## From Sato grassmannian to spectral curve

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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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## 1 Answer

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.

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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 2012 at 22:38
I am not sure what your notation means. Is \phi_k a basis for the point of the Grassmannian? In any case, for a generic point of the Grassmanian the spectral curve will be trivial. – Maarten Bergvelt Jan 14 2012 at 22:44

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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 2012 at 0:04

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@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 2012 at 16:04

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The Kontsevich-Witten tau funtion is in fact a solution of the Kortewg-de Vries hierarchy, not just of KP. This means that the corresponding point of the Grassmannian comes from an element Of the loopgroup of Sl_2, not from an element of the general linear group. Such points always have z^2 as stabilizer, so for trivial reasons we get a P^1 as spectral curve. The "honest" spectral curves for KdV are 2-fold covers of this base curve. See, as always, Segal-Wilson. – Maarten Bergvelt Jan 16 2012 at 2:47
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