Sato theory

수학노트
http://bomber0.myid.net/ (토론)님의 2011년 4월 22일 (금) 07:13 판
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introduction
  • Sato’s Grassmannian and its determinant bundle became a “universal” setting where moduli spaces of curves (or maps or bundles) of arbitrary genus could be mapped and made to interact
  • tau function =  the section of a determinant line bundle over an infinite-dimensional Grassmannian
  • Sato found that character polynomials (Schur functions) solve the KP hierarchy and, based on this observation, he created the theory of the infinite-dimensional (universal) Grassmann manifold
    and showed that the Hirota bilinear equations are nothing but the Plucker relations for this Grassmann manifold.

 

 

KdV equation

\(K(x,t)=1+e^{2a(x-4a^2t+\delta)}\)

\(2(\frac{\partial^2}{\partial x^2})\log K(x,t)\)

\(K(x,t)=1+A_1e^{2a_1(x-4a_1^2t+\delta_1)}+A_2e^{2a_2(x-4a_2^2t+\delta_2)}+A_3e^{2a_1(x-4a_1^2t+\delta_1)+{2a_2(x-4a_2^2t+\delta_2)}\)

\(2(\frac{\partial^2}{\partial x^2})\log K(x,t)\)

 

tau funtions

Speaker: John Harnad, Concordia, CRMLocation: Université de Montréal, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214Abstract:  What do the following have in common?
- Irreducible characters of Lie groups (e.g., Schur functions)
- Riemann's theta function on the Jacobian of a genus g Riemann surface
- Deformation classes of random matrix integrals
- Weights on path spaces of partitions, generating "integrable" random processes
random tilings, and growth processes
- Generating functions for Gromov-Witten invariants
- Generating functions for classical and quantum integrable systems, such as the KP hierarchy

(What have we left out? L-functions? Take their Mellin transforms.) In this talk, I will show how all the above may be seen as special cases of one common object:the "Tau function". This is a family of functions introduced by Sato, Hirota and others,originally in the context of integrable systems. They are parametrized by the points of aninfinite dimensional Grassmann manifold, and depend on an infinite sequence ofvariables (t_1, t_2, ...), real or complex, continuous or discrete. They satisfy aninfinite set of bilinear differential (or difference) relations, which can be interpretedas the Plucker relations defining the embedding of this "universal" Grassmann manifoldinto an exterior product space (called the "Fermi Fock space" by physicists) as a projective variety. This involves the "Bose-Fermi equivalence", which follows from interpreting the t-variables aslinear exponential parameters of an infinite abelian group that acts on the Grassmannian andFock space. A basic tool, which is part and parcel of the Plucker embedding, is the use offermionic "creation" and "annihilation" operators. The tau function is obtained as a"vacuum state matrix element" along orbits of the abelian group. This is language that isfamiliar to all physicists, but little used by mathematicians, except for those, likeKontsevich, Witten, Okounkov (or, in earlier times, Cartan, Chevalley, Weyl), who knowhow to get good use out of it.

 

 

KdV hierarchy

The totality of soliton equations
organized in this way is called a hierarchy of soliton
equations; in the KdV case, it is called the KdV
hierarchy. This notion of hierarchy was introduced by
M Sato. He tried to understand the nature of the
bilinear method of Hirota. First, he counted the
number of Hirota bilinear operators of given degree
for hierarchies of soliton equations. For the number of
bilinear equations,M Sato and Y Sato made extensive
computations and made many conjectures that involve
eumeration of partitions.

 

 

Wronskian determinant

 

 

relation to Kac-Moody algebras
  • the totality of tau-functions of the KdV hierarchy is the group orbit of the highest weight vector (=1) of the basic representation of A_1^1
  • applications of vertex operators are precisely Ba¨cklund transformations
  • This implies that the affine Lie algebra A(1) 1 is the infinitesimal transformation group for solutions of the KdV hierarchy.
  • Frenkel–Kac had already used free fermions to construct basic representations. In this approach, the tau-functions are defined as vacuum expectation values.

 

 

history

 

 

related items

 

 

encyclopedia

 

 

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question and answers(Math Overflow)

 

 

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links
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