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.
\(K(x,t)=1+e^{2a(x-4a^2t+\delta)}\)
\(2(\frac{\partial^2}{\partial x^2})\log K(x,t)\)
\(K(x,t)=1+A_1e^{2a_1(x-4a_1^2t+\delta_1)}+A_2e^{2a_2(x-4a_2^2t+\delta_2)}+A_3e^{2a_1(x-4a_1^2t+\delta_1)+{2a_2(x-4a_2^2t+\delta_2)}\)
\(2(\frac{\partial^2}{\partial x^2})\log K(x,t)\)
tau funtions
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
http://www.math.mcgill.ca/node/1310
Speaker:
John Harnad, Concordia, CRM
Location:
Université de Montréal, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214
Abstract:
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 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.
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.
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 [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.
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
- 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.
- Fermionic construction of tau functions and random processesAuthors: 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.
- 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
- 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)
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