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A. Yu. Orlov, T. Shiota, K. Takasaki
Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions
arXiv:1201.4518v1 [math-ph]
Abstract
We introduce a useful and rather simple class of BKP tau functions which which we shall call "easy tau functions". We consider two versions of BKP hierarchy, one we will call "small BKP hierarchy" (sBKP) related to $O(\infty)$ introduced in Date et al and "large BKP hierarchy" (lBKP) related to $O(2\infty +1)$ introduced in Kac and van de Leur (which is closely related to the large $O(2\infty)$ DKP hierarchy (lDKP) introduced in Jimbo and Miwa). Actually "easy tau functions" of the sBKP hierarchy were already considered in Harnad et al, here we are more interested in the lBKP case and also the mixed small-large BKP tau functions (Kac and van de Leur). Tau functions under consideration are equal to certain sums over partitions and to certain multi-integrals over cone domains. In this way they may be applicable in models of random partitions and models of random matrices. Here is the first part of the paper where sums of Schur and projective Schur functions over partitions are considered.
Kanehisa Takasaki and Takashi Takebe
An hbar-expansion of the Toda hierarchy: a recursive construction of solutions
Analysis and Mathematical Physics 2 (2012), 171-214.
arXiv:1112.0601v1 [math-ph]
Comments: 37 pages, no figures. arXiv admin note: substantial text overlap with arXiv:0912.4867
MSC classes: 37K10, 35Q53
Abstract
A construction of general solutions of the \hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and \bar W. The dressing operators are set in an exponential form as W = e^{X/\hbar} and \bar W = e^{\phi/\hbar}e^{\bar X/\hbar}, and the auxiliary operators X,\bar X and the function \phi are assumed to have \hbar-expansions X = X_0 + \hbar X_1 + ..., \bar X = \bar X_0 + \hbar\bar X_1 + ... and \phi = \phi_0 + \hbar\phi_1 + .... The coefficients of these expansions turn out to satisfy a set of recursion relations. X,\bar X and \phi are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form \Psi = e^{S/\hbar} and \bar\Psi = e^{\bar S/\hbar}, which leads to an \hbar-expansion of the logarithm of the tau function.
Kanehisa Takasaki and Toshio Nakatsu
Thermodynamic limit of random partitions and dispersionless Toda hierarchy
J. Phys. A: Math. Theor. 45 (2012), 025403 (38pp)
arXiv:1110.0657 [math-ph]
Comments: latex2e using amsmath,amssymb,amsthm packages, 55 pages, no figure
MSC-class: 35Q58, 81T13, 82B20
Abstract
We study the thermodynamic limit of random partition models for the instanton
sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical
observables. The physical observables correspond to external potentials in the
statistical model. The partition function is reformulated in terms of the
density function of Maya diagrams. The thermodynamic limit is governed by a
limit shape of Young diagrams associated with dominant terms in the partition
function. The limit shape is characterized by a variational problem, which is
further converted to a scalar-valued Riemann-Hilbert problem. This
Riemann-Hilbert problem is solved with the aid of a complex curve, which may be
thought of as the Seiberg-Witten curve of the deformed U(1) gauge theory. This
solution of the Riemann-Hilbert problem is identified with a special solution
of the dispersionless Toda hierarchy that satisfies a pair of generalized
string equations. The generalized string equations for the 5D gauge theory are
shown to be related to hidden symmetries of the statistical model. The
prepotential and the Seiberg-Witten differential are also considered.
Kanehisa Takasaki
Non-degenerate solutions of universal Whitham hierarchy
invited talk at "7th International Conference on
Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory"
(University of Georgia, Athens, April 5, 2011)
[slide]
Abstract
I will review recent results obtained in joint work with
T. Takebe and L.-P. Teo. In this work,
the notion of ``non-degenerate solutions''
for the dispersionless Toda hierarchy,
is generalied to the universal Whitham hierarchy (of genus zero).
These solutions are characterized by
a nonlinear Riemann--Hilbert problem. One can see
from this characterization that these solutions
are a kind of ``general'' solutions
of these dispersionless integrable hierarchies.
The Riemann--Hilbert problem is translated
to the language of a space of conformal mappings,
and solved by inversion of an infinite dimensional
period map on this space.
I will also argue that a possible dispersive analogue
of these results can be found in a system of multiple
bi-orthogonal polynomials. These multiple
bi-orthogonal polynomials are a generalizations
of bi-orthogonal polynomials studied by Adler and van~Moerbeke
in the context of the Toda hierarchy.
The latter give a special solution of the Toda hierarchy.
This solution may be thought of as a dispersive counterpart
of non-degenerate solution of the dispersionless Toda hierarchy.
I conjecture that a similar interpretation holds
for the multiple bi-orthogonal polynomials
in the framework of our previous work.
Kanehisa Takasaki
Toda tau functions with quantum torus symmetries
colloquium talk at Department of Mathematics,
University of California, Davis (Davis, March 30, 2011)
[slide]
Abstract
The quantum torus algebra plays an important role
in a special class of solutions of the Toda hierarchy.
Typical examples are the solutions related to
the melting crystal model of topological strings
and 5D SUSY gauge theories. The quantum torus algebra
is realized by a 2D complex free fermion system
that underlies the Toda hierarchy, and exhibits mysterious
``shift symmetries''. This talk is based on
collaboration with Toshio Nakatsu.
Kanehisa Takasaki and Takashi Takebe
hbar-Dependent KP hierarchy
Theoretical and Mathematical Physics (to appear)
arXiv:1105.0794v1 [math-ph]
Comments: 12 pages, contribution to the Proceedings of
the "International Workshop on Classical and Quantum Integrable Systems 2011"
(January 24-27, 2011 Protvino, Russia)
Abstract
This is a summary of a recursive construction of solutions of the hbar-dependent KP hierarchy. We give recursion relations for the coefficients X_n of an hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 + ... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). The asymptotic behaviours of (the logarithm of) the wave function and the tau function are also considered.
Kanehisa Takasaki
Toda tau functions with quantum torus symmetries
Acta Polytechnica 51, No.1 (2011), 74-76.
arXiv:1101.4083 [math-ph]
Comments: latex2e using packages amsmath,amssymb,amsthm, 6 pages, no figure,
contribution to "19th International Colloquium on Integrable Systems and Quantum Symmetries"
Abstract
The quantum torus algebra plays an important role in a special class of
solutions of the Toda hierarchy. Typical examples are the solutions related to
the melting crystal model of topological strings and 5D SUSY gauge theories.
The quantum torus algebra is realized by a 2D complex free fermion system that
underlies the Toda hierarchy, and exhibits mysterious "shift symmetries". This
article is based on collaboration with Toshio Nakatsu.
Kanehisa Takasaki
Generalized string equations for double Hurwitz numbers
Journal of Geometry and Physics 62 (2012), 1135--1156
arXiv:1012.5554 [math-ph]
Comments: latex2e using amsmath,amssymb,amsthm, 41 pages, no figure
MSC-class: 35Q58, 14N10, 81R12
Abstract
The generating function of double Hurwitz numbers is known to become a tau
function of the Toda hierarchy. The associated Lax and Orlov-Schulman operators
turn out to satisfy a set of generalized string equations. These generalized
string equations resemble those of $c = 1$ string theory except that the
Orlov-Schulman operators are contained therein in an exponentiated form. These
equations are derived from a set of intertwining relations for fermiom
bilinears in a two-dimensional free fermion system. The intertwiner is
constructed from a fermionic counterpart of the cut-and-join operator. A
classical limit of these generalized string equations is also obtained. The so
called Lambert curve emerges in a specialization of its solution. This seems to
be another way to derive the spectral curve of the random matrix approach to
Hurwitz numbers.
Kanehisa Takasaki
Generalized string equations for Hurwitz numbers
(研究集会「可積分系，ランダム行列，代数幾何と幾何学的不変量」
2010年12月15日〜17日京都大学人間環境学研究科)
[講演スライド]
Abstract
Generating functions of almost-simple and double Hurwitz numbers
for the Riemann sphere are known to become special tau functions
of the KP and Toda hierarchies. This talk presents recent results
on the generating function of duble Hurwitz numbers.
Generalized string equations for the Lax and Orlov--Schulman operators
are derived. These equations turn out to have a meaningful classical limit
in the dispersionless Toda hierarchy. Solving these equations reveals
a relation with Lambert's W-function. This seems to show another
approach to various roles of the W-function in Hurwitz numbers.
高崎金久
フルヴィッツ数に関連する戸田階層の特殊解とその古典極限
(日本数学会2010年秋季総合分科会，2010年9月22日〜25日名古屋大学)
[講演スライド]
要旨
この講演ではフルビッツ数の母函数とその q 類似（ランダム行列の
分配函数に相当する）を特殊値として与えるような戸田階層のτ函数を考察し，
ラックス作用素とオルロフ--シュルマン作用素に対して「一般化弦方程式」を導く．
さらにこの一般化弦方程式の古典極限（無分散極限）の解から
Eynard と Orantin のランダム行列的方法で得られる「スペクトル曲線」
と同じ形の曲線の方程式が現れることを指摘する．
K. Takasaki
Toda tau function with quantum torus symmetries
19th International Colloquium ``Integrable Systems
and Quantum Symmetries'' (Czech Technical University, Prague,
June 17--19, 2010) 講演スライド
[tex|pdf]
Abstract
The quantum torus algebra plays an important role
in a spacial class of solutions of the Toda hierarchy.
Typical examples are the solutions related to
the melting crystal model of 5D SUSY gauge theories,
topological strings on toric Calabi-Yau three folds,
the Hurwitz numbers of the Riemann sphere and
their $q$-analogues. The quantum torus algebra
is realized by a 2D complex fermion system that underlies
the Toda hierarchy, and exhibits a mysterious
``shift symmetry'' that leads to generalized
``string equations''. This talk is partly based on
collaboration with Toshio Nakatsu.
Kanehisa Takasaki, Takashi Takebe and Lee Peng Teo
Non-degenerate solutions of universal Whitham hierarchy
J. Phys. A: Math. Theor. 43 (2010), 325205
arXiv:1003.5767 [math-ph]
Comments: latex2e, using amsmath, amssym and amsthm packages,
32 pages, no figure
Abstract
The notion of non-degenerate solutions for the dispersionless Toda
hierarchy is generalized to the universal Whitham hierarchy of genus
zero with $M+1$ marked points. These solutions are characterized by a
Riemann-Hilbert problem (generalized string equations) with respect to
two-dimensional canonical transformations, and may be thought of as a
kind of general solutions of the hierarchy. The Riemann-Hilbert problem
contains $M$ arbitrary functions $H_a(z_0,z_a)$, $a = 1,\ldots,M$, which
play the role of generating functions of two-dimensional canonical
transformations. The solution of the Riemann-Hilbert problem is
described by period maps on the space of $(M+1)$-tuples $(z_\alpha(p) :
\alpha = 0,1,\ldots,M)$ of conformal maps from $M$ disks of the Riemann
sphere and their complements to the Riemann sphere. The period maps are
defined by an infinite number of contour integrals that generalize the
notion of harmonic moments. The $F$-function (free energy) of these
solutions is also shown to have a contour integral representation.
高崎金久
溶解結晶模型の可積分構造
(日本数学会2010年年会，2010年3月24日〜27日慶應義塾大学矢上キャンパス)
[講演予稿 |
講演スライド]
要旨
溶解結晶模型は５次元超対称ゲージ理論やトーリックCalabi-Yau多様体上の
位相的弦理論と関係する統計力学的模型である，その分配函数に
外部ポテンシャルを導入したものは結合定数に関して１次元戸田階層の
特殊解のτ函数となる．さらに，結晶に一種の非対称性を導入した模型にも
同様の可積分構造（２次元戸田階層の簡約）がある．これらの結果を
基礎の部分から解説する．
Kanehisa Takasaki
KP and Toda tau functions in Bethe ansatz
arXiv:1003.307 [math-ph]
Comments: latex2e, using ws-procs9x6 package, 19 pages, contribution to
the festschrift volume for the 60th anniversary of Tetsuji Miwa
Abstract
Recent work of Foda and his group on a connection between
classical integrable hierarchies (the KP and 2D Toda hierarchies)
and some quantum integrable systems (the 6-vertex model with DWBC,
the finite XXZ chain of spin 1/2, the phase model on a finite chain, etc.)
is reviewed. Some additional information on this issue is also presented.
Kanehisa Takasaki
Two extensions of 1D Toda hierarchy
J. Phys. A: Math. Theor. 43 (2010) 434032
arXiv:1002.4688 [nlin.SI]
Comments: latex2e, usepackage amsmath,amssymb, 19 pages, no figure
Abstract
The extended Toda hierarchy of Carlet, Dubrovin and Zhang is reconsidered
in the light of a 2+1D extension of the 1D Toda hierarchy constructed
by Ogawa. These two extensions of the 1D Toda hierarchy turn out to have
a very similar structure, and the former may be thought of as a kind of
dimensional reduction of the latter. In particular, this explains an origin
of the mysterious structure of the bilinear formalism proposed by Milanov.
Kanehisa Takasaki and Takashi Takebe
hbar-expansion of KP hierarchy: Recursive construction of solutions
arXiv:0912.4867 [math-ph]
Comments: 28 pages
Abstract
The \hbar-dependent KP hierarchy is a formulation of the KP hierarchy that
depends on the Planck constant \hbar and reduces to the dispersionless KP
hierarchy as \hbar -> 0. A recursive construction of its solutions on the basis
of a Riemann-Hilbert problem for the pair (L,M) of Lax and Orlov-Schulman
operators is presented. The Riemann-Hilbert problem is converted to a set of
recursion relations for the coefficients X_n of an \hbar-expansion of the
operator X = X_0 + \hbar X_1 + \hbar^2 X_2 +... for which the dressing operator
W is expressed in the exponential form W = \exp(X/\hbar). Given the lowest
order term X_0, one can solve the recursion relations to obtain the higher
order terms. The wave function \Psi associated with W turns out to have the WKB
form \Psi = \exp(S/\hbar), and the coefficients S_n of the \hbar-expansion S =
S_0 + \hbar S_1 + \hbar^2 S_2 +..., too, are determined by a set of recursion
relations. This WKB form is used to show that the associated tau function has
an \hbar-expansion of the form \log\tau = \hbar^{-2}F_0 + \hbar^{-1}F_1 + F_2 +
..
高崎金久
対数的時間発展による非線形Schroedinger階層とAblowitz-Ladik階層の拡張
研究集会「非線形波動の現状と将来」(九州大学応用力学研究所2009年11月19日〜21日)
一般講演スライド
[tex|pdf]
要旨
Carlet, Dubrovin, Zhang は通常の１次元戸田階層に
「対数的」時間発展を加えて「拡張戸田階層」を構成した．
Milanov はこの拡張戸田階層に対して双線形形式を与えた．
ところで，１次元戸田階層は非線形Schr\"odinger（NLS）階層
のB\"acklund変換列（NLS-戸田階層）とも見なせる．
同様の意味でRuijsenaars-戸田階層（RT階層）は
Ablowitz-Ladik（AL）階層と対応していることが知られている．
本講演では，１次元戸田階層の対数的時間発展を
NLS-戸田階層の言葉に翻訳し，
その結果がAL階層に拡張できることを示す．
高崎金久
対数的時間発展による非線形Schroedinger階層とAblowitz-Ladik階層の拡張
日本数学会2009年秋季総合分科会 (2009年9月24日〜27日大阪大学豊中キャンパス)
一般講演スライド
[tex|pdf]
要旨
Carlet, Dubrovin, Zhang は通常の１次元戸田階層に
「対数的時間発展」を加えて「拡張戸田階層」を構成した．
Milanov はこの拡張戸田階層に対して双線形形式を与えた．
ところで，１次元戸田階層は非線形シュレディンガー（NLS）階層
のB\"acklund変換列（NLS-戸田階層）とも見なせる．
同様の意味でRuijsenaars-戸田階層（RT階層）は
Ablowitz-Ladik（AL）階層と対応している
（Kharchev, Mironov Zedhanov; Suris; Sadakane）．
本講演では，１次元戸田階層の対数的時間発展を
NLS-戸田階層の言葉に翻訳し，
その結果がAL階層に拡張できることを示す．
Kanehisa Takasaki
Auxiliary linear problem, difference Fay identities and
dispersionless limit of Pfaff-Toda hierarchy
SIGMA 5 (2009), paper 109, 34 pages
arXiv:0908.3569 [nlin.SI]
Comments: 49 pages, no figure, usepackage amsmath,amssymb,amsthm
Abstract
Recently the study of Fay-type identities revealed some new features
of the DKP hierarchy (also known as ``the coupled KP hierarchy'' and
``the Pfaff lattice''). Those results are now extended to a Toda version
of the DKP hierarchy (tentatively called ``the Pfaff-Toda hierarchy'') .
Firstly, an auxiliary linear problem of this hierarchy is constructed.
Unlike the case of the DKP hierarchy, building blocks of the auxiliary
linear problem are difference operators. A set of evolution equations
for dressing operators of the wave functions are also obtained.
Secondly, a system of Fay-like identities (difference Fay identities)
are derived. They give a generating functional expression of
auxiliary linear equations. Thirdly, these difference Fay identities
have well defined dispersionless limit (dispersionless Hirota equations).
As in the case of the DKP hierarchy, an elliptic curve is hidden
in these dispersionless Hirota equations. This curve is a kind of
spectral curve, whose defining equation is identified with
the characteristic equation of a subset of all auxiliary linear equations.
The other auxiliary linear equations are related to quasi-classical
deformations of this elliptic spectral curve.
Kanehisa Takasaki
KP and Toda tau functions in Bethe ansatz: a review
Workshop ``Infinite Analysis 09 --- New trends in quantum integrable systems''
(RIMS and Department of Mathematics, Kyoto University, July 27--31, 2009),
slides of talk
[tex|pdf]
Abstract
This is a review of recent work by O. Foda et al.
on KP tau functions that emerge in algebraic Bethe ansatz
of the 6-vertex model with DWBC, the spin 1/2 XXZ model, etc.
高崎金久
２個のq-パラメータをもつランダム平面分割の可積分構造
日本数学会2009年年会 (2009年3月26日〜29日東京大学数理科学研究科)
一般講演スライド [pdf]
要旨
溶解結晶模型と呼ばれる平面分割の統計力学的模型
（５次元の超対称ゲージ理論のインスタントン和という
解釈も有する）に特別な形の外部ポテンシャルを導入するとき，
その分配函数は１次元戸田階層のτ函数に簡単な因子を乗じた
ものになる．この講演ではこの結果の一つの一般化を紹介し，
そこにもう一つの可積分構造として $q$ 差分戸田方程式が
内在していることも指摘する．
高崎金久
Pfaff-Toda階層の差分Fay等式
日本数学会2009年年会 (2009年3月26日〜29日東京大学数理科学研究科)
一般講演スライド
[tex|pdf]
要旨
この講演では筧とWilloxによって導入された可積分階層
(「Pfaffian的」あるいは「Pfaffian化された」戸田階層
という意味で「Pfaff戸田階層」と呼ぶことにする) の
補助線形方程式系と「差分Fay等式」について報告する．
無分散極限についても結果が得られているが，
時間の関係で省略する．
Kanehisa Takasaki
Integrable structure of melting crystal model with two q-parameters
arXiv:0903.2607 [math-ph]
J. Geometry and Physics 59 (2009), 1244-1257
Comments: 27 pages, no figure, latex2e(package amsmath,amssymb,amsthm)
Abstract
This paper explores integrable structures of a generalized
melting crystal model that has two $q$-parameters $q_1,q_2$.
This model, like the ordinary one with a single $q$-parameter,
is formulated as a model of random plane partitions (or,
equivalently, random 3D Young diagrams). The Boltzmann weight
contains an infinite number of external potentials that depend
on the shape of the diagonal slice of plane partitions.
The partition function is thereby a function of an infinite number
of coupling constants $t_1,t_2,\ldots$ and an extra one $Q$.
There is a compact expression of this partition function
in the language of a 2D complex free fermion system, from which
one can see the presence of a quantum torus algebra behind
this model. The partition function turns out to be
a tau function (times a simple factor) of two integrable
structures simultaneously. The first integrable structure
is the bigraded Toda hierarchy, which determine the dependence
on $t_1,t_2,\ldots$. This integrable structure emerges
when the $q$-parameters $q_1,q_2$ take special values.
The second integrable structure is a $q$-difference analogue
of the 1D Toda equation. The partition function satisfies
this $q$-difference equation with respect to $Q$. Unlike
the bigraded Toda hierarchy, this integrable structure
exists for any values of $q_1,q_2$.
Kanehisa Takasaki
Dispersionless Hirota equations and reduction
of universal Whitham hierarchy
研究集会 "Laplacian Growth and Related Topics''
(CRM, University of Montreal, August 18--22, 2008)
講演スライド [pdf]
Abstract
Teo, Takebe and Zabrodin applied dispersionless Hirota equations
to the problem of finite-variable reduction of
the dispersionless KP and Toda hierarchies. We extend
their results to the universal Whitham hierarchy of genus zero.
This is a joint work with Takashi Takebe.
Kanehisa Takasaki and Takashi Takebe
Loewner equations, Hirota equations and
reductions of universal Whitham hierarchy
J. Phys. A: Math. Theor. {\bf 41} (2008), 475206 (27pp)
arXiv:0808.1444 [nlin.SI]
Comments: latex 2e, 39 pages, using packages amsmath,amssymb,amsthm
Abstract
This paper reconsiders finite variable reductions of
the universal Whitham hierarchy of genus zero in the perspective
of dispersionless Hirota equations. In the case of one-variable
reduction, dispersionless Hirota equations turn out to be
a powerful tool for understanding the mechanism of reduction.
All relevant equations describing the reduction (L\"owner-type
equations and diagonal hydrodynamic equations) can be thereby
derived and justified in a unified manner. The case of
multi-variable reductions is not so straightforward.
Nevertheless, the reduction procedure can be formulated
in a general form, and justified with the aid of dispersionless
Hirota equations. As an application, previous results of
Guil, Ma\~{n}as and Mart\'{\i}nez Alonso are reconfirmed
in this formulation.
Toshio Nakatsu and Kanehisa Takasaki
Integrable structure of melting crystal model with external potentials
Advanced Studies in Pure Mathematics, vol. 59 (Mathematical Society of Japan, 2010), pp. 201--223.
arXiv:0807.4970 [math-ph]
Comments: 21 pages, 3 figures, using amsmath,amssymb,amsthm,graphicx packages,
contribution to proceedings of RIMS workshop
"New developments in Algebraic Geometry, Integrable Systems and Mirror symmetry"
(January 7--11, 2008)
Abstract
This is a review of the authors' recent results on an integrable structure
of the melting crystal model with external potentials. The partition function
of this model is a sum over all plane partitions (3D Young diagrams).
By the method of transfer matrices, this sum turns into a sum over
fordinary partitions (Young diagrams), which may be thought of as a model
of q -deformed random partitions. This model can be further translated
to the language of a complex fermion system. A fermionic realization
of the quantum torus Lie algebra is shown to underlie therein.
With the aid of hidden symmetry of this Lie algebra, the partition function
of the melting crystal model turns out to coincide, up to a simple factor,
with a tau function of the 1D Toda hierarchy. Some related issues on
4D and 5D supersymmetric Yang-Mills theories, topological strings
and the 2D Toda hierarchy are briefly discussed.
Toshio Nakatsu, Yui Noma and Kanehisa Takasaki
Extended $5d$ Seiberg-Witten Theory and Melting Crystal
arXiv:0807.0746 [hep-th]
Nucl. Phys. B808 (2009), 411--440
Comments: The solution of the Reimann-Hilbert problem presented here is wrong.
A correct solution, along with a correct curve, can be obtained by solving
a Riemann-Hilbert problem for the primitive function of the Phi potential.
Abstract
We study an extension of the Seiberg-Witten theory of $5d$ $\mathcal{N}=1$
supersymmetric Yang-Mills on $\mathbb{R}^4 \times S^1$. We investigate
correlation functions among loop operators. These are the operators analogous
to the Wilson loops encircling the fifth-dimensional circle and give rise to
physical observables of topological-twisted $5d$ $\mathcal{N}=1$ supersymmetric
Yang-Mills in the $\Omega$ background. The correlation functions are computed
by using the localization technique. Generating function of the correlation
functions of U(1) theory is expressed as a statistical sum over partitions and
reproduces the partition function of the melting crystal model with external
potentials. The generating function becomes a $\tau$ function of 1-Toda
hierarchy, where the coupling constants of the loop operators are interpreted
as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition
function of this model is studied. We solve a Riemann-Hilbert problem that
determines the limit shape of the main diagonal slice of random plane
partitions in the presence of external potentials, and identify a relevant
complex curve and the associated Seiberg-Witten differential.
Toshio Nakatsu, Yui Noma and Kanehisa Takasaki
Integrable Structure of $5d$ $\mathcal{N}=1$ Supersymmetric Yang-Mills
and Melting Crystal
arXiv:0806.3675 [hep-th]
Int. J. Mod. Phys. A23 (2008), 2332-2342.
Comments: The solution of the Reimann-Hilbert problem presented here is wrong.
A correct solution, along with a correct curve, can be obtained by solving
a Riemann-Hilbert problem for the primitive function of the Phi potential.
12 pages, 1 figure, based on an invited talk presented at the
international workshop "Progress of String Theory and Quantum Field Theory"
(Osaka City University, December 7-10, 2007), to be published in the
proceedings
Abstract
We study loop operators of $5d$ $\mathcal{N}=1$ SYM in $\Omega$ background.
For the case of U(1) theory, the generating function of correlation functions
of the loop operators reproduces the partition function of melting crystal
model with external potential. We argue the common integrable structure of $5d$
$\mathcal{N}=1$ SYM and melting crystal model.
Kanehisa Takasaki
Integrable structure in melting crystal model of 5D gauge theory
研究集会 "New developments in Algebraic Geometry,
Integrable Systems and Mirror symmetry" (数理解析研究所2008年1月7日〜11日)
講演スライド[pdf]
Abstract
I will report a recent result on integrable structures in a 5D analogue of Nekrasov's instanton partition function for supersymmetric gauge theories. This partition function is formulated as a statistical model of random plane partition known as ``melting crystal''. By adding new potential terms and viewing the coupling constants as fictitious ``time variables'', the partition function turns out to be a tau function of the 1-Toda hierarchy. A fermionic realization of quantum torus Lie algebra plays a central role in identifying this integrable structure. This is a joint work with Toshio Nakatsu.
Kanehisa Takasaki
Differential Fay identities and auxiliary linear problem of
integrable hiearchies
Advanced Studies in Pure Mathematics vol. 61 (Mathematical Society of Japan, 2011), pp. 387--441.
arXiv:0710.5356 [nlin.SI]
Comments: latex2e, packages "amsmath,amssymb,amsthm", 50 pages, no figure,
contribution to proceedings of conference "Exploration of new structures
and natural constructions in mathematical physics" (Nagoya University,
March, 2007); (v2) a few references added; (v3) final version for publication
Abstract
We review the notion of differential Fay identities and demonstrate,
through case studies, its new role in integrable hierarchies of
the KP type. These identities are known to be a convenient tool
for deriving dispersionless Hirota equations. We show that
differential (or, in the case of the Toda hierarchy, difference)
Fay identities play a more fundamental role. Namely, they are
nothing but a generating functional expression of the full set of
auxiliary linear equations, hence substantially equivalent to
the integrable hierarchies themselves. These results are illustrated
for the KP, Toda, BKP and DKP hierarchies. As a byproduct, we point out
some new features of the DKP hierarchy and its dispersionless limit.
Toshio Nakatsu and Kanehisa Takasaki
Melting Crystal, Quantum Torus and Toda Hierarchy
arXiv:0710.5339 [hep-th]
Commun. Math. Phys. 285 (2009), 445--468
Comments: 30 pages, 4 figures
Abstract
Searching for the integrable structures of supersymmetric
gauge theories and topological strings, we study melting crystal,
which is known as random plane partition, from the viewpoint of
integrable systems. We show that a series of partition functions
of melting crystals gives rise to a tau function of the one-dimensional
Toda hierarchy, where the models are defined by adding
suitable potentials, endowed with a series of coupling constants,
to the standard statistical weight. These potentials can be converted
to a commutative sub-algebra of quantum torus Lie algebra.
This perspective reveals a remarkable connection between
random plane partition and quantum torus Lie algebra, and
substantially enables to prove the statement. Based on the result,
we briefly argue the integrable structures of five-dimensional
$\mathcal{N}=1$ supersymmetric gauge theories and $A$-model
topological strings. The aforementioned potentials correspond
to gauge theory observables analogous to the Wilson loops, and
thereby the partition functions are translated in the gauge theory
to generating functions of their correlators. In topological strings,
we particularly comment on a possibility of topology change caused
by condensation of these observables, giving a simple example.
Kanehisa Takasaki
Fay-type identitties and dispersionless limit
of integrable hierarchies
Workshop ``Exploration of New Structures and
Natural Construction in Mathematical Physics''
(名古屋大学多元数理研究科2007年3月5日〜8日) 講演のスライド
[tex | pdf]
Abstract
Fay-type identities play a significant role
in recent (since 2000) applications of
dispersionless integrable hierarchies
(dKP, dToda, etc.). This talk is an introduction
to theoretical aspects of Fay-type identities.
高崎金久
ダイマー模型とその周辺
数理解析研究所研究集会「可積分系数理の眺望」
(2006年8月21日〜23日研究代表者竹縄知之)講究録原稿
[tex | pdf]
要旨
ダイマー模型は統計物理の模型であり，数学的には
２部グラフの完全マッチングの言葉を用いて定式化される．
この記事ではダイマー模型について基礎的な部分を解説する．
特に，平面的２部グラフの場合に分配函数がパフ式や行列式として
表せる仕組みを丁寧に解説する．周期的ダイマー模型に登場する
アメーバやロンキン函数の概念についても簡単に触れる．
Kanehisa Takasaki
Hamiltonian structure of PI hierarchy
arXiv:nlin.SI/0610073
SIGMA 3 (2007), 042, 32 pages
Comments: This is a contribution to the Vadim Kuznetsov Memorial Issue
on Integrable Systems and Related Topics, published in SIGMA
Abstract
The string equation of type $(2,2g+1)$ may be
thought of as a higher order analogue of
the first Painlev\'e equation that correspond
to the case of $g = 1$. For $g > 1$, this equation
is accompanied with a finite set of commuting
isomonodromic deformations, and they altogether
form a hierarchy called the PI hierarchy.
This hierarchy gives an isomonodromic analogue
of the well known Mumford system. The Hamiltonian
structure of the Lax equations can be formulated
by the same Poisson structure as the Mumford system.
A set of Darboux coordinates, which have been used
for the Mumford system, can be introduced in
this hierarchy as well. The equations of motion
in these Darboux coordinates turn out to take
a Hamiltonian form, but the Hamiltonians are
different from the Hamiltonians of the Lax equations
(except for the lowest one that corresponds to
the string equation itself). The difference
originates in the presence of extra terms in
the isomonodromic Lax equations.
Kanehisa Takasaki and Takashi Takebe
Dispersionless Hirota equations of multi-component integrable hierarchies
slides of talk at MISGRAM workshop "Integrable Systems in Applied Mathematics",
Madrid, September 7--12, 2006
[tex | pdf]
Abstract
I will present recent results and work in progress on
the dispersionless limit of Hirota equations of
various multi-component integrable hierarchies,
such as the multi-component ``charged'' KP hierarchy,
the 2-component BKP hierarchy, the coupled (or $D_\infty'$)
KP/BKP hierarchy, etc. This talk is partially based
on joint work with T. Takebe.
Kanehisa Takasaki and Takashi Takebe
Universal Whitham hierarchy, dispersionless Hirota equations
and multi-component KP hierarchy
arXiv:nlin.SI/0608068
Physica D235, no. 1-2 (2007), 109-125
Comments: latex2e (a4paper, 12pt) using packages "amssymb,amsmath,amsthm",
44 pages, no figure
Abstract
The goal of this paper is to identify
the universal Whitham hierarchy of genus zero
with a dispersionless limit of the multi-component
KP hierarchy. To this end, the multi-component
KP hierarchy is (re)formulated to depend on
several discrete variables called ``charges''.
These discrete variables play the role of
lattice coordinates in underlying Toda field equations.
A multi-component version of the so called
differential Fay identity are derived from
the Hirota equations of the $\tau$-function of
this ``charged'' multi-component KP hierarchy.
These multi-component differential Fay identities
have a well-defined dispersionless limit
(the dispersionless Hirota equations).
The dispersionless Hirota equations turn out
to be equivalent to the Hamilton-Jacobi equations
for the $S$-functions of the universal Whitham
hierarchy. The differential Fay identities
themselves are shown to be a generating functional
expression of auxiliary linear equations for
scalar-valued wave functions of the multi-component
KP hierarchy.
高崎金久
変数分離の簡単な模型
数理解析研究所研究集会「超函数と線型微分方程式2006 数学史とアルゴリズム」
(2006年3月6日〜9日研究代表者戸瀬信之)講究録原稿
[tex | pdf]
要旨
古典力学の場合に話を限り，ある非常に簡単な例を用いて変数分離の
考え方を解説する．これは一種の「おもちゃ模型（toy model）」であるが，
簡単でありながら変数分離の典型的な仕組みを示す含蓄の深い例である
ように思われる．さらに，このおもちゃ模型を動機として行われた
いくつかの研究などを紹介する
Kanehisa Takasaki
Dispersionless Hirota equations of two-component BKP hierarchy
SIGMA Vol. 2 (2006), Paper 057, 22 pages
Comments: 31 pages, no figure, latex2e usepackage amsmath,amssymb
Report-no: nlin.SI/0604003
Abstract
The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy).
Dispersionless limit of this multi-component hierarchy is considered
on the level of the $\tau$-function. The so called dispersionless
Hirota equations are obtained from the Hirota equations of
the $\tau$-function. These dispersionless Hirota equations turn out
to be equivalent to a system of Hamilton-Jacobi equations.
Other relevant equations, in particular, dispersionless Lax equations,
can be derived from these fundamental equations. For comparison,
another approach based on auxiliary linear equations is also presented.
Kanehisa Takasaki and Takashi Takebe
Radial Loewner equation and dispersionless cmKP hierarchy
Comments: Comments: 18 pages, Latex2e (article, amsmath, amssymb, amsthm)
Report-no: nlin.SI/0601063
Abstract
It has been shown that the dispersionless KP hierarchy (or the Benney
hierarchy) is reduced to the chordal L\"owner equation. We show that the radial
L\"owner equation also gives reduction of a dispersionless type integrable
system. The resulting system acquires another degree of freedom and becomes the
dcmKP hierarchy, which is a ``half'' of the dispersionless Toda hierarchy. The
results of this article was announced in nlin.SI/0512008.
Kanehisa Takasaki, Takashi Takebe
Loewner equations and dispersionless hierarchies
Comments: 6 pages (Latex; amsmath, amssymb required),
Contribution to the Proceedings of the XXIII International Conference
of Differential Geometric Methods in Theoretical Physics
(M.-L. Ge and W. Zhang ed., Nankai Tracts in Mathematics
vol. 10, World Scientific, 2006)
Report-no: nlin.SI/0512008
Abstract
Reduction of a dispersionless type integrable system
(dcmKP hierarchy) to the radial Loewner equation is presented.
高崎金久
Hamiltonian structuer of higher flows in string equation
("Kobe workshop on integrable systems and Painleve equations",
神戸大学 2005年11月23日〜25日) 講演のOHP資料
[tex | pdf]
Kanehisa Takasaki
Toy models of separation of variables
("Dynamics: Frontiers in Geommetry, Analysis and Mathematical Physics",
京都大学/ミュンヘン工科大学学術交流行事, 2005年10月7日) 講演のOHP資料
[tex | pdf]
Ryu Sasaki and Kanehisa Takasaki
Explicit solutions of the classical Calogero & Sutherland
systems for any root system
J. Math. Phys. 47 (1) (2006), 012701
Comments: 18 pages, LaTeX, no figure
Report-no: YITP-05-60, hep-th/0510035
Abstract
Explicit solutions of the classical Calogero
(rational with/without harmonic confining potential)
and Sutherland (trigonometric potential) systems is
obtained by diagonalisation of certain matrices of
simple time evolution. The method works for Calogero &
Sutherland systems based on any root system.
It generalises the well-known results by Olshanetsky
and Perelomov for the A type root systems.
Explicit solutions of the (rational and trigonometric)
higher Hamiltonian flows of the integrable hierarchy
can be readily obtained in a similar way for those
based on the classical root systems.
Kanehisa Takasaki
Dispersionless integrable hierarchies revisited
("Riemann-Hilbert problems, integrability and asymptotics",
SISSA, Trieste, September 20--25, 2005) 講演のOHP資料
[tex | pdf]
ソリトン
数理科学事典(丸善)第2版に収録予定
[pdf]
第1版の内容を全面的に書き改めた.
目次：
1. ソリトンの発見,
2. 逆散乱法とラックス表示,
3. 保存量と可積分性,
4. さまざまな方程式・手法,
5. 双線形化法,
6. KP階層,
7. その他の話題.
Kanehisa Takasaki
Tyurin parameters of commuting pairs and
infinite dimensional Grassmannian manifold
M. Noumi and K. Takasaki (ed.), ``Elliptic Integrable Systems'',
Rokko Lectures in Mathematics vol. 18, pp. 289--304
(Kobe University, 2005).
Comment: contribution to proceedings of RIMS workshop
"Elliptic Integrable Systems" (RIMS, 2004)
Report-no: nlin.SI/0505005
Abstract
Commuting pairs of ordinary differential operators are
classified by a set of algebro-geometric data called
``algebraic spectral data''. These data consist of
an algebraic curve (``spectral curve'') $\Gamma$ with
a marked point $\gamma_\infty$, a holomorphic vector bundle
$E$ on $\Gamma$ and some additional data related to
the local structure of $\Gamma$ and $E$ in a neighborhood
of $\gamma_\infty$. If the rank $r$ of $E$ is greater
than $1$, one can use the so called ``Tyurin parameters''
in place of $E$ itself. The Tyurin parameters specify
the pole structure of a basis of joint eigenfunctions
of the commuting pair. These data can be translated to
the language of an infinite dimensional Grassmann manifold.
This leads to a dynamical system of the standard exponential
flows on the Grassmann manifold, in which the role of
Tyurin parameters and some other parameters is made clear.
高崎金久
変形KP階層のq類似とその凖古典極限
日本数学会2005年年会 (2005年3月27日--30日中央大学理工学部）
無限可積分系セッション一般講演予稿.
[tex |
pdf]
要旨
変形KP階層のq類似に対して双線形方程式や線形方程式を導出する。
それに基づいて凖古典極限のHamilton-Jacobi方程式などを論じる。
Takashi Maeda, Toshio Nakatsu, Kanehisa Takasaki, Takeshi Tamakoshi
Free Fermion and Seiberg-Witten Differential in
Random Plane Partitions
Nucl. Phys. B715 (2005) 275-303
Report-no: OU-HET 513, hep-th/0412329
Abstract
A model of random plane partitions which describes
five-dimensional $\mathcal{N}=1$ supersymmetric SU(N)
Yang-Mills is studied. We compute the wave functions
of fermions in this statistical model and investigate
their thermodynamic limits or the semi-classical behaviors.
These become of the WKB type at the thermodynamic limit.
When the fermions are located at the main diagonal of
the plane partition, their semi-classical wave functions
are obtained in a universal form. We further show that
by taking the four-dimensional limit the semi-classical
wave functions turn to live on the Seiberg-Witten curve
and that the classical action becomes precisely the integral
of the Seiberg-Witten differential. When the fermions
are located away from the main diagonal, the semi-classical
wave functions depend on another continuous parameter.
It is argued that they are related with the wave functions
at the main diagonal by the renormalization group flow
of the underlying gauge theory.
Takashi Maeda, Toshio Nakatsu, Kanehisa Takasaki, Takeshi Tamakoshi
Five-Dimensional Supersymmetric Yang-Mills Theories and
Random Plane Partitions
JHEP 03 (2005), 056.
Report-no: OU-HET 512, hep-th/0412327
Abstract
Five-dimensional $\mathcal{N}=1$ supersymmetric Yang-Mills
theories are investigated from the viewpoint of random
plane partitions. It is shown that random plane partitions
are factorizable as q-deformed random partitions so that
they admit the interpretations as five-dimensional
Yang-Mills and as topological string amplitudes.
In particular, they lead to the exact partition functions
of five-dimensional $\mathcal{N}=1$ supersymmetric
Yang-Mills with the Chern-Simons terms. We further
show that some specific partitions, which we call
the ground partitions, describe the perturbative regime
of the gauge theories. We also argue their role in
string theory. The gauge instantons give the deformation
of the ground partition.
Kanehisa Takasaki
$q$-analogue of modified KP hierarchy and its quasi-classical limit
Lett. Math. Phys. 72 (3) (2005), 165--181.
Report-no: nlin.SI/0412067
Abstract
A $q$-analogue of the tau function of the modified
KP hierarchy is defined by a change of independent variables.
This tau function satisfies a system of bilinear
$q$-difference equations. These bilinear equations are
translated to the language of wave functions, which turn out
to satisfy a system of linear $q$-difference equations.
These linear $q$-difference equations are used to formulate
the Lax formalism and the description of quasi-classical limit.
These results can be generalized to a $q$-analogue of
the Toda hierarchy. The results on the $q$-analogue of
the Toda hierarchy might have an application to
the random partition calculus in gauge theories and
topological strings.
高崎金久
弦方程式の時間発展のHamilton構造
数理解析研究所短期共同研究「複素領域における微分方程式の大域解析と漸近解析」
（2003年10月6日--10日）講究録原稿
[tex | pdf]
要旨
いわゆる弦方程式（Douglas方程式）は背後のKdV階層に由来する
可換な時間発展をもつ. 通常の可積分系の変数分離の手法を応用する
ことによって, これらの時間発展を非自励Hamilton系に定式化しなおす
ことができる. Hamiltonianには通常の可積分系や弦方程式自体の
Hamilton形式の場合には存在しない付加項が現れる.
高崎金久
Tyurinパラメータと可換微分作用素環
科学研究費補助金（基盤研究C 2002--2003）成果報告書のための書き下ろし
[tex | pdf]
要旨
KricheverとNovikovの1970年代の研究によって
Tyurinパラメータと可換微分作用素環との関わりが
明らかになった．以下では，ソリトン方程式の
高種数的類似を考える材料として，この研究の
概略を紹介する．可換微分作用素環にはスペクトル曲線
と呼ばれる代数曲線とその上の正則ベクトルなどの
幾何学的構造が付随している．Tyurinパラメータは
代数幾何学で代数曲線上の安定正則ベクトル束の
モジュライの定式化の一つとして提案されたものだが，
可換微分作用素環ではそれが自然な形で登場する．
可換微分作用素環とKP階層による時間発展を組み合わせれば，
KdV方程式などの古典的ソリトン方程式の高種数的類似
とみなすべき方程式が現れる．これらの方程式の定式化
においてもTyurinパラメータが基本的役割を演じる．
Kanehisa Takasaki
Elliptic spectral parameter and infinite dimensional
Grassmann variety
Comments: Contribution to Faro conference
"Infinite dimensional algebras and quantum integrable systems",
Progress in Mathematics vol. 237, pp. 169--197
(Birkhauser, Basel/Switzerland, 2005)
Report-no: nlin.SI/0312016
Abstract
Recent results on the Grassmannian perspective of soliton
equations with an elliptic spectral parameter are presented
along with a detailed review of the classical case with
a rational spectral parameter. The nonlinear Schr\"odinger
hierarchy is picked out for illustration of the classical
case. This system is formulated as a dynamical system on
a Lie group of Laurent series with factorization structure.
The factorization structure induces a mapping to an infinite
dimensional Grassmann variety. The dynamical system on
the Lie group is thereby mapped to a simple dynamical system
on a subset of the Grassmann variety. Upon suitable
modification, almost the same procedure turns out to work
for soliton equations with an elliptic spectral parameter.
A clue is the geometry of holomorphic vector bundles over
the elliptic curve hidden (or manifest) in the zero-curvature
representation.
Kanehisa Takasaki
Landau-Lifshitz hierarchy and infinite dimensional
Grassmann variety
Lett. Math. Phys. 67 (2) (2004), 141-152
Comments: latex2e (usepackage:amssyb), 15 pages, no figure;
(v2) minor changes; (v3) typos corrected
Report-no: lin.SI/0312002
Abstract
The Landau-Lifshitz equation is an example of soliton equations
with a zero-curvature representation defined on an elliptic curve.
This equation can be embedded into an integrable hierarchy of evolution
equations called the Landau-Lifshitz hierarchy. This paper elucidates
its status in Sato, Segal and Wilson's universal description of soliton
equations in the language of an infinite dimensional Grassmann variety.
To this end, a Grassmann variety is constructed from a vector space of
$2 \times 2$ matrices of Laurent series of the spectral parameter $z$.
A special base point $W_0$, called ``vacuum,'' of this Grassmann variety
is chosen. This vacuum is ``dressed'' by a Laurent series $\phi(z)$ to
become a point of the Grassmann variety that corresponds to a general
solution of the Landau-Lifshitz hierarchy. The Landau-Lifshitz hierarchy
is thereby mapped to a simple dynamical system on the set of these
dressed vacua. A higher dimensional analogue of this hierarchy
(an elliptic analogue of the Bogomolny hierarchy) is also presented.
高崎金久
Tyurinパラメータと佐藤理論
日本数学会2003年秋季総合分科会（無限可積分系分科会）一般講演
（2003年9月24日--27日千葉大学）講演予稿．
[tex | pdf]
要旨
最近Kricheverは任意種数の代数曲線上でLax・零曲率方程式の構成法を与えた．
ここでは彼のアイディアを適用して楕円（および任意種数）曲線上の
非線形Schrodinger方程式の類似を構成し，佐藤Grassmann多様体の言葉で
この系の解釈を示す．
Kanehisa Takasaki
Tyurin parameters and elliptic analogue of
nonlinear Schr\"odinger hierarchy
J. Math. Sci. Univ. Tokyo 11 (2004), 91--131
Comments: latex2e, 36 pp, no figure;
(v2) minor changes, mostly typos;
(v3) Title changed, text fully revised with new results;
(v4) serious errors in section 5 corrected;
(v5) proof of main results is improved;
(v6) minor change in proof of Lemma 10 etc
Report-no: nlin.SI/0307030
Abstract
Two ``elliptic analogues'' of the nonlinear Schr\"odinger hiererchy
are constructed, and their status in the Grassmannian perspective
of soliton equations is elucidated. In addition to the usual fields
$u,v$, these elliptic analogues have new dynamical variables called
``Tyurin parameters,'' which are connected with a family of vector
bundles over the elliptic curve in consideration. The zero-curvature
equations of these systems are formulated by a sequence of $2 \times 2$
matrices $A_n(z)$, $n = 1,2,\ldots$, of elliptic functions. In addition
to a fixed pole at $z = 0$, these matrices have several extra poles.
Tyurin parameters consist of the coordinates of those poles and some
additional parameters that describe the structure of $A_n(z)$'s.
Two distinct solutions of the auxiliary linear equations are constructed,
and shown to form a Riemann-Hilbert pair with degeneration points.
The Riemann-Hilbert pair is used to define a mapping to an infinite
dimensional Grassmann variety. The elliptic analogues of the nonlinear
Schr\"odinger hierarchy are thereby mapped to a simple dynamical system
on a special subset of the Grassmann variety.
ソリトン
数学辞典(岩波書店)第4版に収録予定
[tex | pdf]
Hamilton構造、双線形化法、KP階層などの話題も取り入れて、
岩波数学辞典第3版の「ソリトン」を全面的に書き改めた。
高崎金久, 武部尚志
有理型函数対の空間の上の可積分系
日本数学会2003年年会無限可積分系分科会一般講演予稿
（2003年3月23日--26日東京大学数理科学研究科）
[tex | pdf]
要旨 有理関数の空間の空間の上にはAtiyahとHitchinの
シンプレクティック構造に基づいて可積分系の構造が入ることが
知られているが、ここではその構成法を任意種数の複素代数曲線上の
有理型関数の対の空間に拡張する。
高崎金久
佐藤理論から見たLandau-Lifshitz方程式
日本数学会2003年年会無限可積分系分科会一般講演予稿
（2003年3月23日--26日東京大学数理科学研究科）
[tex | pdf]
（2003年11月改訂版）
要旨 Landau-Lifshitz方程式はトーラス上のスペクトルパラメータに
依存する行列によってLax表示される方程式として知られている。ここでは
ソリトン方程式に対する佐藤理論の枠組みでこの方程式を定式化する
方法を示す。
高崎金久
Liouville平面から見た複素WKB法
研究集会「数学解析の望ましい姿を探って」
（九州大学箱崎キャンパス2002年12月12日--14日）講究録原稿
[tex | pdf]
要旨
完全WKBに対するもうひとつのアプローチを提案する。これはいわゆる
「Liouville変換」によって物理空間上の問題を別の空間上の散乱問題に
変換することに基づく。
Kanehisa Takasaki
Spectral curve, Darboux coordinates and
Hamiltonian structure of periodic dressing chains
Commun. Math. Phys. 241 (1) (2003), 111--142
Comments: latex2e, 41 pages no figure;
改訂版（version 3）. 前の版よりもかなり短くなったが
（56pages --> 41 pages）, 結果の一部は改良された.
Report-no: nlin.SI/0206049
Abstract
A chain of one-dimensional Schr\"odinger operators is called
a ``dressing chain'' if they are connected by successive
Darboux transformations. Particularly interesting are
periodic dressing chains; they include finite-band operators
and Painlev\'e equations as a special case. We investigate
the Hamiltonian structure of these nonlinear lattices using
V.~Adler's $2 \times 2$ Lax pair. The Lax equation and the
auxiliary linear problem of this Lax pair contain a shift,
rather than a derivative, in the spectral parameter. Despite
this unusual feature, we can construct a transition matrix
around the periodic chain, an associated ``spectral curve''
and a set of Darboux coordinates (``spectral Darboux
coordinates''). The dressing chain is thereby converted to
a Hamiltonian system in these Darboux coordinates. Moreover,
the Hamiltonian formalism is accompanied by an odd-dimensional
Poisson structure. This induces a quadratic Poisson algebra of
the matrix elements of the transition matrix. As a byproduct,
we show that this Poisson structure is equivalent to another
Poisson structure previously studied by Veselov, Shabat, Noumi
and Yamada.
高崎金久
周期的Darboux鎖と周期的戸田格子の類似性
九州大学応用力学研究所研究集会「非線形波動および非線形力学系に関する最近の話題」
（2002年11月6日--8日）講究録原稿
[tex | pdf]
要旨
周期的Darboux鎖のさまざまな側面を調べて行くと，
Lax表示，スペクトル曲線，Hamilton構造，Poisson構造など
至るところで周期的戸田格子に類似する性質や構造に出会う．
ここではその一端を紹介する．
Kanehisa Takasaki
Integrable systems whose spectral curve is
the graph of a function
Comments: CRM Proceedings and Lecture Notes
vol. 37, pp. 211--222 (AMS, Province, 2004).
Contribution to the proceedings of
the conference "Superintegrability in classical and
quantum systems" (Montreal, September 16--22, 2002),
latex2e, 15 pages, no figure
Report-no: nlin.SI/0211021
Abstract
For some integrable systems, such as the open Toda molecule,
the spectral curve of the Lax representation becomes the
graph $C = \{(\lambda,z) \mid z = A(\lambda)\}$ of a ffunction
$A(\lambda)$. Those integrable systems provide an interesting
``toy model'' of separation of variables. Examples of this type
of integrable systems are presented along with generalizations
for which $A(\lambda)$ lives on a cylinder, a torus or a Riemann
surface of higher genus.
高崎金久
弦方程式のスペクトル曲線とHamilton構造
数理解析研究所短期共同研究「微分方程式の変形と漸近解析」
（2002年6月3日--7日）講究録原稿
[tex | pdf]
要旨
1変数の微分作用素 Q,P に対する交換子方程式 [Q,P] = 1 を
弦方程式という。この論文では Q が2階、P が奇数（2g+1）階の
場合の弦方程式のHamilton構造について得られた結果を紹介する。
この場合の弦方程式は 2 x 2 行列のLax表示をもつ。それを用いて
スペクトル曲線（種数 g の超楕円曲線となる）ならびに 2g 個の
正準変数（Darboux変数）を導入し、弦方程式をHamilton系に書き直す。
高崎金久, 武部尚志
変数分離・代数曲面・Seiberg-Witten理論から見た
有理函数の空間上の可積分系とその拡張
日本数学会2002年年会無限可積分系分科会
（2002年3月28日--31日明治大学駿河台校舎）一般講演予稿
[tex | pdf]
要旨
AtiyahとHitchinは SU(2) Yang-Millsモノポールのモジュライ空間の
ツイスター記述の過程で有理函数の空間の上にシンプレクティック構造
を導入した。この有理函数の極の基本対称式は対応するPoisson構造に
関して可換であり、特にこの空間の上に可積分系が得られる。この可積分系は
Moserが古典的な仕事において非周期的有限戸田格子の別表現として得たものと
同一視できる。本講演では，この（ほとんど自明とも言うべき）
可積分系が表題に掲げた変数分離，代数曲面，ならびにSeiberg-Witten理論
の簡単なモデルでもあることに注目し，それに基づくいくつかの拡張を示す．
Kanehisa Takasaki and Takashi Takebe
An integrable system on the moduli space of rational functions
and its variants
Journal of Geometry and Physics 47 (1) (2003), 1--20
Comments: 25 pages, AMS-LaTeX, no figure
Report-no: nlin.SI/0202042
Abstract
We study several integrable Hamiltonian systems on the moduli spaces
of meromorphic functions on Riemann surfaces (the Riemann sphere,
a cylinder and a torus). The action-angle variables and the separated
variables (in Sklyanin's sense) are related via a canonical transformation,
the generating function of which is the Abel-Jacobi type integral of
the Seiberg-Witten differential over the spectral curve.
高崎金久
dressing chainのスペクトル曲線とHamilton構造
京大数理研・短期共同研究
「可積分系研究における双線形化法とその周辺」
（2001年7月2日--4日）講究録原稿
[tex | pdf]
要旨
dressing chainはShabatらによって導入された概念で，
Darboux変換と定数によるずらしを組み合わせた関係で
結ばれた1次元Schrodinger作用素の列を意味する．
これはもともとSchrodinger方程式のスペクトル的性質を
調べる問題から派生したものだが，周期的境界条件を
課したものは有限帯ポテンシャルやPainleve方程式とも
深く関わり合っている．以下ではまずdressing chainに
ついて一般的なこと（特に二種類のLax形式の存在）を
復習し，周期的な場合の転移行列やスペクトル曲線の
構造を説明してから，Hamilton構造について最近得た
結果を紹介する．
Kanehisa Takasaki
Spectral curve and Hamiltonian structure of isomonodromic
SU(2) Calogero-Gaudin system
J. Math. Phys. 44 (9) (2003), 3979--3999
Comments: J. Math. Phys. vol. 44 に掲載された原稿に
深刻な誤りが見つかった. 訂正版がe-print書庫にarXiv:nlin.SI/0111019 ver.6
として置いてある.
Report-no: nlin.SI/0111019
Abstract
This paper presents an approach to the Hamiltonian structure
of isomonodromic systems of matrix ODE's on a torus from their
spectral curve. An isomonodromic analogue of the so called
$\rmSU(2)$ Calogero-Gaudin system is used for a case study of
this approach. A clue of this approach is a mapping from the
Lax equation to a dynamical system of a finite number of points
on the spectral curve. The coordinates of these moving points
give a new set of canonical variables, which have been used in
the literature for separation of variables of many integrable
systems including the usual $\rmSU(2)$ Calogero-Gaudin system
itself. The same machinery turns out to work for the isomonodromic
system on a trous, though the separability is lost and the
non-autonomous nature of the system causes technical complications.
Strong evidence is shown which suggests that this isomonodromic
system is equivalent to a previously known isomonodromic system
of second order scalar ODE's on a torus.
R. Sasaki and K. Takasaki
Quantum Inozemtsev model, quasi-exact solvability and N-fold
supersymmetry
J.Phys. A34 (2001), 9533--9554
Comments: LaTeX2e 28 pages, no figures
Report-no: hep-th/0109008, YITP-01-60 and KUCP-0191
Abstract
Inozemtsev models are classically integrable multi-particle dynamical systems
related to Calogero-Moser models. Because of the additional q^6 (rational
models) or sin^2(2q) (trigonometric models) potentials, their quantum versions
are not exactly solvable in contrast with Calogero-Moser models. We show that
quantum Inozemtsev models can be deformed to be a widest class of partly
solvable (or quasi-exactly solvable) multi-particle dynamical systems. They
posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A
new method for identifying and solving quasi-exactly solvable systems, the
method of pre-superpotential, is presented.
Saburo Kakei, Takeshi Ikeda and Kanehisa Takasaki
Hierarchy of (2+1)-dimensional nonlinear Schroedinger equation,
self-dual Yang-Mills equation, and toroidal Lie algebras
Ann. Henri Poincare 3 (2002), 817--845.
Comments: 26 pages, Latex2e, uses amsmath, amssymb, amsthm
Report-no: nlin.SI/0107065
Abstract
The hierarchy structure associated with a (2+1)-dimensional Nonlinear
Schroedinger equation is discussed as an extension of the theory of the KP
hierarchy. Several methods to construct special solutions are given. The
relation between the hierarchy and a representation of toroidal Lie algebras
are established by using the language of free fermions. A relation to the
self-dual Yang-Mills equation is also discussed.
高崎金久
有理スペクトル曲線をもつ可積分系
日本数学会2001年年会
（2001年3月26日ー29日慶応義塾大学日吉キャンパス）一般講演予稿
[tex | pdf]
要旨
可積分系のスペクトル曲線と言えば正種数の代数曲線を想像
するのが普通だが，スペクトル曲線が有理曲線になる例もある．
ここではそのような例を紹介する．
高崎金久
Painleve-Calogero correspondence
京大数理研・短期共同研究「離散可積分系の研究の進展」
（2000年8月2日--4日）講究録原稿
[tex | pdf]
要旨
最近，Painlev\'eVI型方程式とCalogero型可積分系の間の
対応関係（Painlev\'e-Calogero対応）がVI型以外の方程式
にも拡張された．また，多自由度のCalogero系に対応して，
Painlev\'e方程式の多自由度への拡張が得られた．これらの
結果を解説する．
高崎金久
変数分離法が甦る
日本応用数理学会2000年度年会オーガナイズドセッション
「力学系の応用数理 -- 幾何学と可積分系の視点から --」
（2000年10月6日ー8日東京工業大学）講演予稿
[tex | pdf]
要旨
最近，19世紀の変数分離法が現代的な可積分系の
視点から見直されている．簡単な可積分系を例に選んで，
現代に復活した変数分離法の考え方を紹介する．
Kanehisa Takasaki
Hyperelliptic Integrable Systems on K3 and Rational Surfaces
Phys. Lett. A283 (2001), 201--208.
Comments: latex2e using packages "amsmath and amssymb", 15 pages
Report-no: KUCP-0161, math.AG/0007073
Abstract:
We show several examples of integrable systems
related to special K3 and rational surfaces (e.g.,
an elliptic K3 surface, a K3 surface given by a
double covering of the projective plane, a rational
elliptic surface, etc.). The construction, based on
Beauvilles's general idea, is considerably simplified
by the fact that all examples are described by
hyperelliptic curves and Jacobians. This also enables
to compare these integrable systems with more classical
integrable systems, such as the Neumann system and the
periodic Toda chain, which are also associated with
rational surfaces. A delicate difference between the
cases of K3 and of rational surfaces is pointed out
therein.
Kanehisa Takasaki
Anti-self-dual Yang-Mills equations on noncommutative spacetime
J. Geom. Phys. 37 (2001), 291 - 306.
Comments: latex2e using packages amsmath and amssymb, 24 pages
Report-no: KUCP-153, hep-th/0005194
Abstract:
By replacing the ordinary product with the so called $\star$-product, one can
construct an analogue of the anti-self-dual Yang-Mills (ASDYM) equations on the
noncommutative $\bbR^4$. Many properties of the ordinary ASDYM equations turn
out to be inherited by the $\star$-product ASDYM equation. In particular, the
twistorial interpretation of the ordinary ASDYM equations can be extended to
the noncommutative $\bbR^4$, from which one can also derive the fundamental
strutures for integrability such as a zero-curvature representation, an
associated linear system, the Riemann-Hilbert problem, etc. These properties
are further preserved under dimensional reduction to the principal chiral field
model and Hitchin's Higgs pair equations. However, some structures relying on
finite dimensional linear algebra break down in the $\star$-product analogues.
Kanehisa Takasaki
Painleve-Calogero Correspondence Revisited
J. Math. Phys. 42 (3) (2001), 1443-1473.
Comments: latex2e using amsmath and amssymb packages,
40 pages, no figure
Report-no: KUCP 149, math.QA/0004118
Abstract:
We extend the work of Fuchs, Painlev\'e and Manin on
a Calogero-like expression of the sixth Painlev\'e
equation (the ``Painlev\'e-Calogero correspondence'')
to the other five Painlev\'e equations. The Calogero
side of the sixth Painlev\'e equation is known to be
a non-autonomous version of the (rank one) elliptic
model of Inozemtsev's extended Calogero systems.
The fifth and fourth Painlev\'e equations correspond
to the hyperbolic and rational models in Inozemtsev's
classification. Those corresponding to the third,
second and first are not included therein. We further
extend the correspondence to the higher rank models,
and obtain a ``multi-component'' version of the
Painlev\'e equations.
Takeshi Ikeda and Kanehisa Takasaki
Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation
International Mathematics Research Notices 7 (2001), 329--369
Comments: LaTeX2e with amsmath and amssymb, 35 pages, no figures
Report-no: nlin.SI/0004015
Abstract:We introduce an extension of the $\ell$-reduced
KP hierarchy, which we call the $\ell$-Bogoyavlensky hierarchy.
Bogoyavlensky's $2+1$-dimensional extension of the KdV equation
is the lowest equation of the hierarchy in case of $\ell=2$.
We present a group-theoretic characterization of this hierarchy
on the basis of the $2$-toroidal Lie algebra ${\fraksl}_\ell^\tor$.
This reproduces essentially the same Hirota bilinear equations as
those recently introduced by Billig and Iohara et al. We can
further derive these Hirota bilinear equation from a Lax formalism
of the hierarchy.This Lax formalism also enables us to construct
a family of special solutions that generalize the
breaking soliton solutions of Bogoyavlensky.
These solutions contain the $N$-soliton solutions,
which are usually constructed by use of vertex operators.
高崎金久
Hyperelliptic integrable systems on elliptic K3 and
rational surfaces
日本数学会2000年年会無限可積分系分科会
（2000年3月27日--30日早稲田大学理工学部）一般講演予稿
[tex | pdf]
Abstract
After the work of Mukai and Beauville, it has come
to be widely recognized that the moduli spaces of
a certain class of sheaves on a K3 surface have the
structure of algebraically integrable Hamiltonian
systems (AIHS). The aim of this talk is to present
a special example of those AIHS's on elliptic K3 (and
rational) surfaces, along with an explicit description
of ``action-angle variables''.
高崎金久
トーラス上の等モノドロミー変形
日本数学会2000年年会無限可積分系分科会
（2000年3月27日--30日早稲田大学理工学部）特別講演予稿
[tex | pdf]
要旨
トーラス上の常微分方程式の等モノドロミー変形ならびに
それらと可積分系・共形場理論・Painlev\'e 方程式との
関係に関する最近の研究の概要を紹介する．
高崎金久
Calogero-Moser系から見たPainleve方程式
数理解析研究所短期共同研究「Paileve系，超幾何系，漸近解析」
（１９９９年６月７日ー１０日）および神戸大学研究集会
「パンルヴェ方程式の大域解析」（１９９９年１０月２６日ー２９日）
での講演に基づく
[tex | pdf]
要旨
ManinによるPainlev\'e VI型方程式の別表現は楕円型
Calogero-Moser系の一種の楕円型Inozemtsev系を非自励系に
変えた形をもつ．最近，各種の楕円型Calogero-Moser系
から同様にして非自励系を構成できること，得られる
非自励系はもとの自励系と同様のLax表示をもつこと，
そこからトーラス上の等モノドロミー変形としての解釈が
導かれること，などが明らかになった．特に，これによって
Maninの非自励系をトーラス上の等モノドロミー系として
特徴づけることができるようになる．この結果と背景
について紹介する．他のPainlev\'e方程式への退化，
Garnier系などへの拡張の可能性など，関連する問題にも
触れる．
高崎金久
非線形波動の変調とWhitham 方程式
数理解析研究所研究会「繰り込み群の数理科学での応用」
（１９９９年７月２１日ー２３日）講究録寄稿
[tex | pdf]
要旨
非線形振動論における Bogoliubov-Krilov-Mitropol'ski の平均化法の
非線形波動への拡張として，Whitham の平均化法 \cite{bib:Whitham}
と呼ばれるものが知られている．これはもともと分散性非線形波動
の変調現象を一般的に扱うために開発されたものであるが，
KdV方程式などの場合には(多重)周期解が(超)楕円函数で厳密に
与えられるため，その変調がRiemann面の幾何学の言葉で記述される，
など数学的に思いがけない広がりを見せる．また，戸田格子などの
格子上の非線形波動や，Painlev\'e型方程式の解の漸近挙動の解析
にも応用されている．
Whithamの平均化法は非線形波動から搬送波の速い振動を
平均操作によって消去して，搬送波のパラメータの緩やかな
時空変化を取り出すものである．類似の考え方は「断熱近似」と
して非線形波動のみならず物理学全般に広く用いられている．
たとえば，量子力学のBorn-Oppenheimer近似もその一種で，
その数学的な取り扱いはWhitham平均化法とほとんど同じである．
これらの手法に共通する「速いモードを消去して遅いモードに
繰り込む」という考え方は繰り込み群にも通じるものがある．
以下では，繰り込み群との関連を期待しつつ，KdV方程式を
主な例として，Whithamの平均化法を解説する．
高崎金久
Elliptic Calogero-Moser systems and isomonodromic
deformations
日本数学会１９９９年度秋季総合分科会
（広島大学１９９９年９月）一般講演予稿
[tex | pdf]
Abstract
We consider a non-autonomous analogue of various
elliptic Calogero-Moser systems. A particularly
interesting case is the Inozemtsev system,
which contains Manin's elliptic (``$\mu$-equation'')
form of the sixth Painlev\'e equation as a special
case. We show that the non-autonomous analogues,
like their autonomous counterparts, have a Lax pair.
This Lax pair enables us to interpret the
non-autonomous systems as isomonodromic deformations
on a torus.
S.P. Khastgir, R. Sasaki and K. Takasaki
Calogero-Moser Models IV: Limits to Toda theory
Prog. Theor. Phys. 102 (4) (1999), 749-776.
Comments: LaTeX2e with amsfonts.sty, 33 pages, no figures
Report-no: YITP-99-20, KUCP-0132, hep-th/9907102
Abstract:
Calogero-Moser models and Toda models are well-known integrable
multi-particle dynamical systems based on root systems associated with Lie
algebras. The relation between these two types of integrable models is
investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs
of Calogero-Moser models are specified by the representations of the
reflection groups, which are not the same as those of the corresponding Lie
algebras. The latter specify the Lax pairs of Toda models. The Hamiltonians of
the elliptic Calogero-Moser models tend to those of Toda models as one of the
periods of the elliptic function goes to infinity, provided the dynamical
variables are properly shifted and the coupling constants are scaled. On the
other hand most of Calogero-Moser Lax pairs, for example, the root type Lax
pairs, do not a have consistent Toda model limit. The minimal type Lax pairs,
which corresponds to the minimal representations of the Lie algebras, tend to
the Lax pairs of the corresponding Toda models.
Kanehisa Takasaki
Whitham Deformations and Tau Functions in
N = 2 Supersymmetric Gauge Theories
Prog. Theor. Phys. Suppl. 135 (1999), 53-74.
Comments: latex2e using amsfonts package
Report-no: KUCP-0136, hep-th/9905224
Abstract:
We review new aspects of integrable systems discovered recently in $N=2$
supersymmetric gauge theories and their topologically twisted versions. The
main topics are (i) an explicit construction of Whitham deformations of the
Seiberg-Witten curves for classical gauge groups, (ii) its application to
contact terms in the $u$-plane integral of topologically twisted theories,
and (iii) a connection between the tau functions and the blowup formula in
topologically twisted theories.
Kanehisa Takasaki
Elliptic Calogero-Moser Systems and
Isomonodromic Deformations
J. Math. Phys. 40 (11) (1999), 5787-5821
Comments: latex2e using amsfonts package
Report-no: KUCP-0133, math.QA/9905101
Abstract:
We show that various models of the elliptic Calogero-Moser systems
are accompanied with an isomonodromic system on a torus. The
isomonodromic partner is a non-autonomous Hamiltonian system
defined by the same Hamiltonian. The role of the time variable
is played by the modulus of the base torus. A suitably chosen
Lax pair (with an elliptic spectral parameter) of the elliptic
Calogero-Moser system turns out to give a Lax representation of
the non-autonomous system as well. This Lax representation
ensures that the non-autonomous system describes isomonodromic
deformations of a linear ordinary differential equation on the
torus on which the spectral parameter of the Lax pair is defined.
A particularly interesting example is the ``extended twisted
$BC_\ell$ model'' recently introduced along with some other models
by Bordner and Sasaki, who remarked that this system is equivalent
to Inozemtsev's generalized elliptic Calogero-Moser system. We
use the ``root type'' Lax pair developed by Bordner et al. to
formulate the associated isomonodromic system on the torus.
Kanehisa Takasaki
Whitham Deformations of Seiberg-Witten Curves
for Classical Gauge Groups
Int. J. Mod. Phys. A15 (23) (2000), 3635-3666
Comments: latex, 39pp, no figure
Report-no: KUCP-0127, hep-th/9901120
Abstract:
Gorsky et al. presented an explicit construction of Whitham
deformations of the Seiberg-Witten curve for the $SU(N+1)$
$\calN = 2$ SUSY Yang-Mills theory. We extend their result
to all classical gauge groups and some other cases such as
the spectral curve of the $A^{(2)}_{2N}$ affine Toda
system. Our construction, too, uses fractional powers of
the superpotential $W(x)$ that characterizes the curve.
We also consider the $u$-plane integral of topologically
twisted theories on four-dimensional manifolds $X$ with
$b_2^{+}(X) = 1$ in the language of these explicitly
constructed Whitham deformations and an integrable hierarchy
of the KdV type hidden behind.
A.J. Bordner, R. Sasaki and K. Takasaki
Calogero-Moser Models II: Symmetries and Foldings
Prog. Thero. Phys. 101 (3) (1999), 487-518.
Comments: 35 pages, LaTeX2e with amsfonts, no-figure
Report-no: YITP-98-60, KUCP-0121, hep-th/9809068
Abstract:
Universal Lax pairs (the root type and the minimal type) are presented for
Calogero-Moser models based on simply laced root systems including (E_8). They
are with and without spectral parameter and they work for all of the four
choices of potentials: the rational, trigonometric, hyperbolic and elliptic.
For the elliptic potential, the discrete symmetries of the simply laced models,
originating from the automorphism of the extended Dynkin diagrams are combined
with the periodicity of the potential to derive a class of Calogero-Moser
models known as the `twisted non-simply laced models'. Among them, a twisted
(BC_n) model is new and it has some novel features. For untwisted non-simply
laced models, two kinds of root type Lax pairs (based on long roots and short
roots) are derived which contain independent coupling constants for the long
and short roots. The (BC_n) model contains three independent couplings, for the
long, middle and short roots. The (G_2) model based on long roots exhibits a
new feature which deserves further study.
高崎金久
Construction of isomonodromic problems on torus
日本数学会１９９８年度秋季総合分科会
（大阪大学, １９９８年９月）一般講演予稿
[tex | pdf]
This talk deals with the problem of constructuting isomonodromic
deformations of a first order matrix system
$\frac{dY}{dz} = L(z) Y$ on the torus. The matrix linear system
is assumed to have regular singularities at $N$ points
$z = t_1,\cdots,t_N$. The problem is to construct isomonodromic
deformations that leave a set of monodromy data invariant while
the position of the poles and the modulus $\tau$ being varied.
I shall present two types of such isomonodromic systems.
Both have an isospectral partner
(the elliptic Calogero-Gaudin system / the ordinary elliptic
Gaudin system), and are also related to a conformal
field theory (the ordinary WZW model / its ``twisted'' version
by Kuroki and Takebe) on a torus.