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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 functions|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. | ||
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| − | + | ==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. | |
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| − | + | ==Wronskian determinant== | |
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| − | + | ==universal Grassmanian manifold== | |
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| − | + | ==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. |
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| − | + | ==role in conformal field theory== | |
| − | * | + | * Kawamoto, Noboru, Yukihiko Namikawa, Akihiro Tsuchiya, 와/과Yasuhiko Yamada. 1988. “Geometric realization of conformal field theory on Riemann surfaces”. <em>Communications in Mathematical Physics</em> 116 (2): 247-308. doi:10.1007/BF01225258. |
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| − | + | ==related items== | |
| − | + | * [[Kadometsev-Petviashvii equation (KP equation)|Kadometsev-Petviashvii (KP hierarchy)]] | |
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| − | + | ==books== | |
| − | * | + | * Discrete Integrable Systems http://dx.doi.org/10.1007/b94662 |
| − | + | * Book review on [http://www.math.ntnu.no/%7Eholden/solitons/BullAMS_book.pdf Soliton equations and their algebro-geometric solutions. Vol. I. (1+1)-dimensional continuous models] | |
| − | * http://www. | ||
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| − | + | ==expositions== | |
| − | * http:// | + | * [http://yokoemon.web.fc2.com/KANT2010/Notes/Yamazaki.pdf Sato theory, p-adic tau function and arithmetic geometry] |
| − | * http:// | + | * Algebraic Geometrical Methods in Hamiltonian Mechanics [http://www.jstor.org/stable/37539 ]http://www.jstor.org/stable/37539 |
| + | * [http://www.math.ucdavis.edu/%7Emulase/texfiles/algebraictheo.pdf 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:[http://dx.doi.org/10.1007/BF02698802 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 | ||
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| − | < | + | ==articles== |
| + | * Letterio Gatto, Parham Salehyan, On Plücker Equations Characterizing Grassmann Cones, http://arxiv.org/abs/1603.00510v1 | ||
| + | * Luu, Martin, and Matej Penciak. “Langlands Parameters of Symmetric Unitary Matrix Models.” arXiv:1511.07466 [math-Ph, Physics:nlin], November 23, 2015. http://arxiv.org/abs/1511.07466. | ||
| + | * Harnad, J., and A. Yu. Orlov. “Fermionic Construction of Tau Functions and Random Processes.” Physica D: Nonlinear Phenomena, Physics and Mathematics of Growing Interfaces In honor of Stan Richardson’s discoveries in Laplacian Growth and related free boundary problem, 235, no. 1–2 (November 2007): 168–206. doi:10.1016/j.physd.2007.05.011. 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:[http://dx.doi.org/10.1093/imrn/rnm140 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”. <em>Journal of Physics A: Mathematical and Theoretical</em> 40 (42): 12661-12675. doi:[http://dx.doi.org/10.1088/1751-8113/40/42/S11 10.1088/1751-8113/40/42/S11]. | ||
| + | * Borodin, Alexei, and Percy Deift. 2002. “'''Fredholm determinants, Jimbo‐Miwa‐Ueno τ‐functions, and representation theory'''.” <em>Communications on Pure and Applied Mathematics</em> 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”. <em>Inverse Problems</em> 5 (4): 613-630. doi:[http://dx.doi.org/10.1088/0266-5611/5/4/012 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”. <em>Physics Letters A</em> 95 (1) (4월 11): 1-3. doi:[http://dx.doi.org/10.1016/0375-9601%2883%2990764-8 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 | ||
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2020년 12월 28일 (월) 05:07 기준 최신판
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.
Wronskian determinant
universal Grassmanian manifold
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.
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
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
- The KP hierarchy and infinite-dimensional Grassmann manifolds M Sato - Theta functions—Bowdoin, 1987
articles
- Letterio Gatto, Parham Salehyan, On Plücker Equations Characterizing Grassmann Cones, http://arxiv.org/abs/1603.00510v1
- Luu, Martin, and Matej Penciak. “Langlands Parameters of Symmetric Unitary Matrix Models.” arXiv:1511.07466 [math-Ph, Physics:nlin], November 23, 2015. http://arxiv.org/abs/1511.07466.
- Harnad, J., and A. Yu. Orlov. “Fermionic Construction of Tau Functions and Random Processes.” Physica D: Nonlinear Phenomena, Physics and Mathematics of Growing Interfaces In honor of Stan Richardson’s discoveries in Laplacian Growth and related free boundary problem, 235, no. 1–2 (November 2007): 168–206. doi:10.1016/j.physd.2007.05.011. 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