■Faculty | Department of Science / Graduate School of Science and Engineering Research |

■Position | Professor |

■Degree | Doctor of Science |

■Commentator Guide | https://www.kindai.ac.jp/meikan/1224-takasaki-kanehisa.html |

■URL | http://www2.yukawa.kyoto-u.ac.jp/~kanehisa.takasaki/index-e.html |

Last Updated :2020/04/03

- 1975 04 - 1979 03 , University of Tokyo, Faculty of Science
- 1979 04 - 1984 03 , University of Tokyo, Graduate School of Sciences

- 2014 04 , - 現在, Professor, Faculty of Science and Engineering, Kinki University
- 2004 01 , - 2014 03 , Professor, Graduate School of Human and Environmental Studies, Kyoto Univeristy
- 2004 01 , - 2014 03 , Professor, Graduate School of Human and Environmental Studies, Kyoto University
- 1991 04 , - 2003 12 , Kyoto University
- 1991 04 , - 2003 12 , Associate Professor, College of Liberal Arts; Faculty of Integrated Human Studies; Graduate School of Human and Environmental studies, Kyoto University
- 1985 10 , - 1991 03 , Research Institute for Mathematical Sciences, Kyoto University
- 1985 09 , - 1991 03 , Research Assistant, Research Institute for Mathematical Sciences, Kyoto University
- 1984 04 , - 1985 09 , Faculty of Science, Saitama University
- 1984 04 , - 1985 08 , Research Assistant, Faculty of Science, Saitama University

- Natural sciences, Mathematical physics and basic theory
- Natural sciences, Basic analysis

- combinatorics, twistor theory, Painleve equation, integrable system, string theory, gauge theory, mathematical physics, algebraic analysis

- Three-partition Hodge integrals and the topological vertex, Toshio Nakatsu, Kanehisa Takasaki, Commun. Math. Phys. (online in 2019), Commun. Math. Phys. (online in 2019), Dec. 2018 , RefereedSummary:A conjecture on the relation between the cubic Hodge integrals and the

topological vertex in topological string theory is resolved. A central role is

played by the notion of generalized shift symmetries in a fermionic realization

of the two-dimensional quantum torus algebra. These algebraic relations of

operators in the fermionic Fock space are used to convert generating functions

of the cubic Hodge integrals and the topological vertex to each other. As a

byproduct, the generating function of the cubic Hodge integrals at special

values of the parameters therein is shown to be a tau function of the

generalized KdV (aka Gelfand-Dickey) hierarchies. - 4D limit of melting crystal model and its integrable structure, Kanehisa Takasaki, J. Geom. Phys. 137C (2019), 184-203., J. Geom. Phys. 137C (2019), 184-203., Aug. 2018 , RefereedSummary:This paper addresses the problems of quantum spectral curves and 4D limit for the melting crystal model of 5D SUSY $U(1)$ Yang-Mills theory on $\mathbb{R}^4\times S^1$. The partition function $Z(\mathbf{t})$ deformed by an infinite number of external potentials is a tau function of the KP hierarchy with respect to the coupling constants $\mathbf{t} = (t_1,t_2,\ldots)$. A single-variate specialization $Z(x)$ of $Z(\mathbf{t})$ satisfies a $q$-difference equation representing the quantum spectral curve of the melting crystal model. In the limit as the radius $R$ of $S^1$ in $\mathbb{R}^4\times S^1$ tends to $0$, it turns into a difference equation for a 4D counterpart $Z_{\mathrm{4D } }(X)$ of $Z(x)$. This difference equation reproduces the quantum spectral curve of Gromov-Witten theory of $\mathbb{CP}^1$. $Z_{\mathrm{4D } }(X)$ is obtained from $Z(x)$ by letting $R \to 0$ under an $R$-dependent transformation $x = x(X,R)$ of $x$ to $X$. A similar prescription of 4D limit can be formulated for $Z(\mathbf{t})$ with an $R$-dependent transformation $\mathbf{t} = \mathbf{t}(\mathbf{T},R)$ of $\mathbf{t}$ to $\mathbf{T} = (T_1,T_2,\ldots)$. This yields a 4D counterpart $Z_{\mathrm{4D } }(\mathbf{T})$ of $Z(\mathbf{t})$. $Z_{\mathrm{4D } }(\mathbf{T})$ agrees with a generating function of all-genus Gromov-Witten invariants of $\mathbb{CP}^1$. Fay-type bilinear equations for $Z_{\mathrm{4D } }(\mathbf{T})$ can be derived from similar equations satisfied by $Z(\mathbf{t})$. The bilinear equations imply that $Z_{\mathrm{4D } }(\mathbf{T})$, too, is a tau function of the KP hierarchy. These results are further extended to deformations $Z(\mathbf{t},s)$ and $Z_{\mathrm{4D } }(\mathbf{T},s)$ by a discrete variable $s \in \mathbb{Z}$, which are shown to be tau functions of the 1D Toda hierarchy.
- Hurwitz numbers and integrable hierarchy of Volterra type, Kanehisa Takasaki, J. Phys. A: Math. Theor. 51 (2018), 43LT01 (9 pages), J. Phys. A: Math. Theor. 51 (2018), 43LT01 (9 pages), Jun. 2018 , RefereedSummary:A generating function of the single Hurwitz numbers of the Riemann sphere

$\mathbb{CP}^1$ is a tau function of the lattice KP hierarchy. The associated

Lax operator $L$ turns out to be expressed as $L = e^{\mathfrak{L } }$, where

$\mathfrak{L}$ is a difference-differential operator of the form $\mathfrak{L}

= \partial_s - ve^{-\partial_s}$. $\mathfrak{L}$ satisfies a set of Lax

equations that form a continuum version of the Bogoyavlensky-Itoh (aka hungry

Lotka-Volterra) hierarchies. Emergence of this underlying integrable structure

is further explained in the language of generalized string equations for the

Lax and Orlov-Schulman operators of the 2D Toda hierarchy. This leads to

logarithmic string equations, which are confirmed with the help of a

factorization problem of operators. - Toda hierarchies and their applications, Kanehisa Takasaki, J. Phys. A: Math. Theor. 51 (2018), 203001 (35pp), J. Phys. A: Math. Theor. 51 (2018), 203001 (35pp), Jan. 2018 , RefereedSummary:The 2D Toda hierarchy occupies a central position in the family of integrable

hierarchies of the Toda type. The 1D Toda hierarchy and the Ablowitz-Ladik (aka

relativistic Toda) hierarchy can be derived from the 2D Toda hierarchy as

reductions. These integrable hierarchies have been applied to various problems

of mathematics and mathematical physics since 1990s. A recent example is a

series of studies on models of statistical mechanics called the melting crystal

model. This research has revealed that the aforementioned two reductions of the

2D Toda hierarchy underlie two different melting crystal models. Technical

clues are a fermionic realization of the quantum torus algebra, special

algebraic relations therein called shift symmetries, and a matrix factorization

problem. The two melting crystal models thus exhibit remarkable similarity with

the Hermitian and unitary matrix models for which the two reductions of the 2D

Toda hierarchy play the role of fundamental integrable structures. - Open string amplitudes of closed topological vertex, Kanehisa Takasaki, Toshio Nakatsu, J. Phys. A: Math. Theor. 49 (2016), 025201 (28 pages), J. Phys. A: Math. Theor. 49 (2016), 025201 (28 pages), Dec. 2015 , RefereedSummary:The closed topological vertex is the simplest ``off-strip'' case of non-compact toric Calabi-Yau threefolds with acyclic web diagrams. By the diagrammatic method of topological vertex, open string amplitudes of topological string theory therein can be obtained by gluing a single topological vertex to an ``on-strip'' subdiagram of the tree-like web diagram. If non-trivial partitions are assigned to just two parallel external lines of the web diagram, the amplitudes can be calculated with the aid of techniques borrowed from the melting crystal models. These amplitudes are thereby expressed as matrix elements, modified by simple prefactors, of an operator product on the Fock space of 2D charged free fermions. This fermionic expression can be used to derive $q$-difference equations for generating functions of special subsets of the amplitudes. These $q$-difference equations may be interpreted as the defining equation of a quantum mirror curve.
- Orbifold melting crystal models and reductions of Toda hierarchy, Kanehisa Takasaki, J. Phys. A: Math. Theor. 48 (2015) , 215201 (34 pages), J. Phys. A: Math. Theor. 48 (2015) , 215201 (34 pages), May 2015 , RefereedSummary:Orbifold generalizations of the ordinary and modified melting crystal models are introduced. They are labelled by a pair $a,b$ of positive integers, and geometrically related to $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifolds of local $\mathbf{CP}^1$ geometry of the $\mathcal{O}(0)\oplus\mathcal{O}(-2)$ and $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ types. The partition functions have a fermionic expression in terms of charged free fermions. With the aid of shift symmetries in a fermionic realization of the quantum torus algebra, one can convert these partition functions to tau functions of the 2D Toda hierarchy. The powers $L^a,\bar{L}^{-b}$ of the associated Lax operators turn out to take a special factorized form that defines a reduction of the 2D Toda hierarchy. The reduced integrable hierarchy for the orbifold version of the ordinary melting crystal model is the bi-graded Toda hierarchy of bi-degree $(a,b)$. That of the orbifold version of the modified melting crystal model is the rational reduction of bi-degree $(a,b)$. This result seems to be in accord with recent work of Brini et al. on a mirror description of the genus-zero Gromov-Witten theory on a $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifold of the resolved conifold.
- Modified melting crystal model and Ablowitz-Ladik hierarchy, Kanehisa Takasaki, J. Phys. A: Math. Theor. 46 (2013), 245202 (23 pages), J. Phys. A: Math. Theor. 46 (2013), 245202 (23 pages), May 2013 , RefereedSummary:This paper addresses the issue of integrable structure in a modified melting

crystal model of topological string theory on the resolved conifold. The

partition function can be expressed as the vacuum expectation value of an

operator on the Fock space of 2D complex free fermion fields. The quantum torus

algebra of fermion bilinears behind this expression is shown to have an

extended set of "shift symmetries". They are used to prove that the partition

function (deformed by external potentials) is essentially a tau function of the

2D Toda hierarchy. This special solution of the 2D Toda hierarc... - Remarks on partition functions of topological string theory on generalized conifolds, Kanehisa Takasaki, RIMS Kokyuroku 1913 (2014), pp.182-201, RIMS Kokyuroku 1913 (2014), pp.182-201, Jan. 2013 Summary:The notion of topological vertex and the construction of topological string

partition functions on local toric Calabi-Yau 3-folds are reviewed.

Implications of an explicit formula of partition functions for the generalized

conifolds are considered. Generating functions of part of the partition

functions are shown to be tau functions of the KP hierarchy. The associated

Baker-Akhiezer functions play the role of wave functions, and satisfy

$q$-difference equations. These $q$-difference equations represent the quantum

mirror curves conjectured by Gukov and Su{\l}kowski. - Old and New Reductions of Dispersionless Toda Hierarchy, Kanehisa Takasaki, SIGMA 8 (2012), 102, 22 pages, SIGMA 8 (2012), 102, 22 pages, Dec. 2012 , RefereedSummary:This paper is focused on geometric aspects of two particular types of

finite-variable reductions in the dispersionless Toda hierarchy. The reductions

are formulated in terms of "Landau-Ginzburg potentials" that play the role of

reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's

trigonometric polynomial. The other is a transcendental function, the logarithm

of which resembles the waterbag models of the dispersionless KP hierarchy. They

both satisfy a radial version of the L\"owner equations. Consistency of these

L\"owner equations yields a radial version of the Gi... - An hbar-expansion of the Toda hierarchy: a recursive construction of solutions, Kanehisa Takasaki, Takashi Takebe, Analysis and Mathematical Physics. 2 (2012), 171-214, Analysis and Mathematical Physics. 2 (2012), 171-214, Dec. 2011 , RefereedSummary: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.
- Thermodynamic limit of random partitions and dispersionless Toda hierarchy, Kanehisa Takasaki, Toshio Nakatsu, J. Phys. A: Math. Theor. 45 (2012), 025403 (38pp), J. Phys. A: Math. Theor. 45 (2012), 025403 (38pp), Dec. 2011 , RefereedSummary: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 pro... - Toda tau functions with quantum torus symmetries, Kanehisa Takasaki, Acta Polytech. 51: 74-76,2011, Acta Polytech. 51: 74-76,2011, Jan. 2011 Summary: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. - Generalized string equations for double Hurwitz numbers, Kanehisa Takasaki, Journal of Geometry and Physics 62 (2012), 1135-1156, Journal of Geometry and Physics 62 (2012), 1135-1156, Dec. 2010 , RefereedSummary: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. ... - KP and Toda tau functions in Bethe ansatz, Kanehisa Takasaki, B. Feigin, M. Jimbo and M. Okado (eds.), ``New Trends in Quantum Integrable Systems'' (World Scientific, Singapore, 2010), B. Feigin, M. Jimbo and M. Okado (eds.), ``New Trends in Quantum Integrable Systems'' (World Scientific, Singapore, 2010), 393 - 392, Mar. 2010 , RefereedSummary: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. - Non-degenerate solutions of universal Whitham hierarchy, Kanehisa Takasaki, Takashi Takebe, Lee Peng Teo, J. Phys. A: Math. Theor. 43 (2010), 325205, J. Phys. A: Math. Theor. 43 (2010), 325205, Mar. 2010 , RefereedSummary: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 $H_a(z_0,z_a)$, $a = 1,...,M$, which play the role of generating functions of two-dimensional canonical transformations. The solution of the Riem...
- Two extensions of 1D Toda hierarchy, Kanehisa Takasaki, J.Phys.A: Math. Theor. 43 (2010), 434032, J.Phys.A: Math. Theor. 43 (2010), 434032, Feb. 2010 , RefereedSummary: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.
- Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy, Kanehisa Takasaki, SIGMA.5 (2009), 109, SIGMA.5 (2009), 109, Aug. 2009 , RefereedSummary: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 (d...
- Integrable structure of melting crystal model with two q-parameters, Kanehisa Takasaki, J. Geom. Phys. 59 (2009), 1244-1257, J. Geom. Phys. 59 (2009), 1244-1257, Mar. 2009 , RefereedSummary: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 $t_1,t_2,...$ and an extra one $Q$. There is a compact expression of this partition fun...
- Loewner equations, Hirota equations and reductions of universal Whitham hierarchy, Kanehisa Takasaki, Takashi Takebe, J. Phys. A: Math. Theor. 41 (2008), 475206 (27pp), J. Phys. A: Math. Theor. 41 (2008), 475206 (27pp), Aug. 2008 , RefereedSummary: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 g...
- Integrable structure of melting crystal model with external potentials, Toshio Nakatsu, Kanehisa Takasaki, Adv. Stud. Pure Math. 59 (2010), 201-223, Adv. Stud. Pure Math. 59 (2010), 201-223, Jul. 2008 , RefereedSummary: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 ordinary 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...
- Extended $5d$ Seiberg-Witten Theory and Melting Crystal, Toshio Nakatsu, Yui Noma, Kanehisa Takasaki, Nucl. Phys. B808 (2009), 411-440, Nucl. Phys. B808 (2009), 411-440, Jul. 2008 , RefereedSummary: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 ove...
- Integrable Structure of $5d$ $\mathcal{N}=1$ Supersymmetric Yang-Mills and Melting Crystal, Toshio Nakatsu, Yui Noma, Kanehisa Takasaki, Int. J. Mod. Phys. A23 (2008), 2332-2342, Int. J. Mod. Phys. A23 (2008), 2332-2342, Jun. 2008 , RefereedSummary: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.
- Differential Fay identities and auxiliary linear problem of integrable hiearchies, Kanehisa Takasaki, Advanced Studies in Pure Mathematics vol. 61 (2011), 387--441, Advanced Studies in Pure Mathematics vol. 61 (2011), 387--441, Oct. 2007 , RefereedSummary: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,...
- Melting Crystal, Quantum Torus and Toda Hierarchy, Toshio Nakatsu, Kanehisa Takasaki, Commun.Math.Phys.285 (2009), 445-468, Commun.Math.Phys.285 (2009), 445-468, Oct. 2007 , RefereedSummary: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 revea...
- Hamiltonian Structure of PI Hierarchy, Kanehisa Takasaki, SIGMA 3 (2007), 042, 32 pages, SIGMA 3 (2007), 042, 32 pages, Oct. 2006 , RefereedSummary:The string equation of type $(2,2g+1)$ may be thought of as a higher order analogue of the first Painlev\'e equation that corresponds 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, ca...
- Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy, Kanehisa Takasaki, Takashi Takebe, Physica D235 (2007), 109-125, Physica D235 (2007), 109-125, Aug. 2006 , RefereedSummary: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 identit...
- Dispersionless Hirota equations of two-component BKP hierarchy, Kanehisa Takasaki, SIGMA 2 (2006), 057, 22 pages, SIGMA 2 (2006), 057, 22 pages, Apr. 2006 , RefereedSummary: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.
- Radial Loewner equation and dispersionless cmKP hierarchy, Kanehisa Takasaki, Takashi Takebe, Mar. 2006 Summary: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.
- Loewner equations and dispersionless hierarchies, Kanehisa Takasaki, Takashi Takebe, Nankai Tracts Math., Nankai Tracts Math., 10(Differential geometry and physics), 438 - 442, 2006 , RefereedSummary:Reduction of a dispersionless type integrable system (dcmKP hierarchy) to the

radial Loewner equation is presented. - Explicit solutions of the classical Calogero & Sutherland systems for any root system, R. Sasaki, K. Takasaki, J.Math.Phys. 47 (2006) 012701, J.Math.Phys. 47 (2006) 012701, Oct. 2005 , RefereedSummary: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. - $q$-analogue of modified KP hierarchy and its quasi-classical limit, Kanehisa Takasaki, Lett.Math.Phys. 72 (2005) 165-181, Lett.Math.Phys. 72 (2005) 165-181, Dec. 2004 , RefereedSummary: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 ...
- Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions, Takashi Maeda, Toshio Nakatsu, Kanehisa Takasaki, Takeshi Tamakoshi, JHEP 0503 (2005) 056, JHEP 0503 (2005) 056, Dec. 2004 , RefereedSummary: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 o...
- Free Fermion and Seiberg-Witten Differential in Random Plane Partitions, Takashi Maeda, Toshio Nakatsu, Kanehisa Takasaki, Takeshi Tamakoshi, Nucl.Phys. B715 (2005) 275-303, Nucl.Phys. B715 (2005) 275-303, Dec. 2004 , RefereedSummary: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-Wit...
- Elliptic spectral parameter and infinite dimensional Grassmann variety, Kanehisa TAKASAKI, "Infinite dimensional algebras and quantum integrable systems", Progress in Mathematics vol. 237, pp. 169--197 (Birkhauser, Basel/Switzerland, 2005), "Infinite dimensional algebras and quantum integrable systems", Progress in Mathematics vol. 237, pp. 169--197 (Birkhauser, Basel/Switzerland, 2005), Dec. 2003 , RefereedSummary: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 parameters. A clue is the geometry of holomorphic vector bundles over the elliptic curve hidden (or manifest) in the zero-curvature representation.
- Landau-Lifshitz hierarchy and infinite dimensional Grassmann variety, Kanehisa TAKASAKI, Lett. Math. Phys. 67 (2) (2004), 141-152, Lett. Math. Phys. 67 (2) (2004), 141-152, Dec. 2003 , RefereedSummary: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.
- Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger hierarchy, Kanehisa TAKASAKI, J. Math. Sci. Univ. Tokyo 11 (2004), 91--131, J. Math. Sci. Univ. Tokyo 11 (2004), 91--131, Jul. 2003 , RefereedSummary: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 $A_n(z)$, $n = 1,2,...$, 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.
- Integrable systems whose spectral curve is the graph of a function, Kanehisa Takasaki, CRM Proceedings and Lecture Notes vol. 37, pp. 211--222 (AMS, Province, 2004)., CRM Proceedings and Lecture Notes vol. 37, pp. 211--222 (AMS, Province, 2004)., Nov. 2002 , RefereedSummary: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 function $A(\lambda)$. 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.
- Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains, Kanehisa TAKASAKI, Commun. Math. Phys. 241 (1) (2003), 111--142, Commun. Math. Phys. 241 (1) (2003), 111--142, Jun. 2002 , RefereedSummary:A chain of one-dimensional Schr\"odinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlev\'e equations . The $N$-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated ``spectral curve'' and a set of Darboux coordinates called ``spectral Darboux coordinates''. The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.
- An integrable system on the moduli space of rational functions and its variants, Kanehisa Takasaki, Takashi Takebe, Journal of Geometry and Physics 47 (1) (2003), 1--20, Journal of Geometry and Physics 47 (1) (2003), 1--20, Feb. 2002 , RefereedSummary: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.
- Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry, R. Sasaki, K. Takasaki, J.Phys.A34:9533-9554,2001; Erratum-ibid.A34:10335,2001, J.Phys.A34:9533-9554,2001; Erratum-ibid.A34:10335,2001, Sep. 2001 , RefereedSummary: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... - Hierarchy of (2+1)-dimensional nonlinear Schroedinger equation, self-dual Yang-Mills equation, and toroidal Lie algebras, Saburo Kakei, Takeshi Ikeda, Kanehisa Takasaki, Annales Henri Poincare 3 (2002) 817-845, Annales Henri Poincare 3 (2002) 817-845, Jul. 2001 , RefereedSummary: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.
- Hyperelliptic integrable systems on K3 and rational surfaces, Kanehisa TAKASAKI, Phys. Lett. A283 (2001), 201--208., Phys. Lett. A283 (2001), 201--208., Jul. 2000 , RefereedSummary: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.
- Anti-self-dual Yang-Mills equations on noncommutative spacetime, Kanehisa Takasaki, J. Geom. Phys. 37 (4) (2001), 291 - 306., J. Geom. Phys. 37 (4) (2001), 291 - 306., May 2000 , RefereedSummary: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.
- Painleve-Calogero correspondence revisited, Kanehisa TAKASAKI, J. Math. Phys. 42 (3) (2001), 1443-1473., J. Math. Phys. 42 (3) (2001), 1443-1473., Apr. 2000 , RefereedSummary: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 seemingly new. We further extend the correspondence to the higher rank models, and obtain a ``multi-component'' version of the Painlev\'e equations.
- Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation, T. Ikeda, K. Takasaki, Internat. Math. Res. Notices 2001, No. 7, 329-369, Internat. Math. Res. Notices 2001, No. 7, 329-369, Apr. 2000 , RefereedSummary: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 sl_\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 ... - Calogero-Moser Models IV: Limits to Toda theory, S. P. Khastgir, R. Sasaki, K. Takasaki, Prog.Theor.Phys. 102 (1999) 749-776, Prog.Theor.Phys. 102 (1999) 749-776, Jul. 1999 , RefereedSummary: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

perio... - Elliptic Calogero-Moser systems and isomonodromic deformations, Kanehisa TAKASAKI, J. Math. Phys. 40 (11) (1999), 5787-5821, J. Math. Phys. 40 (11) (1999), 5787-5821, May 1999 , RefereedSummary: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.
- Whitham Deformations and Tau Functions in N = 2 Supersymmetric Gauge Theories, Kanehisa Takasaki, Prog.Theor.Phys.Suppl.135:53-74,1999, Prog.Theor.Phys.Suppl.135:53-74,1999, May 1999 , RefereedSummary: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. - Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups, Kanehisa Takasaki, Int.J.Mod.Phys. A15 (2000) 3635-3666, Int.J.Mod.Phys. A15 (2000) 3635-3666, Jan. 1999 , RefereedSummary: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 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

def... - Calogero-Moser Models II: Symmetries and Foldings, A. J. Bordner, R. Sasaki, K. Takasaki, Prog.Theor.Phys. 101 (1999) 487-518, Prog.Theor.Phys. 101 (1999) 487-518, Sep. 1998 , RefereedSummary: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

exist 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'. Fo... - Integrable hierarchies and contact terms in u-plane integrals of topologically twisted supersymmetric gauge theories, Kanehisa Takasaki, Int. J. Mod. Phys. A 14 (7) (1999), 1001-1013., Int. J. Mod. Phys. A 14 (7) (1999), 1001-1013., Mar. 1998 , RefereedSummary:The $u$-plane integrals of topologically twisted $N = 2$ supersymmetric gauge theories generally contain contact terms of nonlocal topological observables. This paper proposes an interpretation of these contact terms from the point of view of integrable hierarchies and their Whitham deformations. This is inspired by Mari\~no and Moore's remark that the blowup formula of the $u$-plane integral contains a piece that can be interpreted as a single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multi-time version without spoiling the modular invariance of the blowup formula. The multi-time tau function is comprised of a Gaussian factor $e^{Q(t_1,t_2,...)}$ and a theta function. The time variables $t_n$ play the role of physical coupling constants of 2-observables $I_n(B)$ carried by the exceptional divisor $B$. The coefficients $q_{mn}$ of the Gaussian part are identified to be the contact terms of these 2-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential.
- Gaudin model, KZ Equation, and isomonodromic problem on torus, Kanehisa TAKASAKI, Lett. Math. Phys. 44 (2) (1998), 143-156., Lett. Math. Phys. 44 (2) (1998), 143-156., Nov. 1997 , RefereedSummary:This paper presents a construction of isospectral problems on the torus. The construction starts from an SU(n) version of the XYZ Gaudin model recently studied by Kuroki and Takebe in the context of a twisted WZW model. In the classical limit, the quantum Hamiltonians of the generalized Gaudin model turn into classical Hamiltonians with a natural $r$-matrix structure. These Hamiltonians are used to build a non-autonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from an elliptic analogue of the KZ equation for the twisted WZW model. Finally, a geometric interpretation of this isomonodromic problem is discussed in the language of a moduli space of meromorphic connections.
- Dual isomonodromic problems and Whitham equations, Kanehisa Takasaki, Lett. Math. Phys. 43 (2) (1998), 123-135., Lett. Math. Phys. 43 (2) (1998), 123-135., May 1997 , RefereedSummary:The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter $\epsilon$. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed, and shown to possess a number of properties similar to the Seiberg- Witten solutions of low energy supersymmetric gauge theories.
- Dispersionless hierarchies, Hamilton-Jacobi theory and twistor correspondences, Partha Guha, Kanehisa Takasaki, J. Geom. Phys. 25 (3-4) (1998), 326-340., J. Geom. Phys. 25 (3-4) (1998), 326-340., May 1997 , RefereedSummary:The dispersionless KP and Toda hierarchies possess an underlying twistorial structure. A twistorial approach is partly implemented by the method of Riemann-Hilbert problem. This is however still short of clarifying geometric ingredients of twistor theory, such as twistor lines and twistor surfaces. A more geometric approach can be developed in a Hamilton-Jacobi formalism of Gibbons and Kodama.
- Spectral curves and Whitham equations in isomonodromic problems of Schlesinger type, K. Takasaki, Asian J.Math. 2 (4) (1998), 1049-1078., Asian J.Math. 2 (4) (1998), 1049-1078., Apr. 1997 , RefereedSummary:It has been known since the beginning of this century that isomonodromic problems --- typically the Painlev\'e transcendents --- in a suitable asymptotic region look like a kind of ``modulation'' of isospectral problem. This connection between isomonodromic and isospectral problems is reconsidered here in the light of recent studies related to the Seiberg-Witten solutions of $N = 2$ supersymmetric gauge theories. A general machinary is illustrated in a typical isomonodromic problem, namely the Schlesinger equation, which is reformulated to include a small parameter $\epsilon$. In the small-$\epsilon$ limit, solutions of this isomonodromic problem are expected to behave as a slowly modulated finite-gap solution of an isospectral problem. The modulation is caused by slow deformations of the spectral curve of the finite-gap solution. A modulation equation of this slow dynamics is derived by a heuristic method. An inverse period map of Seiberg-Witten type turns out to give general solutions of this modulation equation. This construction of general solution also reveals the existence of deformations of Seiberg-Witten type on the same moduli space of spectral curves. A prepotential is also constructed in the same way as the prepotential of the Seiberg-Witten theory.
- Isomonodromic deformations and supersymmetric Yang-Mills theories, K. Takasaki, T. Nakatsu, Int. J. Mod. Phys. A 11 (38) (1996), 5505-5518., Int. J. Mod. Phys. A 11 (38) (1996), 5505-5518., Mar. 1996 , RefereedSummary:Seiberg-Witten solutions of four-dimensional supersymmetric gauge theories possess rich but involved integrable structures. The goal of this paper is to show that an isomonodromy problem provides a unified framework for understanding those various features of integrability. The Seiberg-Witten solution itself can be interpreted as a WKB limit of this isomonodromy problem. The origin of underlying Whitham dynamics (adiabatic deformation of an isomonodromy problem), too, can be similarly explained by a more refined asymptotic method (multiscale analysis). The case of $N=2$ SU($s$) supersymmetric Yang-Mills theory without matter is considered in detail for illustration. The isomonodromy problem in this case is closely related to the third Painlev\'e equation and its multicomponent analogues. An implicit relation to $t\tbar$ fusion of topological sigma models is thereby expected.
- Quasi-classical limit of KP hierarchy, W-symmetries and free fermions, Kanehisa Takasaki, Takashi Takebe, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 235(Differ. Geom. Gruppy Li i Mekh. 15-2), 295 - 303, 1996 , RefereedSummary:This paper deals with the dispersionless KP hierarchy from the point of view

of quasi-classical limit. Its Lax formalism, W-infinity symmetries and general

solutions are shown to be reproduced from their counterparts in the KP

hierarchy in the limit of $\hbar \to 0$. Free fermions and bosonized vertex

operators play a key role in the description of W-infinity symmetries and

general solutions, which is technically very similar to a recent free fermion

formalism of $c=1$ matrix models.

translation in J. Math. Sci. (New York) 94 (1999), no. 4, 1635--1641 - Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory, T. Nakatsu, K. Takasaki, Mod. Phys. Lett. A11 (2) (1996), 157-168., Mod. Phys. Lett. A11 (2) (1996), 157-168., Sep. 1995 , RefereedSummary:The exact solution of $N=2$ supersymmetric $SU(N)$ Yang-Mills theory is studied in the framework of the Whitham hierarchies. The solution is identified with a homogeneous solution of a Whitham hierarchy. This integrable hierarchy (Whitham-Toda hierarchy) describes modulation of a quasi-periodic solution of the (generalized) Toda lattice hierarchy associated with the hyperelliptic curves over the quantum moduli space. The relation between the holomorphic pre-potential of the low energy effective action and the $\tau$ function of the (generalized) Toda lattice hierarchy is also clarified.
- Toda lattice hierarchy and generalized string equations, K. Takasaki, Commun. Math. Phys. 181 (1) (1996), 131-156., Commun. Math. Phys. 181 (1) (1996), 131-156., Jun. 1995 , RefereedSummary:String equations of the $p$-th generalized Kontsevich model and the compactified $c = 1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c \le 1$ string models is discussed in this new framework.
- Quantum and Classical Aspects of Deformed $c=1$ Strings., T. NAKATSU, K. TAKASAKI, S. TSUJIMARU, Nucl.Phys. B443 (1995) 155-200, Nucl.Phys. B443 (1995) 155-200, Jan. 1995 , RefereedSummary:The quantum and classical aspects of a deformed $c=1$ matrix model proposed by

Jevicki and Yoneya are studied. String equations are formulated in the

framework of Toda lattice hierarchy. The Whittaker functions now play the role

of generalized Airy functions in $c<1$ strings. This matrix model has two

distinct parameters. Identification of the string coupling constant is thereby

not unique, and leads to several different perturbative interpretations of this

model as a string theory. Two such possible interpretations are examined. In

both cases, the classical limit of the string equations, w... - Symmetries and tau function of higher dimensional dispersionless integrable hierarchies, K. Takasaki, J. Math. Phys. 36 (1995), 3574-3607., J. Math. Phys. 36 (1995), 3574-3607., Jul. 1994 , RefereedSummary:A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional ``phase space'' variables $(k,x)$ of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions compactified to a two (or any even) dimensional torus. Integrability of this hierarchy and the existence of an infinite dimensional of ``additional symmetries'' are ensured by an underlying twistor theoretical structure (or a nonlinear Riemann-Hilbert problem). An analogue of the tau function, whose logarithm gives the $F$ function (``free energy'' or ``prepotential'' in the contest of matrix models and topological conformal field theories), is constructed. The infinite dimensional symmetries can be extended to this tau (or $F$) function. The extended symmetries, just like those of the dispersionless KP hierarchy, obey an anomalous commutation relations.
- Integrable hierarchies and dispersionless limit, K. Takasaki, T. Takebe, Reviews in Mathematical Physics 7 (1995), 743-808., Reviews in Mathematical Physics 7 (1995), 743-808., May 1994 , RefereedSummary:Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quatization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessary technical details.
- Dispersionless Toda hierarchy and two-dimensional string theory, Kanehisa Takasaki, Commun.Math.Phys. 170 (1995) 101-116, Commun.Math.Phys. 170 (1995) 101-116, Mar. 1994 , RefereedSummary:The dispersionless Toda hierarchy turns out to lie in the heart of a recently

proposed Landau-Ginzburg formulation of two-dimensional string theory at

self-dual compactification radius. The dynamics of massless tachyons with

discrete momenta is shown to be encoded into the structure of a special

solution of this integrable hierarchy. This solution is obtained by solving a

Riemann-Hilbert problem. Equivalence to the tachyon dynamics is proven by

deriving recursion relations of tachyon correlation functions in the machinery

of the dispersionless Toda hierarchy. Fundamental ingredients of the

... - Nonabelian KP hierarchy with Moyal algebraic coefficients, Kanehisa Takasaki, Journal of Geometry and Physics 14 (1994), 332-364., Journal of Geometry and Physics 14 (1994), 332-364., May 1993 , RefereedSummary:A higher dimensional analogue of the KP hierarchy is presented. Fundamental

constituents of the theory are pseudo-differential operators with Moyal

algebraic coefficients. The new hierarchy can be interpreted as large-$N$ limit

of multi-component ($\gl(N)$ symmetric) KP hierarchies. Actually, two different

hierarchies are constructed. The first hierarchy consists of commuting flows

and may be thought of as a straightforward extension of the ordinary and

multi-component KP hierarchies. The second one is a hierarchy of noncommuting

flows, and related to Moyal algebraic deformations of selfdua... - Integrable hierarchy underlying topological Landau-Ginzburg models of D-type, Kanehisa Takasaki, Lett. Math. Phys. 29 (1993), 111-121., Lett. Math. Phys. 29 (1993), 111-121., May 1993 , RefereedSummary:A universal integrable hierarchy underlying topological Landau-Ginzburg

models of D-tye is presented. Like the dispersionless Toda hierarchy, the new

hierarchy has two distinct (``positive" and ``negative") set of flows. Special

solutions corresponding to topological Landau-Ginzburg models of D-type are

characterized by a Riemann-Hilbert problem, which can be converted into a

generalized hodograph transformation. This construction gives an embedding of

the finite dimensional small phase space of these models into the full space of

flows of this hierarchy. One of flat coordinates in the smal... - Quasi-classical limit of Toda hierarchy and W-infinity symmetries, K. Takasaki, T. Takebe, Lett. Math. Phys. 28 (1993), 165-176., Lett. Math. Phys. 28 (1993), 165-176., Jan. 1993 , RefereedSummary:Previous results on quasi-classical limit of the KP hierarchy and its W-infinity symmetries are extended to the Toda hierarchy. The Planck constant $\hbar$ now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions and tau function, are redefined. $W_{1+\infty}$ symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo-Kashiara-Miwa vertex operators. These symmetries are contracted to $w_{1+\infty}$ symmetries of the dispersionless hierarchy through their action on the tau function. (A few errors in the earlier version is corrected.)
- Quasi-classical limit of BKP hierarchy and W-infinity symmeties, Kanehisa Takasaki, Lett.Math.Phys. 28 (1993) 177-186, Lett.Math.Phys. 28 (1993) 177-186, Jan. 1993 , RefereedSummary:Previous results on quasi-classical limit of the KP and Toda hierarchies are

now extended to the BKP hierarchy. Basic tools such as the Lax representation,

the Baker-Akhiezer function and the tau function are reformulated so as to fit

into the analysis of quasi-classical limit. Two subalgebras $\WB_{1+\infty}$

and $\wB_{1+\infty}$ of the W-infinity algebras $W_{1+\infty}$ and

$w_{1+\infty}$ are introduced as fundamental Lie algebras of the BKP hierarchy

and its quasi-classical limit, the dispersionless BKP hierarchy. The quantum

W-infinity algebra $\WB_{1+\infty}$ emerges in symmetries of t...

(Comments: The contents of this paper concerning the BKP hierarchy itself turned out to be completely wrong, though the results on the dispersionless limit seem to remain intact. The main source of errors is (14), which should read M* = - M + L. This led to a wrong definition, (53), of the generators Aij of symmetries. A correct reformulation of the results of this paper is presented by M.-H. Tu, Lett. Math. Phys. 81 (2007), 91-105 (arXiv:nlin.SI/0611053). ) - Dressing operator approach to Moyal algebraic deformation of selfdual gravity, Kanehisa Takasaki, J. Geom. Phys. 14 (1994) 111-120, J. Geom. Phys. 14 (1994) 111-120, Dec. 1992 , RefereedSummary:Recently Strachan introduced a Moyal algebraic deformation of selfdual

gravity, replacing a Poisson bracket of the Plebanski equation by a Moyal

bracket. The dressing operator method in soliton theory can be extended to this

Moyal algebraic deformation of selfdual gravity. Dressing operators are defined

as Laurent series with coefficients in the Moyal (or star product) algebra, and

turn out to satisfy a factorization relation similar to the case of the KP and

Toda hierarchies. It is a loop algebra of the Moyal algebra (i.e., of a

$W_\infty$ algebra) and an associated loop group that charact... - Quasi-classical limit of KP hierarchy, W-symmetries and free fermions, K. Takasaki, T. Takebe, V.E. Matveev (ed), Proceedings of Lobachevsky Semester of Euler International Institute, 1992, St. Petersburg. Zapiski Nauchnykh Seminarov POMI 235 (1996), 295 - 303., V.E. Matveev (ed), Proceedings of Lobachevsky Semester of Euler International Institute, 1992, St. Petersburg. Zapiski Nauchnykh Seminarov POMI 235 (1996), 295 - 303., Jul. 1992 Summary:This paper deals with the dispersionless KP hierarchy from the point of view of quasi-classical limit. Its Lax formalism, W-infinity symmetries and general solutions are shown to be reproduced from their counterparts in the KP hierarchy in the limit of $\hbar \to 0$. Free fermions and bosonized vertex operators play a key role in the description of W-infinity symmetries and general solutions, which is technically very similar to a recent free fermion formalism of $c=1$ matrix models.
- Volume-preserving diffeomorphisms in integrable deformations of selfdual gravity, Kanehisa Takasaki, Phys.Lett. B285 (1992) 187-190, Phys.Lett. B285 (1992) 187-190, Mar. 1992 , RefereedSummary:A group of volume-preserving diffeomorphisms in 3D turns out to play a key

role in an Einstein-Maxwell theory whose Weyl tensor is selfdual and whose

Maxwell tensor has algebraically general anti-selfdual part. This model was

first introduced by Flaherty and recently studied by Park as an integrable

deformation of selfdual gravity. A twisted volume form on the corresponding

twistor space is shown to be the origin of volume-preserving diffeomorphisms.

An immediate consequence is the existence of an infinite number of symmetries

as a generalization of $w_{1+\infty}$ symmetries in selfdual gra... - SDiff(2) Toda equation --- hierarchy, tau function and symmetries, K. Takasaki, T. Takebe, Lett. Math. Phys. 23 (1991), 205-214., Lett. Math. Phys. 23 (1991), 205-214., Dec. 1991 , RefereedSummary:A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum Einstein equation and its hyper-K\"ahler version, however now based upon a symplectic structure and the group SDiff(2) of area preserving diffeomorphisms on a cylinder $S^1 \times \R$. An analogue of the Toda lattice tau function is introduced. The existence of hidden SDiff(2) symmetries are derived from a Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function turn out to have commutator anomalies, hence give a representation of a central extension of the SDiff(2) algebra.
- Structure and duality of ${\cal D}$-modules related to KP hierarchy, T. Miyajima, A. Nakayashiki, K. Takasaki, J. Math. Soc. Japan 43 (1991), 751-773., J. Math. Soc. Japan 43 (1991), 751-773., 1991 , Refereed
- Integrable systems in gauge theory, K\"ahler geometry and super KP hierarchy --- symmetries and algebraic point of view, K. Takasaki, Proc. International Congress of Mathematicians, Kyoto, 1990, pp. 1205-1214 (Springer-Verlag, 1991)., Proc. International Congress of Mathematicians, Kyoto, 1990, pp. 1205-1214 (Springer-Verlag, 1991)., 1991
- Symmetries of the super KP hierarchy, K. Takasaki, Lett. Math. Phys. 17 (1989), 351-357., Lett. Math. Phys. 17 (1989), 351-357., 1989 , Refereed
- Geometry of universal Grassmann manifold from algebraic point of view, K. Takasaki, Reviews in Math. Phys. 1 (1989), 1-46., Reviews in Math. Phys. 1 (1989), 1-46., 1989 , Refereed
- An infinite number of hidden variables in hyper-K\"{a}hler metrics, K. Takasaki, J. Math. Phys. 30 (1989), 1515-1521., J. Math. Phys. 30 (1989), 1515-1521., 1989 , Refereed
- Integrable systems as deformations of ${\cal D}$-modules, K. Takasaki, Proc. Symp. Pure Math. 49 , Part I, pp. 143-168 (American Mathematical Society, 1989)., Proc. Symp. Pure Math. 49 , Part I, pp. 143-168 (American Mathematical Society, 1989)., 1989 , Refereed
- Aspects of integrability in self-dual Einstein metrics and related Equations, K. Takasaki, Publ. RIMS, Kyoto Univ., 22 (1986), 949-990., Publ. RIMS, Kyoto Univ., 22 (1986), 949-990., 1986 , RefereedSummary:Comments: The last paragraph of section 5 of this paper is wrong. Namely, as opposed to the statement therein, the time evolution can be defined only on the orbit of a loop group action, not on the whole Grassmannian introduced therein. An improved reformulation can be found in the appendix of "Symmetries of hyper-K\"{a}hler (or Poisson gauge field) hierarchy", J. Math. Phys. 31 (1990), 1877-1888.
- A new approach to the self-dual Yang-Mills equations II, K. Takasaki, Saitama Math. J. 3 (1985), 11-40., Saitama Math. J. 3 (1985), 11-40., 1985 Summary:Comments: The last paragraph of section 2.6 is wrong. Namely, a rational initial data can give rise to non-rational solution.
- A new approach to the self-dual Yang-Mills equations, K. Takasaki, Commun. Math. Phys. 94 (1984), 35-59., Commun. Math. Phys. 94 (1984), 35-59., 1984 , Refereed
- Initial value problem for the Toda lattice hierarchy, K. Takasaki, Advanced Studies in Pure Math. 4, pp. 139-163 (Kinokuniya, Tokyo, 1984)., Advanced Studies in Pure Math. 4, pp. 139-163 (Kinokuniya, Tokyo, 1984)., 1984 , Refereed
- Toda lattice hierarchy, K. Ueno, K. Takasaki, Advanced Studies in Pure Math. 4, pp. 1-94 (Kinokuniya, Tokyo, 1984)., Advanced Studies in Pure Math. 4, pp. 1-94 (Kinokuniya, Tokyo, 1984)., 1984 , RefereedSummary:Comments: 1) The contents of section 2.4 are wrong except that the tau functions for s = 0,1 can be defined and satisfy the bilinear equaztions of the 2-component BKP herarrchy. 2) The second line of (A.41) should be multiplied by the inverse of lambda, cf. (A.44).
- On the structure of solutions to the self-dual Yang-Mills equations, K. Takasaki, Proc. Japan Acad. 59A (1983), 418-421., Proc. Japan Acad. 59A (1983), 418-421., 1983
- Toda lattice hierarchy II, Toda lattice hierarchy I, Proc. Japan Acad. 59A (1983), 215-218., Proc. Japan Acad. 59A (1983), 215-218., 1983
- Toda lattice hierarchy I, K. Ueno, K. Takasaki, Proc. Japan Acad. 59A (1983), 167-170., Proc. Japan Acad. 59A (1983), 167-170., 1983
- A class of solutions to the self-dual Yang-Mills equations, K. Takasaki, Proc. Japan Acad. 59A (1983), 308-311., Proc. Japan Acad. 59A (1983), 308-311., 1983
- Singular Cauchy problems for a class of weakly hyperbolic differential operators, K. Takasaki, Comm. Partial Differential Equations 7 (1982), 1151-1188., Comm. Partial Differential Equations 7 (1982), 1151-1188., 1982 , Refereed

- 線形代数とネットワーク, 高崎金久, 単著, 2017 , 9784535788299
- 学んでみよう! 記号論理, 高崎金久, 単著, 日本評論社, 2014 08 , 4535787603
- 復刊 可積分系の世界 ―戸田格子とその仲間―, 高崎 金久, 単著, 共立出版, 2013 07 , 4320110420
- 線形代数と数え上げ, 高崎 金久, 単著, 日本評論社, 2012 06 , 4535786801
- 現代数理科学事典 第2版, 現代数理科学事典編集委員会, 広中 平祐, 甘利 俊一, 伊理 正夫, 巌佐 庸, 楠岡 成雄, 一松 信, 室田 一雄, 和達 三樹, 分担執筆, ソリトン, 丸善, 2009 12 , 462108125X
- 岩波数学辞典, 日本数学会, 相川弘明, 日本数学会, 分担執筆, ソリトン, 岩波書店, 2007 03 , 4000803093
- Elliptic integrable systems, Masatoshi Noumi, Kanehisa Takasaki, 共編者(共編著者), Department of Mahtematics, Faculty of Science, Kobe University, 2005 , 4907719183
- 可積分系の世界―戸田格子とその仲間, 高崎 金久, 単著, 共立出版, 2001 03 , 4320016696
- コマの幾何学―可積分系講義, 高崎 金久, 単訳, 共立出版, 2000 03 , 4320016556

- Equivariant Gromov-Witten theory of CP1 and equivariant Toda hierarchy, Kanehisa Takasaki, 2020 Annual meeting of Mathematical Society of Japan, 2020 03 19 Summary:The annual meeting was cancelled because of the COVID-19 epidemic, but the talk was authorized to be delivered.
- Integrable structures of cubic Hodge integrals, Kanehisa TAKASAKI, Mathematical structures of integrable systems, its deepening and expansion, 2019 09 10 , 招待有り, Research Inistitute of Mathematics, Kyoto UniversitySummary:The cubic Hodge integrals are intersection numbers with Hodge classes on the moduli space of complex stable curves. These numbers are labelled by one, two or three integer partitions, and closely related to the topological vertex of topological string theory. A combinatorial expression was found by researches arround 2003. Moreover, the generating functions of these numbers in the one- and two-partition cases were shown to be tau functions of the KP and Toda hierarchies. In this talk, we reconsider the two-partition case, and point out that the tau function is related to the Volterra- and KdV-type integrable hierarchies when a parameter of the Hodge integrals takes values in a set of rational numbers. This research is based on collaboration with Toshio Nakatsu (Setsunan University).
- Volterra-type hierarchies for specialized hypergeometric tau functions, Kanehisa TAKASAKI, China-Japan Joint Workshop on Integrable Systems 2019, 2019 08 20 , 招待有り, Daisuke Takahashi (Waseda University) et al.Summary:An important example of Orlov and Scherbin's hypergeometric tau functions is the generating function of the double Hurwitz numbers introduced by Okounkov. Specialization of the second set of the 2D Toda time variables to particular values yields generating functions of the single Hurwitz numbers and the cubic Hodge integrals. These specialized hypergeometric tau functions turn out to be related to the Bogoyavlensky-Itoh hierarchies (aka the hugry Lotka-Volterra hierarchy). The Volterra-type hierarchies are derived as reductions of the lattice KP hierarchy.
- Integrable structures of cubic Hodge integrals, Kanehisa TAKASAKI, 2nd IBS-CGP Workshop on integrable systems and applications, 2019 05 07 , 招待有り, Alexander AleksandrovSummary:Around 2003, C.-C. Mellissa Liu, Kefeng Liu and Jian Zhou presented a combinatorial description of two-partition cubic Hodge integrals that generalizes the Marino-Vafa formula for one-partition cubic Hodge integrals. Moreover, Zhou pointed out that generating functions of those combinatorial expressions become tau functions of the (one-and two-component) KP and 2D Toda hierarchies. I reconsider these tau functions in the case where a parameter of the cubic Hodge integrals is specialized to a set of particular discrete values. The tau functions therein turn out to be related to the generalized KdV hierarchies or the hungry Lotka-Volterra (aka Bogoyavlensky-Itoh) hierarchies depending on the value of the parameter. This talk is based on collaboration with Toshio Nakatsu.
- Quantum mirror curves of topological string theory, Kanehisa TAKASAKI, 72nd Encounter with Mathematics, 2019 01 12 , 招待有り, Department of Mathematics, Chuo University
- Toda and q-Toda equations for Nekrasov partition functions, Kanehisa TAKASAKI, SIDE13, 2018 11 13 , 招待有り, SIDE13 organizing committeeSummary:Some results on Toda-type equations and Nekrasov partitions functions are presented.
- Hurwitz numbers and integrable hierarchy of Volterra type, Kanehisa TAKASAKI, AIMS Conferencer 2018, 2018 07 07 , 招待有り, American Institute of Mathematical SciencesSummary:This talk presents recent results on integrable structures in generating functions of Hurwitz numbers. Ref: arXiv:1807.00085T.
- 3D Young diagrams and Gromov-Witten theory of CP1, Kanehisa TAKASAKI, Seminar of IBS Center of Geometry and Physics, 2018 03 27 , 招待有り, IBS Center of Geometry and PhysicsSummary:The melting crystal model is a model of statistical mechanics for random 3D Young diagrams. The partition function of this model may be thought of as a $q$-deformation of the generating function of stationary Gromov-Witten invariants of $\mathbf{C}\mathbf{P}^1$ studied by Okounkov and Pandharipande. We consider these generating functions in the perspectives of integrable systems and quantum spectral curves. A main issue is how to capture the limit to the Gromov-Witten theory of $\mathbf{C}\mathbf{P}^1$ as $q \to 1$.
- 3D Young diagrams and Gromov-Witten theory of CP1, Kanehisa TAKASAKI, The 13th Kagoshima Algebra-Analysis- Geometry Seminar, 2018 02 16 , 招待有り, Shoji Yokura, Kiyoshi Takeuchi, Shunichi Kimura, Masaaki Murakami, Hiroyuki NakaokaSummary:The melting crystal model is a model of statistical mechanics for random 3D Young diagrams. The partition function of this model may be thought of as a q-deformation of the generating function of stationary Gromov-Witten invariants of CP1 studied by Okounkov and Pandharipande. We consider these generating functions in the perspectives of integrable systems and quantum spectral curves. A main issue is how to capture the limit to the Gromov-Witten theory of CP1 as q → 1.
- Melting crystal model and its 4D limit, Kanehisa TAKASAKI, Physics and Mathematics of Nonlinear Phenomena, 2017 06 19 , 招待有り, B. Konopelchenko et al.Summary:The melting crystal model is a toy model of Nekrasov's instanton partition functions for 5D supersymmetric gauge theories on R4 x S1. Its deformation by an infinite set of external potentials is known to become a tau function of the KP hierarchy. Its 4D counterpart Z4D(t) is known to coincide with a generating function of Gromov-Witten theory of CP1. We formulate a precise prescription of 4D limit of the deformed partition as the radius of S1 tends to 0. We can thereby re-derive the quantum spectral curve of CP1 theory (recently constructed by Dunin-Barkowski et al.) from the the quantum spectral curve of the melting crystal model. We further show that bilinear equations of the Fay type survive the 4D limit. This leads to yet another proof of the fact (proved by Getzler, Dubrovin-Zhang and Milanov by geometric methods) that Z4D(t) is a tau function of the KP hierarchy.
- Quantum spectral curve of melting crystal model and its 4D limit, Kanehisa TAKASAKI, Geometry, Analysis and Mathematical Physics, 2017 02 16 , 招待有り, Hiroshi Iritani (Kyoto University) Kohei Iwaki (Nagoya University) Yukiko Konishi (Kyoto University) Atsushi Takahashi (Osaka University)Summary:The melting crystal model is a statistical model of 3D Young diagrams. Its partition function can be identified with the instanton partition function of 5D N = 1 supersymmetric U(1) Yang-Mills theory on R^4 × S^1. Its deformation by a set of external potentials is known to be a tau function of the KP hierarchy. We can derive an associated quantum spectral curve by the method of q-difference Kac-Schwarz operators (arxiv:1609.00882). In the 4D limit where the radius of S^1 tends to 0, this quantum curve turns into the quantum spectral curve of Gromov-Witten theory on CP^1 derived by Dunin-Barkowski, Mulase, Norbury, Popolitov and Shadrin (arxiv:1312.5336). The partition function itself, too, has a natural limit to a generating function of the Gromov-Witten invariants. We can thus reproduce the results of Dunin-Barkowski et al. from a different approach.
- Quantum mirror curve of closed topological vertex and Kac-Schwarz operator of $q$-difference type, Kanehisa TAKASAKI, MSJ Autumn Meeting 2016, 2016 09 16 , Mathematical Society of JapanSummary:We show that the quantum mirror curve of closed topological vertex can be interpreted as Kac-Schwarz operators of $q$-difference type. This is also the case for topological string theory in strip geometry including the resolved conifold.
- Integrable hierarchies in melting crystal models and topological vertex, Kanehisa TAKASAKI, Infinite Analysis 16 Summer School, 2016 08 29 , 招待有り, Rei Inoue (Chiba) Atsuo Kuniba (Tokyo) Masato Okado (Osaka City) Tomoki Nakanishi (Nagoya) Yoshihiro Takeyama (Tsukuba)Summary:The melting crystal models are statistical models of random partitions that stems from the instanton sum of a 5D supersymmetric gauge theory. The topological vertex is a diagrammatic method for constructing the partition functions (or amplitudes) of topological string theory on non-compact toric Calabi-Yau threefolds. I will review aspects of integrable hierarchies hidden in these combinatorial models of mathematical physics. Main tools are Schur functions, free fermions, a 2D quantum torus algebra and quantum dilogarithmic functions. The contents of these lectures are based on joint work with Toshio Nakatsu.
- Integrable hierarchies in melting crystal models and topological vertex, Kanehisa TAKASAKI, KIAS Workshop on integrable systems and related topics, 2016 06 21 , 招待有り, Jinsung Park (KIAS)Summary:The melting crystal models are statistical models of random partitions that stems from the instanton sum of a 5D supersymmetric gauge theory. The topological vertex is a diagrammatic method for constructing the partition functions, or amplitudes, of topological string theory on non-compact toric Calabi-Yau threefolds. I will review aspects of integrable hierarchies hidden in these combinatorial models of mathematical physics.
- Matrix factorization and reductions of Kostant-Toda hierarchy, Kanehisa TAKASAKI, Workshop on Applied Analysis, 2016 05 20 , 招待有りSummary:Gekhtman, Shapiro and Veinstein constructed Toda-type integrable systems on the double Bruhat cells G^{u,v} of Coxeter elements u,v of S_n. This is an application of Berenstein, Fomin and Zelevinsky's matrix factorization, and presents a generalization of Feybusovich and Gekhtman's ``elementary Toda orbits''. I review elementary part of this topic.
- Topological vertex and quantum mirror curves, Kanehisa TAKASAKI, Rikkyo MathPhys 2016, 2016 01 09 , 招待有り, Rikkyo Research Center for Mathematical PhysicsSummary:Topological vertex is a diagrammatic method for constructing the partition functions of topological string theory on non-compact toric Calabi-Yau threefolds. We present a few cases, including the so called ``closed topological vertex'', where open string amplitudes can be computed explicitly by this method. These expressions of open string amplitudes can be used to derive ``quantum mirror curves''. This is a joint work with Toshio Nakatsu.
- Topological vertex and quantum mirror curves, Kanehisa TAKASAKI, Quantization of Spectral Curves, 2015 11 06 , 招待有り, Osaka City Universitiy Advanced Mathematical InstituteSummary:Topological vertex is a diagrammatic method for constructing the partition functions (or amplitudes) of topological string theory on non-compact toric Calabi-Yau threefolds. This talk is focussed on two cases where open string amplitudes can be computed explicitly by this method. One is the case of ``on-strip'' geometry, and the other is the so called ``closed topological vertex''. For both cases, generating functions of open string amplitudes turn out to satisfy a linear q-difference equation. This equation may be thought of as quantization of the equation of the mirror curve.
- Open string amplitudes of closed topological vertex, Kanehisa TAKASAKI, Mathematical Society of Japan Autumn Meeting, 2015 09 13 , Mathematical Society of JapanSummary:We compute ppen string amplitudes of closed topological vertex by the method of topological vertex. Single-variate generating functions of these amplitudes turn out to satisfty a q-difference equation. This equation may be thought of as a quantization of the mirror curve.
- Integrable structure of various melting crystal models, Kanehisa TAKASAKI, Recent Progress of Integrable Systems, 2015 04 12 , 招待有り, Academia Sinica, TaiwanSummary:My recent work on integrable structure of melting crystal models is reviewd. The simplest model is a statistical model of random 3D Young diagrams. A variant of this model originates in topological string theory on the resolved conifold. These two models are related to the 1D Toda hierarchy and the Ablowitz- Ladik hierarchy, respectively. These results can be further extended to ``orbifold'' models. The relevant integrable systems are particular reductions of the 2D Toda hierarchy.
- Integrable structure of various melting crystal models, Kanehisa TAKASAKI, Curves, Moduli and Integrable Systems', 2015 02 18 , 招待有り, Atsushi NakayashikiSummary:My recent work on integrable structure of melting crystal models is reviewd. The simplest model is a statistical model of random 3D Young diagrams. A variant of this model originates in topological string theory on the resolved conifold. These two models are related to the 1D Toda hierarchy and the Ablowitz- Ladik hierarchy, respectively. These results can be further extended to ``orbifold'' models. The relevant integrable systems are particular reductions of the 2D Toda hierarchy.
- Twistor theory of gravity and integrable systems, Kanehisa Takasaki, 16th Singularity workshop, 2015 01 10 , 招待有り, Singularity WorkshopSummary:I. Aspects of anti-selfduality of four dimensional spacetimes 1. Levi-Civita connection in terms of moving frames 2. Spinor bundles and spinor connecions 3. Anti-selfdual spacetimes II. Construction and inverse-construction of twistor spaces 1. Twistor space of anti-selfdual spacetime 2. Twistor space of right-flat spacetime 3. Generation of right-flat spacetime 4. Example of generation III. Dimensional reductions and integrable systems 1. Example of dimensional reduction: dispersionless Toda equation 2. Example of dimensional reduction: dispersionless KP equation 3. Three-dimensional Einstein-Weyl spaces
- Integrable structure of melting crystal models, Kanehisa TAKASAKI, Various aspects of nonlinear mathematical models: continuous, discrete, ultra-discrete, and beyond, 2014 08 08 , 招待有り, IMI, Kyushu University,Summary:The melting crystal model is known as a statistical model of 3D Young diagrams, and generalized in various ways in the context of toplogical string theory and topological invariants. An integrable structure of the Toda type is hidden in this model. Recently a variant of this model is pointed out to be to related to the Ablowitz-Ladik (or relativistic Toda) hierarchy.
- Generalized Ablowitz-Ladik hierarchy in topological string theory, Kanehisa TAKASAKI, Mathematical Society of Japan Annual meeeting, 2014 03 13 , Mathematical Society of JapanSummary:A generalization of the Ablowitz-Ladik hierarchy is shown to be an integrable structure of topological string theory on a special class of toric Calabi-Yau threefolds called generalized conifolds.
- Historical aspects of linear algebra, Kanehisa Takasaki, 2014 03 06 , Human and Envronmental Studdies, Kyoto UniversitySummary:My last lecture at Kyoto University. Presents historical aspects of linear algebra.
- Introduction to Sato's theory on soliton equations, Kanehisa TAKASAKI, Around Sato's Theory on Soliton Equations, 2013 12 14 , 招待有り, Atsushi NakayashikiSummary:This talk is a review of Sato’s theory on soliton equations. Most part will be focused on the progress in the early 80’s where the KP hierarchy was invented and used as a master system of various soliton equations. Sato’s most remarkable discovery therein was the fact that the KP hierarchy can be converted to a dynamical system on an infinite dimensional Grassmann manifold. This led, for example, to a beautiful explanation of the origin of bilinear equations for the tau function. I will review these well known topics as well as Sato’s nowadays almost forgotten approach by D-modules.
- Melting crystal models and integrable hierarchies, Kanehisa TAKASAKI, Summer School on Integrability in Quantum and Statistical Systems, 2013 08 07 , 招待有り, National Taiwan Univerisity
- Modified melting crystal model and Ablowitz-Ladik hierarchy, Kanehisa TAKASAKI, Physics and Mathematic of Nonlinear Phenomena, 2013 06 24 , 招待有り, Research group on Nonlinear Physics of Università del SalentoSummary:This talk presents my recent work on the integrable structure of a modified melting crystal model (arXiv:1208.4497 [math-ph], arXiv:1302.6129 [math-ph]). Contents: 1.Melting crystal model --- 3D Young diagram, plane partitions, partition function, diagonal slicing, partial sums, final answer 2.Integrable structure in deformed models --- undeformed models, deformation by external potentials, summary of previous result, summary of new result 3. Fermionic approach to partition functions --- fermions, fermionic representation of partition functions, previous result, new result, technical clue 4. Integrable structure in Lax formalism --- Lax formalism of 2D Toda hierarchy, reduction to 1D Toda hierarchy, reduction to Ablowitz-Ladik hierarchy, result, technical clue
- Melting crystal model and Ablowitz-Ladik hierarchy, Kanehisa TAKASAKI, Mathematical Society of Japan Anual Meeting, 2013 03 22 , Mathematical Society of JapanSummary:I report a result on a melting crytal model that originates in topological string amplitudes of the resolved conifold. I show, for both the tau function and the Lax formalism, that this model correspond to a special solution of the Ablowitz-Ladik hierarchy (or, equivalently, the Ruijsenaars-Toda hierarchy).
- Integrable structure of modified melting crystal model, Kanehisa TAKASAKI, Integrability in Gauge and String Theory, 2012 08 20 , Institut für Theoretische PhysikSummary:Our previous work on a hidden integrable structure of the melting crystal model (the $U(1)$ Nekrasov function) is extended to a modified crystal model. As in the previous case, ``shift symmetries'' of a quantum torus algebra plays a central role. With the aid of these algebraic relations, the partition function of the modified model is shown to be a tau function of the 2D Toda hierarchy. We conjecture that this tau function belongs to a class of solutions (the so called Toeplitz reduction) related to the Ablowitz-Ladik hierarchy.
- Combinatorial properties of toric topological string partition function, Kanehisa TAKASAKI, Algebraic combinatorics related to Young diagrams and statistical physics, 2012 08 09 , 招待有り, RIMS, Kyoto UniversitySummary:The notion of topological vertex and the construction of topological string partition functions on local toric Calabi-Yau 3-folds are reviewed. Implications of an explicit formula of partition functions for the generalized conifolds are considered. Generating functions of part of the partition functions are shown to be tau functions of the KP hierarchy. The associated Baker-Akhiezer functions play the role of wave functions, and satisfy $q$-difference equations. These $q$-difference equations represent the quantum mirror curves conjectured by Gukov and Sulkowski.
- Non-degenerate solutions of universal Whitham hierarchy, Kanehisa TAKASAKI, 7th International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory, 2011 04 05 , 招待有り, Institute for Mathematics and Computer ScienceSummary: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.
- Toda tau functions with quantum torus symmetries, Kanehisa TAKASAKI, 2011 03 30 , 招待有り, Department of Mathematics, University of California, DavisSummary: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.

- Cubic Hodge integrals and integrable hierarchies of Volterra type, Kanehisa Takasaki, 2019 09 , 招待有り, http://arxiv.org/abs/1909.13095v1Summary:A tau function of the 2D Toda hierarchy can be obtained from a generating

function of the two-partition cubic Hodge integrals. The associated Lax

operators turn out to satisfy an algebraic relation. This algebraic relation

can be used to identify a reduced system of the 2D Toda hierarchy that emerges

when the parameter $\tau$ of the cubic Hodge integrals takes a special value.

Integrable hierarchies of the Volterra type are shown to be such reduced

systems. They can be derived for positive rational values of $\tau$. In

particular, the discrete series $\tau = 1,2,\ldots$ correspond to the Volterra

lattice and its hungry generalizations. This provides a new explanation to the

integrable structures of the cubic Hodge integrals observed by Dubrovin et al.

in the perspectives of tau-symmetric integrable Hamiltonian PDEs. - Integrable systems and Calabi-Yau varieties, Kanehisa TAKASAKI, Suri Kagaku No. 666, pp. 61-66, 2018 10 , 招待有り
- Integrable structure of modified melting crystal model, Kanehisa Takasaki, poster presentation at conference "Integrability in Gauge and String Theory" (Zurich, August 20-24, 2012), 2012 08 , http://arxiv.org/abs/1208.4497Summary:Our previous work on a hidden integrable structure of the melting crystal

model (the U(1) Nekrasov function) is extended to a modified crystal model. As

in the previous case, "shift symmetries" of a quantum torus algebra plays a

central role. With the aid of these algebraic relations, the partition function

of the modified model is shown to be a tau function of the 2D Toda hierarchy.

We conjecture that this tau function belongs to a class of solutions (the so

called Toeplitz reduction) related to the Ablowitz-Ladik hierarchy. - hbar-Dependent KP hierarchy, Kanehisa Takasaki, Takashi Takebe, Theoretical and Mathematical Physics 171 (2) (2012), 683-690, 2011 05 , http://arxiv.org/abs/1105.0794Summary: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.
- Integrable system and N=2 supersymmetric Yang-Mills theory, K. Takasaki, T. Takebe, H. Itoyama et al. (eds.), {\it Frontiers in Quantum Field Theory}, 325-330 (World Scientific, Singapore, 1996)., 1996 03 , Refereed, 招待有り, http://arxiv.org/abs/hep-th/9603129Summary:The exact solutions (Seiberg-Witten type) of $N=2$ supersymmetric Yang-Mills theory are discussed from the view of Whitham-Toda hierarchy.
- Integrable hierarchies, dispersionless limit and string equations, K. Takasaki, M. Morimoto and T. Kawai (eds.), {\it Structures of Solutions of Differential Equations\/}, pp. 457-481 (World Scientific, Singapore, 1996)., 1996 , 招待有りSummary:The notion of string equations was discovered in the end of the eighties, and has been studied in the language of integrable hierarchies. String equations in the KP hierarchy are nowadays relatively well understood. Meanwhile, systematic studies of string equations in the Toda hierarchy started rather recently. This article presents the state of art of these issues from the author's point of view.
- W algebra, twistor, and nonlinear integrable systems, Kanehisa Takasaki, RIMS Kokyuroku 810 (September, 1992)., 1992 06 , 招待有り, http://arxiv.org/abs/hep-th/9206030v3Summary:W algebras arise in the study of various nonlinear integrable systems such as: self-dual gravity, the KP and Toda hierarchies, their quasi-classical (or dispersionless) limit, etc. Twistor theory provides a geometric background for these algebras. Present state of these topics is overviewed. A few ideas on possible deformations of self-dual gravity (including quantum deformations) are presented.
- Area-Preserving Diffeomorphisms and Nonlinear Integrable Systems, Kanehisa Takasaki, J. Mickelsson and O. Pekonen (eds.), {\it Topological and geometrical methods in field theory\/}, Turku, Finland, May 26 - June 1, 1991 pp. 383-397 (World Scientific, Singapore, 1992)., 1991 12 , 招待有り, http://arxiv.org/abs/hep-th/9112041v1Summary:Present state of the study of nonlinear ``integrable" systems related to the

group of area-preserving diffeomorphisms on various surfaces is overviewed.

Roles of area-preserving diffeomorphisms in 4-d self-dual gravity are reviewed.

Recent progress in new members of this family, the SDiff(2) KP and Toda

hierarchies, is reported. The group of area-preserving diffeomorphisms on a

cylinder plays a key role just as the infinite matrix group GL($\infty$) does

in the ordinary KP and Toda lattice hierarchies. The notion of tau functions is

also shown to persist in these hierarchies, and gives rise... - Hidden symmetries of integrable system in Yang-Mills theory and K\"ahler geometry, K. Takasaki, Seminaire des Equations aux D\'eriv\'ees Partielles, 1990-1991, Expos\'e n$^o$ VIII, 22 Janvier 1991 (Ecole Polytechnique, 1991)., 1991 , 招待有り
- Analytic expression of Voros coefficients and its application to WKB connection problem, K. Takasaki, M. Kashiwara and T. Miwa (eds.), {\it Special functions\/}, ICM-90 Satellite Conference Proceedings, pp. 294-315 (Springer-Verlag, Berlin-New York-Tokyo, 1991)., 1991 Summary:Usually, the WKB method starts from formal solutions (WKB or Liouville-Green solutions) expanded in powers of the Planck constant, and connects these solutions by asymptotic matching at turning points. A resummation prescription of these formal calculations was proposed by A. Voros, after an idea of Balian and Bloch, and illustrated for a homogeneous quartic oscillator. Voros argued that his results should be deeply related with J. Ecalle's theory of ``resurgent functions." Further progress along that line has been made by F. Pham and his coworkers. I shall report another approach based upon a classical idea of F. Olver.
- An infinite number of Hamiltonian flows arising from hyper-K\"{a}hler metric, K. Takasaki, Y. Saint-Aubin and L. Vinet (eds.), {\it the XVIIth International Colloquium on Group Theoretical Methods in Physics}, Sainte-Ad\`{e}le 1988, pp. 516-519. (World Scientific, Sigapore, 1989)., 1989
- Issues of multi-dimensional integrable systems, K. Takasaki, M. Kashiwara and T. Kawai (eds.), {\it Algebraic Analysis}, Vol. II, pp. 853-866 (Academic Press, 1988)., 1988 , 招待有り

- 1998 , Daiwa Anglo-Japan Foundation, Daiwa Adrian Prize (jointly awarded)

- Ministry of Education, Culture, Sports, Science, Grant-in-Aid for Scientific Research, Integrable structures related to Gromov-Witten invariants
- Ministry of Education, Culture, Sports, Science, Grant-in-Aid for Scientific Research, Integrable structures in mathematical physics and combinatorics
- Ministry of EduMinistry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research, Theory of integrable hierarchies and their applitations to mathematical physics
- Ministry of Education, Culture, Sports, Science and Technology, Grants-in-Aid for Scientific Research(基盤研究(C)), Search for new connection of integrable systems and mathematical physics
- Ministry of Education, Culture, Sports, Science and Technology, Grants-in-Aid for Scientific Research(基盤研究(B)), Geometric structure and integrable systems in mathematical physics
- Ministry of Education, Culture, Sports, Science and Technology, Grants-in-Aid for Scientific Research(基盤研究(C)), Integrable systems with higher genus spectral parameter
- Ministry of Education, Culture, Sports, Science and Technology, Grants-in-Aid for Scientific Research(基盤研究(C)), Classical and quantum theory of finite-dimensional integrable systems, 1. The dressing chains are known to be an important nonlinear differential equation that includes the Painleve equations. These equations have a Lax representation by a second order square matrix, and the associated spectral curve becomes hyperelliptic. This enabled us to apply the technique of separation of variables, and to obtain a Hamiltonian representation of the dressing chains under a periodic boundary condition.2. A non-autonomous version of the SU(2) Calogero-Gaudin system was taken up an example of isomonodromic deformations on a torus. We applied the technique of separation of va...
- Ministry of Education, Culture, Sports, Science and Technology, Grants-in-Aid for Scientific Research(基盤研究(C)), Finite dimensional integrable structure in systems with infinite degree of freedom, This research project aimed to search for various finite dimensional integrable systems in systems with infinite degrees of freedom, and to elucidate their mathematical structures. Related issues on geometry and non-integrable systems were also investigated.The head investigator obtained interesting results on Whitham deformation equations, isomonodromic deformations, systems arising in supersymmetric/topological gauge theories and Calogero-Moser systems, all of which are mutually related. Firsly, he could find an explicit form of the Whitham deformation equations for asymptoci description ...
- Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (重点領域研究, Study on nonlinear integrable systems related to moduli spaces
- Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (一般研究(C)), Study on nonlinear problems by method of asymptotic analysis
- Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (一般研究(C)), Study on problems of asymptotic analysis in mathematical sciences
- Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (重点領域研究, Study on problems of nonlinear integrable systems in mathematical sciences
- Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (奨励研究(A)), Asymptotic and remormalization group analyses in mathematical physics
- Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (重点領域研究), Search for new integrable systems in mathematical physics and applied analysis