Sato theory

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 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.

 

 

 

 

 

 

 

 

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.

 

 

 

role in conformal field theory

• Kawamoto, Noboru, Yukihiko Namikawa, Akihiro Tsuchiya, 와/과Yasuhiko Yamada. 1988. “Geometric realization of conformal field theory on Riemann surfaces”. Communications in Mathematical Physics 116 (2): 247-308. doi:10.1007/BF01225258.

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• Kadometsev-Petviashvii (KP hierarchy)

 

 

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://eom.springer.de
• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

• Discrete Integrable Systems http://dx.doi.org/10.1007/b94662
• Book review on Soliton equations and their algebro-geometric solutions. Vol. I. (1+1)-dimensional continuous models
• 2011년 books and articles
• http://library.nu/search?q=
• http://library.nu/search?q=

 

 

expositions

• Sato theory, p-adic tau function and arithmetic geometry
• Algebraic Geometrical Methods in Hamiltonian Mechanics [1]http://www.jstor.org/stable/37539
• Algebraic theory of the KP equations, M Mulase - Perspectives in mathematical physics, 1994
• Segal, Graeme, and George Wilson. 1985. Loop groups and equations of KdV type. Publications Mathématiques de L’Institut des Hautes Scientifiques 61, no. 1 (12): 5-65. doi:10.1007/BF02698802.
• Sato interview
  
  • http://www.ams.org/notices/200702/fea-sato-2.pdf
  • http://www.ams.org/notices/200702/comm-schapira.pdf
• The KP hierarchy and infinite-dimensional Grassmann manifolds M Sato - Theta functions—Bowdoin, 1987

 

 

articles

• Fermionic construction of tau functions and random processes Authors: John Harnad, Alexander Yu. Orlov http://dx.doi.org/10.1016/j.physd.2007.05.011
• Eilbeck, J. C., V. Z. Enolski, S. Matsutani, Y. Onishi, and E. Previato. 2010. Abelian Functions for Trigonal Curves of Genus Three. International Mathematics Research Notices (7). doi:10.1093/imrn/rnm140. http://imrn.oxfordjournals.org/content/2007/rnm140.short. 
• Kajiwara, Kenji, Marta Mazzocco, 와/과Yasuhiro Ohta. 2007. “A remark on the Hankel determinant formula for solutions of the Toda equation”. Journal of Physics A: Mathematical and Theoretical 40 (42): 12661-12675. doi:10.1088/1751-8113/40/42/S11.
• Borodin, Alexei, and Percy Deift. 2002. “Fredholm determinants, Jimbo‐Miwa‐Ueno τ‐functions, and representation theory.” Communications on Pure and Applied Mathematics 55 (9) (September 1): 1160-1230. doi:10.1002/cpa.10042.
• Poppe, C. 1989. “General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy”. Inverse Problems 5 (4): 613-630. doi:10.1088/0266-5611/5/4/012.
• Freeman, N. C., 와/과J. J. C. Nimmo. 1983. “Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique”. Physics Letters A 95 (1) (4월 11): 1-3. doi:10.1016/0375-9601(83)90764-8
• M. Sato and Y. Sato, Soliton equations as dynamical systems on infi- nite dimensional Grassmann manifold, in Nonlinear Partial Differential. Equations in Applied Science
• http://dx.doi.org/10.1016/0375-9601(94)90040-X

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/