral curve

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?

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Maarten, thank you for your ansver. I think I have to re-read Segal-Wilson. Just to be sure: assume I know the coefficients $a_{km}$ explicitely. Is there a step-by-step procedure to construct a spectral curve? – Sasha Jan 14 '12 at 22:38
Well, look for all g such that $g\phi_k$ is a linear combination of the basis elements. You get a bunch of equations for the coefficients of g. Yes, it is an interesting exercise to see if the Kontsevich-Witten point has a spctral curve. I do not know the answer, but you would expect that the curve would be trivial, no? It would be very strange if the genus of this curve would be any finite number. I am not sure if you can get infinite genus curves in this way. – Maarten Bergvelt Jan 15 '12 at 0:04
@Sasha, you and David may have a different definition of what it means for a solution to be "described" by a spectral curve. Krichever's definition (explained e.g. in Segal-Wilson) is that the stabilizer of $W$ (the algebra of all $g$) is sufficiently big, for example $W$ is a rank 1 module over it. All string solutions to KdV have the minimal possible stabilizer $\mathbf{C}[z^2]$ (and so $W$ has rank 2). – Pavel Safronov Jan 15 '12 at 16:04