"Sato theory"의 두 판 사이의 차이

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.

related items

Kadometsev-Petviashvii (KP hierarchy)

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
- 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

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

@@ 1번째 줄: / 1번째 줄: @@
-<h5>introduction</h5>
+==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
+* [[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.
-[[KdV equation]]
+==KdV hierarchy==
-<math>K(x,t)=1+e^{2a(x-4a^2t+\delta)}</math>
+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.
-<math>2(\frac{\partial^2}{\partial x^2})\log K(x,t)</math>
-<math>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)}</math>
-<math>2(\frac{\partial^2}{\partial x^2})\log K(x,t)</math>
+==Wronskian determinant==
-<h5>tau funtions</h5>
-Tau functions, built from a determinant bundle over a grassmannian, incorporate many known functions (e.g. theta, Schur) and appear in a variety of subjects (integrable systems, isomonodromic deformations, matrix models, moduli problems). The reason why is a bit of a mystery; the talk gives no coherent reason why, but gives a tour of these various contexts. http://events.iupui.edu/event/?event_id=1415
+==universal Grassmanian manifold==
-http://www.math.mcgill.ca/node/1310
-Speaker:<br> John Harnad, Concordia, CRM<br> Location:<br> Université de Montréal, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
+==relation to Kac-Moody algebras==
-Abstract:
-What do the following have in common?<br> - Irreducible characters of Lie groups (e.g., Schur functions)<br> - Riemann's theta function on the Jacobian of a genus g Riemann surface<br> - Deformation classes of random matrix integrals<br> - Weights on path spaces of partitions, generating "integrable" random processes, random tilings, and growth processes<br> - Generating functions for Gromov-Witten invariants<br> - Generating functions for classical and quantum integrable systems, such as the KP hierarchy<br> (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 an infinite dimensional Grassmann manifold, and depend on an infinite sequence ofvariables (t_1, t_2, ...), real or complex, continuous or discrete. They satisfy an infinite set of bilinear differential (or difference) relations, which can be interpreted as the Plucker relations defining the embedding of this "universal" Grassmann manifold into 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 as linear exponential parameters of an infinite abelian group that acts on the Grassmannian and Fock space. A basic tool, which is part and parcel of the Plucker embedding, is the use of fermionic "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 is familiar to all physicists, but little used by mathematicians, except for those, like Kontsevich, Witten, Okounkov (or, in earlier times, Cartan, Chevalley, Weyl), who know how to get good use out of it.
-<h5>KdV hierarchy</h5>
-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<br> 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<br> 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<br> computations and made many conjectures that involve eumeration of partitions.
-<h5>Wronskian determinant</h5>
-<h5>universal Grassmanian manifold</h5>
-<h5>relation to Kac-Moody algebras</h5>
 * 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
@@ 76번째 줄: / 40번째 줄: @@
 * Frenkel–Kac had already used free fermions to construct basic representations. In this approach, the tau-functions are defined as vacuum expectation values.
-<h5>role in conformal field theory</h5>
+==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.
-<h5>history</h5>
+==related items==
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
-<h5>related items</h5>
 * [[Kadometsev-Petviashvii equation (KP equation)|Kadometsev-Petviashvii (KP hierarchy)]]
+==books==
-<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
-* http://en.wikipedia.org/wiki/
-* http://www.scholarpedia.org/
-* [http://eom.springer.de/ http://eom.springer.de]
-* http://www.proofwiki.org/wiki/
-* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
-<h5>books</h5>
 * 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]
-* [[2011년 books and articles]]
-* http://library.nu/search?q=
-* http://library.nu/search?q=
-<h5>expositions</h5>
+==expositions==
 * [http://yokoemon.web.fc2.com/KANT2010/Notes/Yamazaki.pdf Sato theory, p-adic tau function and arithmetic geometry]
-* Algebraic Geometrical Methods in Hamiltonian Mechanics http://www.jstor.org/stable/37539
+* 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/fea-sato-2.pdf
+** http://www.ams.org/notices/200702/comm-schapira.pdf
-http://www.ams.org/notices/200702/comm-schapira.pdf
 * The KP hierarchy and infinite-dimensional Grassmann manifolds M Sato - Theta functions—Bowdoin, 1987
-<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
+==articles==
+* Letterio Gatto, Parham Salehyan, On Plücker Equations Characterizing Grassmann Cones, http://arxiv.org/abs/1603.00510v1
-* Eilbeck, J C, V Z Enolski, and J Gibbons. 2010. Sigma, tau and Abelian functions of algebraic curves. Journal of Physics A: Mathematical and Theoretical 43, no. 45 (11): 455216. doi:[http://dx.doi.org/10.1088/1751-8113/43/45/455216 10.1088/1751-8113/43/45/455216].
+* 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.
-* Fermionic construction of tau functions and random processesAuthors: John Harnad, Alexander Yu. Orlov http://dx.doi.org/10.1016/j.physd.2007.05.011
+* 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.<br>
+* 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.
-* Matsutani, Shigeki. 2000. Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions. nlin/0007001 (July 1). doi:doi:[http://dx.doi.org/10.1088/0305-4470/34/22/312 10.1088/0305-4470/34/22/312]. http://arxiv.org/abs/nlin/0007001.
-* Nakamura, Yoshimasa. 1994. “A tau-function of the finite nonperiodic Toda lattice”. <em>Physics Letters A</em> 195 (5-6) (12월 12): 346-350. doi:[http://dx.doi.org/10.1016/0375-9601%2894%2990040-X 10.1016/0375-9601(94)90040-X].
 * 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]<br>  <br>
+* 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
-* http://dx.doi.org/10.1016/0375-9601(94)90040-X
-<h5>question and answers(Math Overflow)</h5>
-* http://mathoverflow.net/search?q=
-* http://mathoverflow.net/search?q=
-<h5>blogs</h5>
-*  구글 블로그 검색<br>
-**  http://blogsearch.google.com/blogsearch?q=<br>
-** http://blogsearch.google.com/blogsearch?q=
-* http://ncatlab.org/nlab/show/HomePage
-<h5>experts on the field</h5>
-* http://arxiv.org/
-<h5>links</h5>
-* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
+[[분류:개인노트]]
-* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
+[[분류:integrable systems]]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
+[[분류:math and physics]]
-* http://functions.wolfram.com/
+[[분류:migrate]]