"Sato theory"의 두 판 사이의 차이
18번째 줄: | 18번째 줄: | ||
<math>2(\frac{\partial^2}{\partial x^2})\log K(x,t)</math> | <math>2(\frac{\partial^2}{\partial x^2})\log K(x,t)</math> | ||
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+ | <h5>tau funtions</h5> | ||
+ | Speaker: John Harnad, Concordia, CRMLocation: Université de Montréal, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214Abstract: | ||
+ | <em>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<br> 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><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 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.</em> | ||
2011년 4월 22일 (금) 07:13 판
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.
\(K(x,t)=1+e^{2a(x-4a^2t+\delta)}\)
\(2(\frac{\partial^2}{\partial x^2})\log K(x,t)\)
\(K(x,t)=1+A_1e^{2a_1(x-4a_1^2t+\delta_1)}+A_2e^{2a_2(x-4a_2^2t+\delta_2)}+A_3e^{2a_1(x-4a_1^2t+\delta_1)+{2a_2(x-4a_2^2t+\delta_2)}\)
\(2(\frac{\partial^2}{\partial x^2})\log K(x,t)\)
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
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 http://www.jstor.org/stable/37539
- 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.
articles
- 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:10.1088/1751-8113/43/45/455216.
- 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.
- 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: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”. Physics Letters A 195 (5-6) (12월 12): 346-350. doi:10.1016/0375-9601(94)90040-X.
- 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
- http://dx.doi.org/10.1016/0375-9601(94)90040-X
question and answers(Math Overflow)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field