### Recent Writing

Kanehisa Takasaki
Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy
arXiv:2211.11353
Comments: latex2e using amsmath,amssymb,amsthm, 30 pages, no figure
MSC-class: 14N35, 37K10
Abstract The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer $N$, can be formulated as reductions of the lattice KP hierarchy. The integrability of the lattice KP hierarchy is inherited by these reduced systems. In particular, all solutions can be captured by a factorization problem of difference operators. A special solution among them is obtained from Okounkov and Pandharipande's dressing operators for the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. This indicates a hidden link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also related to the lattice Gelfand-Dickey (GD) hierarchy and its extension by logarithmic flows. The logarithmic flows can be derived from the generalized ILW hierarchy by a scaling limit as a parameter of the system tends to $0$. This explains an origin of the logarithmic flows. A similar scaling limit of the equivariant Toda hierarchy yields the extended 1D/bigraded Toda hierarchy.

Kanehisa Takasaki
Extended lattice Gelfand-Dickey hierarchy
J. Phys. A: Math. Theor. 55 (2022), 305203 (14pp)
doi:10.1088/1751-8121/ac7ca2
arXiv:2203.06621
MSC classes: 14N35, 37K10
Abstract The lattice Gelfand-Dickey hierarchy is a lattice version of the Gelfand-Dickey hierarchy. A special case is the lattice KdV hierarchy. Inspired by recent work of Buryak and Rossi, we propose an extension of the lattice Gelfand-Dickey hierarchy. The extended system has an infinite number of logarithmic flows alongside the usual flows. We present the Lax, Sato and Hirota equations and a factorization problem of difference operators that captures the whole set of solutions. The construction of this system resembles the extended 1D and bigraded Toda hierarchy, but exhibits several novel features as well.
Kanehisa Takasaki
Origin of Matrix-Tree theorem (in Japanese)
Slide [pdf file] and report [pdf file] of invited talk at the 31st Symposium of History of Mathematics (October 16-17, 2021, Tsudajuku University, On-line)

Kanehisa Takasaki
Dressing operators in equivariant Gromov-Witten theory of $\mathbb{CP}^1$
Phys. A: Math. Theor. (to appear), https://doi.org/10.1088/1751-8121/ac1828
J. Phys. A: Math. Theor. 54 (2021), 35LT02
doi: 10.1088/1751-8121/ac1828
arXiv:2103.10666
Comments: latex2e using packages amsmath,amssymb,amsthm; (v2) A typo in the definition of E_k(z) on page 6 corrected
MSC-class: 14N35, 37K10
Abstract Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. A technical clue of their method is a pair of dressing operators on the Fock space of 2D charged free fermion fields. We reformulate these operators as difference operators in the Lax formalism of the 2D Toda hierarchy. This leads to a new explanation to the question of why the equivariant Toda hierarchy emerges in the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. Moreover, the non-equivariant limit of these operators turns out to capture the integrable structure of the non-equivariant Gromov-Witten theory correctly.

Kanehisa Takasaki
Toda hierarchies in string and gauge theories (in Japanese)
Slide file of invited talk at on-line workshop "Quantum Geometry in Gauge Theory and Strings" (November 21, 2020) [pdf file]
Abstract The roles of the Toda hierarchies in gauge and string theories since 1990's are reviewed.
Kanehisa Takasaki
Equivariant Gromov-Witten theory of CP1 and equivariant Toda hierarchy (in Japanese)
Abstract of talk at MSJ annual meeting (Nihon University, March 16--19, 2020) cancelled because of the COVID-19 epidemic [pdf file (partly corrected)]
Abstract Okounkov and Pandharipande considered a dressing operator'' in the description of the equivariant Gromov-Witten theory of CP1. We find a new role of this operator.

Kanehisa Takasaki
Integrable structures of specialized hypergeometric tau functions
RIMS Kokyuroku Bessatsu B87 (2021), 057--078.
arXiv:2002.00660
Comments: latex2e, 21pages, no figure, submitted to proceedings of RIMS workshop "Mathematical structures of integrable systems, its deepening and expansion" (September 9-11, 2019)
MSC-class: 05E10, 14N10, 37K10
Abstract Okounkov's generating function of the double Hurwitz numbers of the Riemann sphere is a hypergeometric tau function of the 2D Toda hierarchy in the sense of Orlov and Scherbin. This tau function turns into a tau function of the lattice KP hierarchy by specializing one of the two sets of time variables to constants. When these constants are particular values, the specialized tau functions become solutions of various reductions of the lattice KP hierarchy, such as the lattice Gelfand-Dickey hierarchy, the Bogoyavlensky-Itoh-Narita lattice and the Ablowitz-Ladik hierarchy. These reductions contain previously unknown integrable hierarchies as well.

Kanehisa Takasaki
Cubic Hodge integrals and integrable hierarchies of Volterra type
Proceedings of Symposia in Pure Mathematics, vol. 103.1, Amer. Math. Soc., Providence, RI, 2021, pp. 481-502
arXiv:1909.13095
Comments: latex2e, amsmath,amssymb,amsthm, 29pp, no figure, contribution to Boris Dubrovin memorial volume, American Mathematical Society
Abstract 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.
Kanehisa Takasaki
Integrable structures of cubic Hodge integrals
Invited talk at RIMS workshop Mathematical structures of integrable systems, its deepening and expansion'' (September 9-11, 2019) [slide]
Abstract 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).
Kanehisa Takasaki
Volterra-type hierarchies for specialized hypergeometric tau functions
Invited talk at China-Japan Joint Workshop on Integrable Systems 2019 (Shonan Village Center, Hayama, Japan, August 18-23, 2019) [slide]
Abstract 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.
Toshio Nakatsu and Kanehisa Takasaki
Three-partition Hodge integrals and the topological vertex
Comm. Math. Phys. 376(1) (2020), 201-234
https://doi.org/10.1007/s00220-019-03648-5
arXiv:1812.11726
Abstract 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.
Kanehisa Takasaki
Toda and q-Toda equations for Nekrasov partition functions
Invited talk at SIDE13 conference (Hakata, japan, November 13, 2018) [slide]
Abstract Some results on Toda-type equations and Nekrasov partitions functions are presented.
Contents 1. Introduction
2. Deriving q-Toda equations from Toda hierarchy
3. Toda-like equations for U(1) Nekrasov functions
4. Dual partition function of 5D SU(N) theory
5. Conclusion
Kanehisa Takasaki
4D limit of melting crystal model and its integrable structure
(previous title: Quantum curve and 4D limit of melting crystal model)
Journal of Geometry and Physics 137 (2019), 184--203
DOI: 10.1016/j.geomphys.2018.12.012
arXiv:1704.02750 [math-ph]
Date (revised v2): Tue, 28 Aug 2018 21:26:40 GMT (21kb)
Comments: latex2e using packages amsmath,amssymb,amsthm, 35 pages, no figure; (v2) the title is changed, and an appendix on the relevance to the Toda hierarchy is added; (v3) texts in Introduction and Sect. 4.2 are modified, a few typos are corrected, final version for publication
MSC-class: 14N35, 37K10, 39A13
Abstract 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.

Kanehisa Takasaki
Hurwitz numbers and integrable hierarchy of Volterra type
invited talk at Special session 49, AIMS Conferencer 2018 (Taipei, Taiwan, July 7, 2018) [slide]
Abstract Recent results on an underlying integrable structure of Hurwitz numbers (arXiv:1704.02750) are presented.
Contents 1. Generating functions of Hurwitz numbers
2. Lax equations in single Hurwitz sector
3. Perspective from generalized string equations

Kanehisa Takasaki
Hurwitz numbers and integrable hierarchy of Volterra type
J. Phys. A: Math. Theor. 51 (2018), 43LT01 (9 pages)
doi: 10.1088/1751-8121/aae10b
arXiv:1807.00085
Comments: latex2e, amsmath,amssymb,amsthm, 12 pages, no figure
MSC-class: 14N10, 37K10
Abstract 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.

Kanehisa Takasaki
3D Young diagrams and Gromov-Witten theory of CP1
invited talk at IBS Center of Geometry and Physics, Pohang, Korea (March 27, 2018) [slide]
Abstract 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 $q \to 1$.

Kanehisa Takasaki
Toda hierarchies and their applications
J. Phys. A: Math. Theor. 51 (2018), 203001 (35pp)
doi: 10.1088/1751-8121/aabc14
arXiv:1801.09924 [math-ph]
Comments: 46 pages, no figure, contribution to JPhysA Special Issue "Fifty years of the Toda lattice"
MSC classes: 17B65, 37K10, 82B20
Abstract 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.

Kanehisa Takasaki
Melting crystal model and its 4D limit
invited talk at Conference Physics and Mathematics of Nonlinear Phenomena'' (Gallipoli, Italy, June 17-24, 2017) [slide]
Abstract 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.

Kanehisa Takasaki and Toshio Nakatsu
Quantum mirror curve of closed topological vertex and $q$-differene Kac-operators (in Japanese)
Talk at the autumn meeting of the Mathematical Society of Japan (Kansai University, September 15--18, 2016) [slide]
Abstract 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.

Kanehisa Takasaki and Toshio Nakatsu
$q$-difference Kac-Schwarz operators in topological string theory
SIGMA 13 (2017), 009, 28 pages
doi:10.3842/SIGMA.2017.009
arXiv:1609.00882 [math-ph]
Comments: Contribution to the Special Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology, Quantisation, and Beyond
MSC-class: 37K10, 39A13, 81T30
Key words: topological vertex; mirror symmetry; quantum curve; q-difference equation; KP hierarchy; Kac-Schwarz operator
Abstract The perspective of Kac-Schwarz operators is introduced to the authors' previous work on the quantum mirror curves of topological string theory in strip geometry and closed topological vertex. Open string amplitudes on each leg of the web diagram of such geometry can be packed into a multi-variate generating function. This generating function turns out to be a tau function of the KP hierarchy. The tau function has a fermionic expression, from which one finds a vector $|W\rangle$ in the fermionic Fock space that represents a point $W$ of the Sato Grassmannian. $|W\rangle$ is generated from the vacuum vector $|0\rangle$ by an operator $g$ on the Fock space. $g$ determines an operator $G$ on the space $V = \mathbb{C}((x))$ of Laurent series in which $W$ is realized as a linear subspace. $G$ generates an admissible basis $\{\Phi_j(x)\}_{j=0}^\infty$ of $W$. $q$-difference analogues $A,B$ of Kac-Schwarz operators are defined with the aid of $G$. $\Phi_j(x)$'s satisfy the linear equations $A\Phi_j(x) = q^j\Phi_j(x)$, $B\Phi_j(x) = \Phi_{j+1}(x)$. The lowest equation $A\Phi_0(x) = \Phi_0(x)$ reproduces the quantum mirror curve in the authors' previous work.

Kanehisa Takasaki
Matrix factorization and reductions of Kostant-Toda hierarchy (in Japanese)
Invited Talk at Workshop on Applied Analysis (Osaka, May 19-21, 2016) [slide]
Gekhtman, Shapiro and Veinstein constructed Toda-type integrable systems on the double Bruhat cells $G^{u,v} \subset G = \mathrm{GL}(n,\CC)$ 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.

Kanehisa Takasaki
Topological vertex and quantum mirror curves
Invited talk at international symposium Rikkyo MathPhys 2016'' (Rikkyo University, January 9--11, 2016) [slide]
Abstract 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.
Kanehisa Takasaki
Topological vertex and quantum mirror curves
Invited talk at workshop Quantization of Spectral Curves'' (Osaka City University, November 2--6, 2015) [slide]
Abstract 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.

Kanehisa Takasaki and Toshio Nakatsu
Open string amplitude of closed topological vertex (in Japanese)
Talk at the autumn meeting of the Mathematical Society of Japan (Kyoto Sangyo University, September 13--16, 2015) [slide]
Abstract We compute open string amplitudes of closed topological vertex by the method of topological vertex, and show that one-variate generating functions of these amplitudes satisfy a kind of $q$-difference equation. This $q$-difference equations may be thought of as quantization of the mirror curve.

Kanehisa Takasaki and Toshio Nakatsu
Open string amplitudes of closed topological vertex
J. Phys. A: Math. Theor. 49 (2016), 025201 (28pp)
doi:10.1088/1751-8113/49/2/025201
arXiv:1507.07053 [math-ph]
Comments: latex2e, package amsmath,amssymb,amsthm,graphicx, 10 figures
MSC-class: 17B81, 33E20, 81T30
Abstract 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.

Kanehisa Takasaki
Integrable structure of various melting crystal models
Invited talk at workshop Recent Progress of Integrable Systems'', National Taiwan University, April 10-12, 2015 [slide]
This is an expanded version of my talk at Tsuda College, Tokyo, in February, 2015.

Kanehisa Takasaki
Integrable structure of various melting crystal models
Invited talk at workshop Curves, Moduli and Integrable Systems'', Tsuda College, Tokyo, February 17-19, 2015 [slide]
Abstract 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 relevand integrable systems are particular reductions of the 2D Toda hierarchy.

Kanehisa Takasaki
Twistor theory of gravitational fields and integrable systems (in Japanese)
Invited talks in the 16th "Singularity Workshop" (Nagoya University, January 10-12, 2015) [slides]
Contents 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

Kanehisa Takasaki
Orbifold melting crystal models and reductions of Toda hierarchy
J. Phys. A: Math. Theor. 48 (2015), 215201 (34 pages)
doi:10.1088/1751-8113/48/21/215201
arXiv:1410.5060 [math-ph]
MSC-class: 17B65, 35Q55, 81T30, 82B20
Abstract 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.

Kanehisa Takasaki
Integrable structure of melting crystal models (in Japanese)
Invited talk at workshop "Various aspects of nonlinear mathematical models: continuous, discrete, ultra-discrete, and beyond" (IMI, Kyushu University, August 6-8, 2014) [slide | report]
Abstract 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.

Kanehisa Takasaki
Generalized Ablowitz-Ladik hierarchy in topological string theory
Talk at MSJ annual meeeting (Gakushuin University, March 13, 2014) [abstract | slide]
Abstract 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.

Kanehisa Takasaki
Modified melting crystal model and Ablowitz-Ladik hierarchy
J. Phys.: Conf. Ser. 482 (2014), 012041 [open access]
doi:10.1088/1742-6596/482/1/012041
arXiv:1312.7276 [math-ph]
Comments: 10 pages, 4 figures, contribution to proceedings of the conference "Physics and Mathematic of Nonlinear Phenomena", Gallipoli, Italy, June 23-28, 2013
MSC-class: 17B65, 35Q55, 81T30, 82B20
Abstract This is a review of recent results on the integrable structure of the ordinary and modified melting crystal models. When deformed by special external potentials, the partition function of the ordinary melting crystal model is known to become essentially a tau function of the 1D Toda hierarchy. In the same sense, the modified model turns out to be related to the Ablowitz-Ladik hierarchy. These facts are explained with the aid of a free fermion system, fermionic expressions of the partition functions, algebraic relations among fermion bilinears and vertex operators, and infinite matrix representations of those operators.

Kanehisa Takasaki
Generalized Ablowitz-Ladik hierarchy in topological string theory
J. Phys. A: Math. Theor. 47 (2014), 165201 (20 pages)
doi:10.1088/1751-8113/47/16/165201
arXiv:1312.7184 [math-ph]
MSC-class: 17B80, 35Q55, 81T30
Abstract This paper addresses the issue of integrable structure in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. The generating function of these amplitudes with respect to the product of two independent Schur functions become a tau function of the 2D Toda hierarchy. The associated Lax operators turn out to have a particular factorized form. This factorized form of the Lax operators characterizes a generalization of the Ablowitz-Ladik hierarchy embedded in the 2D Toda hierarchy. The generalized Ablowitz-Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.

Kanehisa Takasaki
Modified melting crystal model and Ablowitz-Ladik hierarchy
Invited talk at "Physics and Mathematic of Nonlinear Phenomena", Gallipoli, Italy, June 23--28, 2013 [slide]
Abstract 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

Kanehisa Takasaki
Melting crystal model and Ablowitz-Ladik hierarchy (in Japanese)
talk presented at the Annual Meeting of the Mathematical Society of Japan (Kyoto University, March 22, 2013) [abstract| slide]
Abstract 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).

Kanehisa Takasaki
Modified melting crystal model and Ablowitz-Ladik hierarchy
J. Phys. A: Math. Theor. 46 (2013), 245202 (23 pages)
arXiv:1302.6129 [math-ph]
Comments: latex2e, 33 pages, no figure
MSC classes: 17B65, 35Q55, 81T30, 82B20
doi:10.1088/1751-8113/46/24/245202
Abstract 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 hierarchy can be characterized by a factorization problem of $\ZZ\times\ZZ$ matrices as well. The associated Lax operators turn out to be quotients of first order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz-Ladik (equivalently, relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.

Kanehisa Takasaki
Remarks on partition functions of topological string theory on generalized conifolds
RIMS Kokyuroku No. 1913 (2014), 182--201
arXiv:1301.4548 [math-ph]
Comments: 20 pages, 3 figures, contribution to the proceedings of the RIMS camp-style seminar "Algebraic combinatorics related to Young diagrams and statistical physics", August, 2012, International Institute for Advanced Studies, Kyoto, organized by M. Ishikawa, S. Okada and H. Tagawa
MSC-class: 05E05, 37K10, 81T30
Abstract 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.

Kanehisa Takasaki
Integrable structure of modified melting crystal model
Poster presentation at conference "Integrability in Gauge and String Theory", ETH Zurich, August 20--24, 2012 [poster]
arXiv version arXiv:1208.4497 [math-ph]]
MSC-class: 17B65, 35Q58, 82B20
Abstract 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.

Kanehisa Takasaki
Combinatorial properties of toric topological string partition functions
Invited talk at RIMS camp-style seminar "Algebraic Combinatorics Related to Young Diagrams and Statistical Physics", IIAS, Kyoto, August 6 - 10, 2012 [slide]
Index
1. Topological vertex and web diagrams
2. Generalized conifolds
3. Partition functions of generalized conifolds
4. Simplest examples
5. General rules
6. Quantum mirror curve

Kanehisa Takasaki
Old and new reductions of dispersionless Toda hierarchy
SIGMA 8 (2012), 102, 22 pages
arXiv:1206.1151 [math-ph]
Contribution to SIGMA Special Issue on Geometrical Methods in Mathematical Physics
DOI: 10.3842/SIGMA.2012.102
MSC classes: 35Q58, 37K10, 53B50, 53D45
Abstract Two types of finite-variable reductions of the dispersionless Toda hierarchy are considered in the geometric perspectives. 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 intended to be a Toda version of the waterbag model of the dispersionless KP hierarchy. The two types of Landau-Ginzburg potentials are shown to satisfy (a radial version of) the L\"onwer equations with respect to the critical values of the Landau-Ginzburg potentials. Integrability conditions of these L\"owner equations are (a radial version of) the Gibbons-Tsarev equations. These equations are used to formulate hodograph solutions of the reduced hierarchy. Geometric aspects of the Gibbons-Tsarev equations are explained in the language of classical differential geometry (Darboux equations, Egorov metrics and Combescure transformations). Frobenius structures on the parameter space of the Landau-Ginzburg potentials are introduced, and flat coordinates are constructed explicitly.

A. Yu. Orlov, T. Shiota, K. Takasaki
Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions
arXiv:1201.4518v1 [math-ph]
Abstract We introduce a useful and rather simple class of BKP tau functions which which we shall call "easy tau functions". We consider two versions of BKP hierarchy, one we will call "small BKP hierarchy" (sBKP) related to $O(\infty)$ introduced in Date et al and "large BKP hierarchy" (lBKP) related to $O(2\infty +1)$ introduced in Kac and van de Leur (which is closely related to the large $O(2\infty)$ DKP hierarchy (lDKP) introduced in Jimbo and Miwa). Actually "easy tau functions" of the sBKP hierarchy were already considered in Harnad et al, here we are more interested in the lBKP case and also the mixed small-large BKP tau functions (Kac and van de Leur). Tau functions under consideration are equal to certain sums over partitions and to certain multi-integrals over cone domains. In this way they may be applicable in models of random partitions and models of random matrices. Here is the first part of the paper where sums of Schur and projective Schur functions over partitions are considered.

Kanehisa Takasaki and Takashi Takebe
An hbar-expansion of the Toda hierarchy: a recursive construction of solutions
Analysis and Mathematical Physics 2 (2012), 171-214.
arXiv:1112.0601v1 [math-ph]
Comments: 37 pages, no figures. arXiv admin note: substantial text overlap with arXiv:0912.4867
MSC classes: 37K10, 35Q53
Abstract A construction of general solutions of the \hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and \bar W. The dressing operators are set in an exponential form as W = e^{X/\hbar} and \bar W = e^{\phi/\hbar}e^{\bar X/\hbar}, and the auxiliary operators X,\bar X and the function \phi are assumed to have \hbar-expansions X = X_0 + \hbar X_1 + ..., \bar X = \bar X_0 + \hbar\bar X_1 + ... and \phi = \phi_0 + \hbar\phi_1 + .... The coefficients of these expansions turn out to satisfy a set of recursion relations. X,\bar X and \phi are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form \Psi = e^{S/\hbar} and \bar\Psi = e^{\bar S/\hbar}, which leads to an \hbar-expansion of the logarithm of the tau function.

Kanehisa Takasaki and Toshio Nakatsu
Thermodynamic limit of random partitions and dispersionless Toda hierarchy
J. Phys. A: Math. Theor. 45 (2012), 025403 (38pp)
arXiv:1110.0657 [math-ph]
Comments: latex2e, 55 pages, no figure
MSC-class: 35Q58, 81T13, 82B20
doi:10.1088/1751-8113/45/2/025403
Abstract We study the thermodynamic limit of random partition models for the instanton sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical observables. The physical observables correspond to external potentials in the statistical model. The partition function is reformulated in terms of the density function of Maya diagrams. The thermodynamic limit is governed by a limit shape of Young diagrams associated with dominant terms in the partition function. The limit shape is characterized by a variational problem, which is further converted to a scalar-valued Riemann-Hilbert problem. This Riemann-Hilbert problem is solved with the aid of a complex curve, which may be thought of as the Seiberg-Witten curve of the deformed U(1) gauge theory. This solution of the Riemann-Hilbert problem is identified with a special solution of the dispersionless Toda hierarchy that satisfies a pair of generalized string equations. The generalized string equations for the 5D gauge theory are shown to be related to hidden symmetries of the statistical model. The prepotential and the Seiberg-Witten differential are also considered.

Kanehisa Takasaki
Non-degenerate solutions of universal Whitham hierarchy
Invited talk at "7th International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory" (University of Georgia, Athens, April 5, 2011) [slide]
Abstract I will review recent results obtained in joint work with T. Takebe and L.-P. Teo. In this work, the notion of non-degenerate solutions'' for the dispersionless Toda hierarchy, is generalied to the universal Whitham hierarchy (of genus zero). These solutions are characterized by a nonlinear Riemann--Hilbert problem. One can see from this characterization that these solutions are a kind of general'' solutions of these dispersionless integrable hierarchies. The Riemann--Hilbert problem is translated to the language of a space of conformal mappings, and solved by inversion of an infinite dimensional period map on this space.
I will also argue that a possible dispersive analogue of these results can be found in a system of multiple bi-orthogonal polynomials. These multiple bi-orthogonal polynomials are a generalizations of bi-orthogonal polynomials studied by Adler and van~Moerbeke in the context of the Toda hierarchy. The latter give a special solution of the Toda hierarchy. This solution may be thought of as a dispersive counterpart of non-degenerate solution of the dispersionless Toda hierarchy. I conjecture that a similar interpretation holds for the multiple bi-orthogonal polynomials in the framework of our previous work.

Kanehisa Takasaki
Toda tau functions with quantum torus symmetries
colloquium talk at Department of Mathematics, University of California, Davis (Davis, March 30, 2011) [slide]
Abstract The quantum torus algebra plays an important role in a special class of solutions of the Toda hierarchy. Typical examples are the solutions related to the melting crystal model of topological strings and 5D SUSY gauge theories. The quantum torus algebra is realized by a 2D complex free fermion system that underlies the Toda hierarchy, and exhibits mysterious shift symmetries''. This talk is based on collaboration with Toshio Nakatsu.

Kanehisa Takasaki and Takashi Takebe
An h-bar dependent formulation of the Kadomtsev-Petviashvili hierarchy
Theoretical and Mathematical Physics 171 (2) (2012), 683-690.
arXiv:1105.0794v1 [math-ph]
Comments: 12 pages, contribution to the Proceedings of the "International Workshop on Classical and Quantum Integrable Systems 2011" (January 24-27, 2011 Protvino, Russia)
Abstract This is a summary of a recursive construction of solutions of the hbar-dependent KP hierarchy. We give recursion relations for the coefficients X_n of an hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 + ... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). The asymptotic behaviours of (the logarithm of) the wave function and the tau function are also considered.

Kanehisa Takasaki
Toda tau functions with quantum torus symmetries
Acta Polytechnica 51, No.1 (2011), 74-76.
arXiv:1101.4083 [math-ph]
Comments: latex2e using packages amsmath,amssymb,amsthm, 6 pages, no figure, contribution to "19th International Colloquium on Integrable Systems and Quantum Symmetries"
Abstract The quantum torus algebra plays an important role in a special class of solutions of the Toda hierarchy. Typical examples are the solutions related to the melting crystal model of topological strings and 5D SUSY gauge theories. The quantum torus algebra is realized by a 2D complex free fermion system that underlies the Toda hierarchy, and exhibits mysterious "shift symmetries". This article is based on collaboration with Toshio Nakatsu.

Kanehisa Takasaki
Generalized string equations for double Hurwitz numbers
arXiv:1012.5554 [math-ph]
Journal of Geometry and Physics 62 (2012), 1135--1156
Comments: latex2e using amsmath,amssymb,amsthm, 41 pages, no figure
MSC-class: 35Q58, 14N10, 81R12
Abstract The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov-Schulman operators turn out to satisfy a set of generalized string equations. These generalized string equations resemble those of $c = 1$ string theory except that the Orlov-Schulman operators are contained therein in an exponentiated form. These equations are derived from a set of intertwining relations for fermiom bilinears in a two-dimensional free fermion system. The intertwiner is constructed from a fermionic counterpart of the cut-and-join operator. A classical limit of these generalized string equations is also obtained. The so called Lambert curve emerges in a specialization of its solution. This seems to be another way to derive the spectral curve of the random matrix approach to Hurwitz numbers.

Kanehisa Takasaki
Generalized string equations for Hurwitz numbers
(Workshop Integrable Systems, Random Matrices, Algebraic Geometry and Geometric Invariants'', Kyoto University, December 15-17, 2010) [slides of talk]
Abstract Generating functions of almost-simple and double Hurwitz numbers for the Riemann sphere are known to become special tau functions of the KP and Toda hierarchies. This talk presents recent results on the generating function of duble Hurwitz numbers. Generalized string equations for the Lax and Orlov--Schulman operators are derived. These equations turn out to have a meaningful classical limit in the dispersionless Toda hierarchy. Solving these equations reveals a relation with Lambert's W-function. This seems to show another approach to various roles of the W-function in Hurwitz numbers.

K. Takasaki
Special solution of Toda hierarchy related to Hurwitz numbers, and its classical limit (in Japanese)
(MSJ Autumn Meeting, Nagoya University, September 22-25, 2010) [slides of talk]
Abstract This talk deals with a tau funtion of the Toda hierarchy related to Hurwitz numbers and their q-analogues. Generalized string equations'' for the Lax and Orlov-Schulman operators are derived. A curve of the same form as the spectral curve'' in Eynard and Orantin's random matrix approach emerges in the classical (dispersionless) limit of the string equations.

K. Takasaki
Toda tau function with quantum torus symmetries
19th International Colloquium Integrable Systems and Quantum Symmetries'' (Czech Technical University, Prague, June 17--19, 2010) slides of talk [pdf]
Abstract The quantum torus algebra plays an important role in a spacial class of solutions of the Toda hierarchy. Typical examples are the solutions related to the melting crystal model of 5D SUSY gauge theories, topological strings on toric Calabi-Yau three folds, the Hurwitz numbers of the Riemann sphere and their $q$-analogues. The quantum torus algebra is realized by a 2D complex fermion system that underlies the Toda hierarchy, and exhibits a mysterious shift symmetry'' that leads to generalized string equations''. This talk is partly based on collaboration with Toshio Nakatsu.

Kanehisa Takasaki, Takashi Takebe and Lee Peng Teo
Non-degenerate solutions of universal Whitham hierarchy
J. Phys. A: Math. Theor. 43 (2010), 325205
arXiv:1003.5767 [math-ph]
Comments: latex2e, using amsmath, amssym and amsthm packages, 32 pages, no figure
Abstract The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with $M+1$ marked points. These solutions are characterized by a Riemann-Hilbert problem (generalized string equations) with respect to two-dimensional canonical transformations, and may be thought of as a kind of general solutions of the hierarchy. The Riemann-Hilbert problem contains $M$ arbitrary functions $H_a(z_0,z_a)$, $a = 1,\ldots,M$, which play the role of generating functions of two-dimensional canonical transformations. The solution of the Riemann-Hilbert problem is described by period maps on the space of $(M+1)$-tuples $(z_\alpha(p) : \alpha = 0,1,\ldots,M)$ of conformal maps from $M$ disks of the Riemann sphere and their complements to the Riemann sphere. The period maps are defined by an infinite number of contour integrals that generalize the notion of harmonic moments. The $F$-function (free energy) of these solutions is also shown to have a contour integral representation.

Kanehisa Takasaki
Integrable structure of melting crystal models (in Japanese)
(MSJ spring meeting, University of Keio, March 24--27, 2010) [abstract of talk | slides of talk]
Abstract Melting crystal models are models of statistical mechanics that are related to 5D supersymmetric gauge theories and strings on toric Calabi-Yau varieties. If a set of external potentials are inserted, the partition function becomes the tau function of a special solution of the 1D Toda hierarchy with respect to the coupling constants. Moreover, a model modified by introducing a kind of asymmetry, too, has a similar integrable structure, which is a reduction of the 2D Toda hierarchy. These resuls are reviewed along with basic knowledge.

Kanehisa Takasaki
KP and Toda tau functions in Bethe ansatz
B. Feigin, M. Jimbo and M. Okado (eds.), New Trends in Quantum Integrable Systems'', Proceedings of the Infinite Analysis 09, Kyoto, Japan 27-31 July 2009 (World Sci. Publ., Hackensack, NJ), pp. 373--391.
arXiv:1003.307 [math-ph]
Comments: latex2e, using ws-procs9x6 package, 19 pages, contribution to the festschrift volume for the 60th anniversary of Tetsuji Miwa
Abstract Recent work of Foda and his group on a connection between classical integrable hierarchies (the KP and 2D Toda hierarchies) and some quantum integrable systems (the 6-vertex model with DWBC, the finite XXZ chain of spin 1/2, the phase model on a finite chain, etc.) is reviewed. Some additional information on this issue is also presented.

Kanehisa Takasaki
Two extensions of 1D Toda hierarchy
J. Phys. A: Math. Theor. 43 (2010) 434032
arXiv:1002.4688 [nlin.SI]
Comments: latex2e, usepackage amsmath,amssymb, 19 pages, no figure
Abstract The extended Toda hierarchy of Carlet, Dubrovin and Zhang is reconsidered in the light of a 2+1D extension of the 1D Toda hierarchy constructed by Ogawa. These two extensions of the 1D Toda hierarchy turn out to have a very similar structure, and the former may be thought of as a kind of dimensional reduction of the latter. In particular, this explains an origin of the mysterious structure of the bilinear formalism proposed by Milanov.

Kanehisa Takasaki and Takashi Takebe
hbar-expansion of KP hierarchy: Recursive construction of solutions
arXiv:0912.4867 [math-ph]
Abstract The \hbar-dependent KP hierarchy is a formulation of the KP hierarchy that depends on the Planck constant \hbar and reduces to the dispersionless KP hierarchy as \hbar -> 0. A recursive construction of its solutions on the basis of a Riemann-Hilbert problem for the pair (L,M) of Lax and Orlov-Schulman operators is presented. The Riemann-Hilbert problem is converted to a set of recursion relations for the coefficients X_n of an \hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 +... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). Given the lowest order term X_0, one can solve the recursion relations to obtain the higher order terms. The wave function \Psi associated with W turns out to have the WKB form \Psi = \exp(S/\hbar), and the coefficients S_n of the \hbar-expansion S = S_0 + \hbar S_1 + \hbar^2 S_2 +..., too, are determined by a set of recursion relations. This WKB form is used to show that the associated tau function has an \hbar-expansion of the form \log\tau = \hbar^{-2}F_0 + \hbar^{-1}F_1 + F_2 + ..

Kanehisa Takasaki
Extension of nonlinear Schrodinger and Ablowitz-Ladic hierarchies by logarithmic time evolutions (in Japanese)
Slides of talk at RIAM workshop Currrent and Future Research on Nonlinear Waves'' (RIAM, Kyushu University, October 19--21, 2009) [pdf]
Abstract Carlet, Dubrovin and Zhang constructed an "extended Toda hierarchy" by adding logarithmic time evolutions to the ordinary 1D Toda hierarchy. Milanov presented a bilinear formalism of this hierarchy. By the way, the 1D Toda hierarchy may be thought of as a sequence of Backlund transformations of the nonlinear Schrodinger (NLS) hierarchy. In the same sense, the Ruijsenaars-Toda hierarchy corresponds to the Ablowitz-Ladik (AL) hierarchy (Kharchev, Mironov Zedhanov; Suris; Sadakane). In this talk, I translate the logarithmic flows of the 1D Toda hierarchy to the language of the NLS hierarchy, and show that this result can be generalized to the AL hierarchy.

Kanehisa Takasaki
Extension of nonlinear Schrodinger and Ablowitz-Ladic hierarchies by logarithmic time evolutions (in Japanese)
Slides of talk at MSJ autumn meeting (Osaka University, September 24--27, 2009) [pdf]
Abstract Carlet, Dubrovin and Zhang constructed an "extended Toda hierarchy" by adding logarithmic time evolutions to the ordinary 1D Toda hierarchy. Milanov presented a bilinear formalism of this hierarchy. By the way, the 1D Toda hierarchy may be thought of as a sequence of Backlund transformations of the nonlinear Schrodinger (NLS) hierarchy. In the same sense, the Ruijsenaars-Toda hierarchy corresponds to the Ablowitz-Ladik (AL) hierarchy (Kharchev, Mironov Zedhanov; Suris; Sadakane). In this talk, I translate the logarithmic flows of the 1D Toda hierarchy to the language of the NLS hierarchy, and show that this result can be generalized to the AL hierarchy.

Kanehisa Takasaki
Auxiliary linear problem, difference Fay identities and dispersionless limit of Pfaff-Toda hierarchy
SIGMA 5 (2009), paper 109, 34 pages
arXiv:0908.3569 [nlin.SI]
Comments: 49 pages, no figure, usepackage amsmath,amssymb,amsthm
Abstract Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as the coupled KP hierarchy'' and the Pfaff lattice''). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called the Pfaff-Toda hierarchy'') . Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve.

Kanehisa Takasaki
KP and Toda tau functions in Bethe ansatz: a review
Workshop Infinite Analysis 09 --- New trends in quantum integrable systems'' (RIMS and Department of Mathematics, Kyoto University, July 27--31, 2009), slides of talk [pdf]
Abstract This is a review of recent work by O. Foda et al. on KP tau functions that emerge in algebraic Bethe ansatz of the 6-vertex model with DWBC, the spin 1/2 XXZ model, etc.

Kanehisa Takasaki
Integrable structure of random plane partitions with two q-parameters (in Japanese)
slides of talk at MSJ spring meeting 2009 (Graduate School of Mathematical Science, University of Tokyo, March 26--29, 2009) [pdf]
Abstract The melting crytal model is a model of random plane partitions in statitical mehanics (being also interpreted as an instanton sum of 5D supersymmetric gauge theory). Upon introducing special external potentials, the partition function of this model becomes a tau function of the 1D Toda hierarchy multiplied by a simple factor. This talk presents a generalization of this results, and point out the relevance of the q-difference Toda equation as another integrable struture of this model.

Kanehisa Takasaki
Difference Fay identities of Pfaff-Toda hierarchy (in Japanese)
slides of talk at MSJ spring meeting 2009 (Graduate School of Mathematical Science, University of Tokyo, March 26--29, 2009) [pdf]
Abstract This talk presents recent results on auxiliary linear equations and difference Fay identities'' of an integrable hierarchy that was first introduced by Kakei and Willox.

Kanehisa Takasaki
Integrable structure of melting crystal model with two q-parameters
arXiv:0903.2607 [math-ph]
J. Geometry and Physics 59 (2009), 1244-1257
Comments: 27 pages, no figure, latex2e(package amsmath,amssymb,amsthm)
Abstract This paper explores integrable structures of a generalized melting crystal model that has two $q$-parameters $q_1,q_2$. This model, like the ordinary one with a single $q$-parameter, is formulated as a model of random plane partitions (or, equivalently, random 3D Young diagrams). The Boltzmann weight contains an infinite number of external potentials that depend on the shape of the diagonal slice of plane partitions. The partition function is thereby a function of an infinite number of coupling constants $t_1,t_2,\ldots$ and an extra one $Q$. There is a compact expression of this partition function in the language of a 2D complex free fermion system, from which one can see the presence of a quantum torus algebra behind this model. The partition function turns out to be a tau function (times a simple factor) of two integrable structures simultaneously. The first integrable structure is the bigraded Toda hierarchy, which determine the dependence on $t_1,t_2,\ldots$. This integrable structure emerges when the $q$-parameters $q_1,q_2$ take special values. The second integrable structure is a $q$-difference analogue of the 1D Toda equation. The partition function satisfies this $q$-difference equation with respect to $Q$. Unlike the bigraded Toda hierarchy, this integrable structure exists for any values of $q_1,q_2$.

Kanehisa Takasaki
Dispersionless Hirota equations and reduction of universal Whitham hierarchy
slides of talk at workshop Laplacian Growth and Related Topics'' (CRM, University of Montreal, August 18--22, 2008) [pdf]
Abstract Teo, Takebe and Zabrodin applied dispersionless Hirota equations to the problem of finite-variable reduction of the dispersionless KP and Toda hierarchies. We extend their results to the universal Whitham hierarchy of genus zero. This is a joint work with Takashi Takebe.

Kanehisa Takasaki and Takashi Takebe
Loewner equations, Hirota equations and reductions of universal Whitham hierarchy
J. Phys. A: Math. Theor. {\bf 41} (2008), 475206 (27pp)
arXiv:0808.1444 [nlin.SI]
Comments: latex 2e, 39 pages, using packages amsmath,amssymb,amsthm
Abstract This paper reconsiders finite variable reductions of the universal Whitham hierarchy of genus zero in the perspective of dispersionless Hirota equations. In the case of one-variable reduction, dispersionless Hirota equations turn out to be a powerful tool for understanding the mechanism of reduction. All relevant equations describing the reduction (L\"owner-type equations and diagonal hydrodynamic equations) can be thereby derived and justified in a unified manner. The case of multi-variable reductions is not so straightforward. Nevertheless, the reduction procedure can be formulated in a general form, and justified with the aid of dispersionless Hirota equations. As an application, previous results of Guil, Ma\~{n}as and Mart\'{\i}nez Alonso are reconfirmed in this formulation.

Toshio Nakatsu and Kanehisa Takasaki
Integrable structure of melting crystal model with external potentials
Advanced Studies in Pure Mathematics, vol. 59 (Mathematical Society of Japan, 2010), pp. 201--223.
arXiv:0807.4970 [math-ph]
Comments: 21 pages, 3 figures, using amsmath,amssymb,amsthm,graphicx packages, contribution to proceedings of RIMS workshop "New developments in Algebraic Geometry, Integrable Systems and Mirror symmetry" (January 7--11, 2008)
Abstract This is a review of the authors' recent results on an integrable structure of the melting crystal model with external potentials. The partition function of this model is a sum over all plane partitions (3D Young diagrams). By the method of transfer matrices, this sum turns into a sum over fordinary partitions (Young diagrams), which may be thought of as a model of q -deformed random partitions. This model can be further translated to the language of a complex fermion system. A fermionic realization of the quantum torus Lie algebra is shown to underlie therein. With the aid of hidden symmetry of this Lie algebra, the partition function of the melting crystal model turns out to coincide, up to a simple factor, with a tau function of the 1D Toda hierarchy. Some related issues on 4D and 5D supersymmetric Yang-Mills theories, topological strings and the 2D Toda hierarchy are briefly discussed.

Toshio Nakatsu, Yui Noma and Kanehisa Takasaki
Extended $5d$ Seiberg-Witten Theory and Melting Crystal
arXiv:0807.0746 [hep-th]
Nucl. Phys. B808 (2009), 411--440
Comments: The solution of the Reimann-Hilbert problem presented here is wrong. A correct solution, along with a correct curve, can be obtained by solving a Riemann-Hilbert problem for the primitive function of the Phi potential.
Abstract We study an extension of the Seiberg-Witten theory of $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills on $\mathbb{R}^4 \times S^1$. We investigate correlation functions among loop operators. These are the operators analogous to the Wilson loops encircling the fifth-dimensional circle and give rise to physical observables of topological-twisted $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills in the $\Omega$ background. The correlation functions are computed by using the localization technique. Generating function of the correlation functions of U(1) theory is expressed as a statistical sum over partitions and reproduces the partition function of the melting crystal model with external potentials. The generating function becomes a $\tau$ function of 1-Toda hierarchy, where the coupling constants of the loop operators are interpreted as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition function of this model is studied. We solve a Riemann-Hilbert problem that determines the limit shape of the main diagonal slice of random plane partitions in the presence of external potentials, and identify a relevant complex curve and the associated Seiberg-Witten differential.

Toshio Nakatsu, Yui Noma and Kanehisa Takasaki
Integrable Structure of $5d$ $\mathcal{N}=1$ Supersymmetric Yang-Mills and Melting Crystal
arXiv:0806.3675 [hep-th]
Int. J. Mod. Phys. A23 (2008), 2332-2342.
Comments: The solution of the Reimann-Hilbert problem presented here is wrong. A correct solution, along with a correct curve, can be obtained by solving a Riemann-Hilbert problem for the primitive function of the Phi potential.
12 pages, 1 figure, based on an invited talk presented at the international workshop "Progress of String Theory and Quantum Field Theory" (Osaka City University, December 7-10, 2007), to be published in the proceedings
Abstract We study loop operators of $5d$ $\mathcal{N}=1$ SYM in $\Omega$ background. For the case of U(1) theory, the generating function of correlation functions of the loop operators reproduces the partition function of melting crystal model with external potential. We argue the common integrable structure of $5d$ $\mathcal{N}=1$ SYM and melting crystal model.

Kanehisa Takasaki
Integrable structure in melting crystal model of 5D gauge theory
slides of talk at workshop "New developments in Algebraic Geometry, Integrable Systems and Mirror symmetry" (RIMS, January 7 - 11, 2008) [pdf]
Abstract I will report a recent result on integrable structures in a 5D analogue of Nekrasov's instanton partition function for supersymmetric gauge theories. This partition function is formulated as a statistical model of random plane partition known as melting crystal''. By adding new potential terms and viewing the coupling constants as fictitious time variables'', the partition function turns out to be a tau function of the 1-Toda hierarchy. A fermionic realization of quantum torus Lie algebra plays a central role in identifying this integrable structure. This is a joint work with Toshio Nakatsu.

Kanehisa Takasaki
Differential Fay identities and auxiliary linear problem of integrable hierarchies
Advanced Studies in Pure Mathematics vol. 61 (Mathematical Society of Japan, 2011), pp. 387--441.
arXiv:0710.5356 [nlin.SI]
Comments: latex2e, packages "amsmath,amssymb,amsthm", 50 pages, no figure, contribution to proceedings of conference "Exploration of new structures and natural constructions in mathematical physics" (Nagoya University, March, 2007); (v2) a few references added; (v3) final version for publication
Abstract We review the notion of differential Fay identities and demonstrate, through case studies, its new role in integrable hierarchies of the KP type. These identities are known to be a convenient tool for deriving dispersionless Hirota equations. We show that differential (or, in the case of the Toda hierarchy, difference) Fay identities play a more fundamental role. Namely, they are nothing but a generating functional expression of the full set of auxiliary linear equations, hence substantially equivalent to the integrable hierarchies themselves. These results are illustrated for the KP, Toda, BKP and DKP hierarchies. As a byproduct, we point out some new features of the DKP hierarchy and its dispersionless limit.

Toshio Nakatsu and Kanehisa Takasaki
Melting Crystal, Quantum Torus and Toda Hierarchy
arXiv:0710.5339 [hep-th]
Commun. Math. Phys. 285 (2009), 445--468
Abstract Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories and $A$-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.

Kanehisa Takasaki
Fay-type identitties and dispersionless limit of integrable hierarchies
slides of talk at Workshop Exploration of New Structures and Natural Construction in Mathematical Physics'' (Nagoya University, March 5--8, 2007) [pdf]
Abstract Fay-type identities play a significant role in recent (since 2000) applications of dispersionless integrable hierarchies (dKP, dToda, etc.). This talk is an introduction to theoretical aspects of Fay-type identities.

Kanehisa Takasaki
Dimer models and related topics (in Japanese)
Contribution to proceedings of RIMS workshop Prospects of theories of integrable systems'' (August 21--23, 2006) [pdf]
Abstract Dimer models'' are a kind of models in statistical physics. Mathematically, they are formulated in the language of perfect matchings of a bipartite graph. This article presents a basic exposition of dimer models. The main part is devoted to the mechanism that yields a Pfaffian or determinantal formula of the partition function. The notions of amoeba and Ronkin functions, which play a role in periodic dimers, are also briefly mentioned.

Kanehisa Takasaki
Hamiltonian structure of PI hierarchy
arXiv:nlin.SI/0610073
SIGMA 3 (2007), 042, 32 pages
Contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics
Abstract The string equation of type $(2,2g+1)$ may be thought of as a higher order analogue of the first Painlev\'e equation that correspond to the case of $g = 1$. For $g > 1$, this equation is accompanied with a finite set of commuting isomonodromic deformations, and they altogether form a hierarchy called the PI hierarchy. This hierarchy gives an isomonodromic analogue of the well known Mumford system. The Hamiltonian structure of the Lax equations can be formulated by the same Poisson structure as the Mumford system. A set of Darboux coordinates, which have been used for the Mumford system, can be introduced in this hierarchy as well. The equations of motion in these Darboux coordinates turn out to take a Hamiltonian form, but the Hamiltonians are different from the Hamiltonians of the Lax equations (except for the lowest one that corresponds to the string equation itself). The difference originates in the presence of extra terms in the isomonodromic Lax equations.

Kanehisa Takasaki and Takashi Takebe
Dispersionless Hirota equations of multi-component integrable hierarchies
slides of talk at MISGRAM workshop "Integrable Systems in Applied Mathematics", Madrid, September 7--12, 2006 [pdf]
Abstract I will present recent results and work in progress on the dispersionless limit of Hirota equations of various multi-component integrable hierarchies, such as the multi-component charged'' KP hierarchy, the 2-component BKP hierarchy, the coupled (or $D_\infty'$) KP/BKP hierarchy, etc. This talk is partially based on joint work with T. Takebe.

Kanehisa Takasaki and Takashi Takebe
Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy
arXiv:nlin.SI/0608068
Physica D235, no. 1-2 (2007), 109-125
Comments: latex2e (a4paper, 12pt) using packages "amssymb,amsmath,amsthm", 44 pages, no figure
Abstract The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multi-component KP hierarchy. To this end, the multi-component KP hierarchy is (re)formulated to depend on several discrete variables called charges''. These discrete variables play the role of lattice coordinates in underlying Toda field equations. A multi-component version of the so called differential Fay identity are derived from the Hirota equations of the $\tau$-function of this charged'' multi-component KP hierarchy. These multi-component differential Fay identities have a well-defined dispersionless limit (the dispersionless Hirota equations). The dispersionless Hirota equations turn out to be equivalent to the Hamilton-Jacobi equations for the $S$-functions of the universal Whitham hierarchy. The differential Fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar-valued wave functions of the multi-component KP hierarchy.

Kanehisa Takasaki
Simple model of separation of variables (in Japanese)
contribution to proceedings of RIMS workshop Hyperfunctions and Linear Differential Equations 2006'' (March 6--9, 2006) [pdf]
Abstract Employing a kind of toy model'', we illustrate the basic idea of separation of variables in classical mechanics. Though being very simple, this remarkable model exhibits a typical machinary of separation of variables. We present a few researches that were inspired by this model.

Kanehisa Takasaki
Dispersionless Hirota equations of two-component BKP hierarchy
SIGMA Vol. 2 (2006), Paper 057, 22 pages
Comments: 31 pages, no figure, latex2e usepackage amsmath,amssymb
Report-no: nlin.SI/0604003
Abstract The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy). Dispersionless limit of this multi-component hierarchy is considered on the level of the $\tau$-function. The so called dispersionless Hirota equations are obtained from the Hirota equations of the $\tau$-function. These dispersionless Hirota equations turn out to be equivalent to a system of Hamilton-Jacobi equations. Other relevant equations, in particular, dispersionless Lax equations, can be derived from these fundamental equations. For comparison, another approach based on auxiliary linear equations is also presented.

Kanehisa Takasaki and Takashi Takebe
Radial Loewner equation and dispersionless cmKP hierarchy
Report no: nlin.SI/0601063
Abstract It has been shown that the dispersionless KP hierarchy (or the Benney hierarchy) is reduced to the chordal L\"owner equation. We show that the radial L\"owner equation also gives reduction of a dispersionless type integrable system. The resulting system acquires another degree of freedom and becomes the dcmKP hierarchy, which is a half'' of the dispersionless Toda hierarchy. The results of this article was announced in nlin.SI/0512008.

Kanehisa Takasaki, Takashi Takebe
Loewner equations and dispersionless hierarchies
Comments: 6 pages (Latex; amsmath, amssymb required), Contribution to the Proceedings of the XXIII International Conference of Differential Geometric Methods in Theoretical Physics (M.-L. Ge and W. Zhang ed., Nankai Tracts in Mathematics vol. 10, World Scientific, 2006)
Report-no: nlin.SI/0512008
Abstract Reduction of a dispersionless type integrable system (dcmKP hierarchy) to the radial Loewner equation is presented.

Kanehisa Takasaki
Hamiltonian structuer of higher flows in string equation ("Kobe workshop on integrable systems and Painleve equations", Kobe University, November 23 -- 25, 2005) OHP transparencies of talk [pdf]

Kanehisa Takasaki
Toy models of separation of variables
("Dynamics: Frontiers in Geommetry, Analysis and Mathematical Physics", Academic exchange events of Kyoto University and Technical University of Munchen, October 7, 2005) OHP transparencies of talk [pdf]

Ryu Sasaki and Kanehisa Takasaki
Explicit solutions of the classical Calogero & Sutherland systems for any root system
J. Math. Phys. 47 (1) (2006), 012701
Comments: 18 pages, LaTeX, no figure
Report-no: YITP-05-60, hep-th/0510035
Abstract Explicit solutions of the classical Calogero (rational with/without harmonic confining potential) and Sutherland (trigonometric potential) systems is obtained by diagonalisation of certain matrices of simple time evolution. The method works for Calogero & Sutherland systems based on any root system. It generalises the well-known results by Olshanetsky and Perelomov for the A type root systems. Explicit solutions of the (rational and trigonometric) higher Hamiltonian flows of the integrable hierarchy can be readily obtained in a similar way for those based on the classical root systems.

Kanehisa Takasaki
Dispersionless integrable hierarchies revisited
("Riemann-Hilbert problems, integrability and asymptotics", SISSA, Trieste, September 20--25, 2005) OHP transparencies of talk [pdf]

Soliton(in Japanese)
To appear in "Encyclopedia of Mathematical Sciences" Ver. 2 (Maruzen, Tokyo) [pdf]
A fully revised version of "Soliton" in Ver. 1. Contents: 1. Discovery of soliotns, 2. Inverse scattering and Lax form, 3. Conserved quantities and integrability, 4. Various equations and methods, 5. Bilinearization method, 6. KP hierarchy, 7. Other topis.
Kanehisa Takasaki
Tyurin parameters of commuting pairs and infinite dimensional Grassmannian manifold
M. Noumi and K. Takasaki (ed.), Elliptic Integrable Systems'', Rokko Lectures in Mathematics vol. 18, pp. 289--304 (Kobe University, 2005).
Comment: contribution to proceedings of RIMS workshop "Elliptic Integrable Systems" (RIMS, 2004)
Report-no: nlin.SI/0505005
Abstract Commuting pairs of ordinary differential operators are classified by a set of algebro-geometric data called algebraic spectral data''. These data consist of an algebraic curve (spectral curve'') $\Gamma$ with a marked point $\gamma_\infty$, a holomorphic vector bundle $E$ on $\Gamma$ and some additional data related to the local structure of $\Gamma$ and $E$ in a neighborhood of $\gamma_\infty$. If the rank $r$ of $E$ is greater than $1$, one can use the so called Tyurin parameters'' in place of $E$ itself. The Tyurin parameters specify the pole structure of a basis of joint eigenfunctions of the commuting pair. These data can be translated to the language of an infinite dimensional Grassmann manifold. This leads to a dynamical system of the standard exponential flows on the Grassmann manifold, in which the role of Tyurin parameters and some other parameters is made clear.

Kanehisa Takasaki
q-analogue of modified KP hierarchy and its quasi-classical limit (in Japanese)
Abstract of talk at the MSJ spring meeging 2005 (Chuo University, Tokyo, March 23--30, 2005) [pdf]
Abstract Bilinear and linear equations for a q-analogue of the modified KP hierarchy are derived. Hamilton-Jacobi equations etc. in quasi-classical limit are discussed.

Takashi Maeda, Toshio Nakatsu, Kanehisa Takasaki, Takeshi Tamakoshi
Free Fermion and Seiberg-Witten Differential in Random Plane Partitions
Nucl. Phys. B715 (2005) 275-303
Report-no: OU-HET 513, hep-th/0412329
Abstract A model of random plane partitions which describes five-dimensional $\mathcal{N}=1$ supersymmetric SU(N) Yang-Mills is studied. We compute the wave functions of fermions in this statistical model and investigate their thermodynamic limits or the semi-classical behaviors. These become of the WKB type at the thermodynamic limit. When the fermions are located at the main diagonal of the plane partition, their semi-classical wave functions are obtained in a universal form. We further show that by taking the four-dimensional limit the semi-classical wave functions turn to live on the Seiberg-Witten curve and that the classical action becomes precisely the integral of the Seiberg-Witten differential. When the fermions are located away from the main diagonal, the semi-classical wave functions depend on another continuous parameter. It is argued that they are related with the wave functions at the main diagonal by the renormalization group flow of the underlying gauge theory.

Takashi Maeda, Toshio Nakatsu, Kanehisa Takasaki, Takeshi Tamakoshi
Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions
JHEP 03 (2005), 056. Report-no: OU-HET 512, hep-th/0412327
Abstract Five-dimensional $\mathcal{N}=1$ supersymmetric Yang-Mills theories are investigated from the viewpoint of random plane partitions. It is shown that random plane partitions are factorizable as q-deformed random partitions so that they admit the interpretations as five-dimensional Yang-Mills and as topological string amplitudes. In particular, they lead to the exact partition functions of five-dimensional $\mathcal{N}=1$ supersymmetric Yang-Mills with the Chern-Simons terms. We further show that some specific partitions, which we call the ground partitions, describe the perturbative regime of the gauge theories. We also argue their role in string theory. The gauge instantons give the deformation of the ground partition.

Kanehisa Takasaki
$q$-analogue of modified KP hierarchy and its quasi-classical limit
Lett. Math. Phys. 72 (3) (2005), 165--181.
Report-no: nlin.SI/0412067
Abstract A $q$-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear $q$-difference equations. These bilinear equations are translated to the language of wave functions, which turn out to satisfy a system of linear $q$-difference equations. These linear $q$-difference equations are used to formulate the Lax formalism and the description of quasi-classical limit. These results can be generalized to a $q$-analogue of the Toda hierarchy. The results on the $q$-analogue of the Toda hierarchy might have an application to the random partition calculus in gauge theories and topological strings.

Kanehisa Takasaki
Hamiltonian structure of time evolutions of string equation (in Japanese)
Contributution to the proceedings of the RIMS workshop "Global and asymptotic analysis of differential equations in complex domain" (RIMS, Kyoto University, October 6--10, 2003) [pdf]
Abstract The so called strin eqution (the Douglas equation) has a commuting set of time evolutions that stem from the underlying KdV hierarchy . Applying the method of separation of variables for usual integrable systems, one can reformulate these time evolution as a nonautonomous Hamiltonian system. The Hamiltonians contain extra terms that do not exist in the Hamiltonian formulation of usual integrable systems and of the string equation itself.

Kanehisa Takasaki
Tyurin parameters and commutative rings of differential operators (in Japanese)
Article written as report for Grant-in-Aid for Scientific Research (2002--2003) [pdf]
Abstract Krichever and Novikov's research in 1970's revealed a link between Tyurin parameters and commutative rings of differential operators. In this atricle, we review an outline of this research as a clue to search for higher genus analogues of soliton equations. A commutative ring of differential operators is acompanied by an algebraic curve (called the spectral curve) and some other geometric data such as a holomorphic vector bundle. Tyurin parameters are a notion proposed in algebraid geometry as a kind of moduli of stable holomorphic vector bundles over algebraic curves; they arize here in a very natural way. If the ring of commutative differential operators is combined with time evolutions of the KP hierarchy, one obtains a higher genus analogue of classical soliton equations. Tyurin parameters plays a fundamental role in the formulation of this equation as well.

Kanehisa Takasaki
Elliptic spectral parameter and infinite dimensional Grassmann variety
Comments: Contribution to Faro conference "Infinite dimensional algebras and quantum integrable systems", Progress in Mathematics vol. 237, pp. 169--197 (Birkhauser, Basel/Switzerland, 2005)
Report-no: nlin.SI/0312016
Abstract Recent results on the Grassmannian perspective of soliton equations with an elliptic spectral parameter are presented along with a detailed review of the classical case with a rational spectral parameter. The nonlinear Schr\"odinger hierarchy is picked out for illustration of the classical case. This system is formulated as a dynamical system on a Lie group of Laurent series with factorization structure. The factorization structure induces a mapping to an infinite dimensional Grassmann variety. The dynamical system on the Lie group is thereby mapped to a simple dynamical system on a subset of the Grassmann variety. Upon suitable modification, almost the same procedure turns out to work for soliton equations with an elliptic spectral parameter. A clue is the geometry of holomorphic vector bundles over the elliptic curve hidden (or manifest) in the zero-curvature representation.

Kanehisa Takasaki
Landau-Lifshitz hierarchy and infinite dimensional Grassmann variety
Lett. Math. Phys. 67 (2) (2004), 141-152
Comments: latex2e (usepackage:amssyb), 15 pages, no figure; (v2) minor changes; (v3) typos corrected
Report-no: lin.SI/0312002
Abstract The Landau-Lifshitz equation is an example of soliton equations with a zero-curvature representation defined on an elliptic curve. This equation can be embedded into an integrable hierarchy of evolution equations called the Landau-Lifshitz hierarchy. This paper elucidates its status in Sato, Segal and Wilson's universal description of soliton equations in the language of an infinite dimensional Grassmann variety. To this end, a Grassmann variety is constructed from a vector space of $2 \times 2$ matrices of Laurent series of the spectral parameter $z$. A special base point $W_0$, called vacuum,'' of this Grassmann variety is chosen. This vacuum is dressed'' by a Laurent series $\phi(z)$ to become a point of the Grassmann variety that corresponds to a general solution of the Landau-Lifshitz hierarchy. The Landau-Lifshitz hierarchy is thereby mapped to a simple dynamical system on the set of these dressed vacua. A higher dimensional analogue of this hierarchy (an elliptic analogue of the Bogomolny hierarchy) is also presented.

Kanehisa Takasaki
Tyurin parameters and Sato theory (in Japanese)
Abstract of talk at Mathematical Society Autumn Meeting 2004 (Chiba University, September 24--27, 2003). [pdf] Removed because of a serious error in the contents (Nov. 2003)
Abstract Recently Krichever presented a general construction of Lax and zero curvature equations on algebraic curves of an arbitrary genus. We apply his idea to construct an elliptic (and higher genus) analogue of the nonlinear Schrodinger hierarchy, and show an interpretation of this system in the language of the Sato Grassmannian.

Kanehisa Takasaki
Tyurin parameters and elliptic analogue of nonlinear Schr\"odinger hierarchy
J. Math. Sci. Univ. Tokyo 11 (2004), 91--131
Comments: latex2e, 36 pp, no figure; (v2) minor changes, mostly typos; (v3) Title changed, text fully revised with new results; (v4) serious errors in section 5 corrected; (v5) proof of main results is improved; (v6) minor change in proof of Lemma 10 etc Report-no: nlin.SI/0307030
Abstract Two elliptic analogues'' of the nonlinear Schr\"odinger hiererchy are constructed, and their status in the Grassmannian perspective of soliton equations is elucidated. In addition to the usual fields $u,v$, these elliptic analogues have new dynamical variables called Tyurin parameters,'' which are connected with a family of vector bundles over the elliptic curve in consideration. The zero-curvature equations of these systems are formulated by a sequence of $2 \times 2$ matrices $A_n(z)$, $n = 1,2,\ldots$, of elliptic functions. In addition to a fixed pole at $z = 0$, these matrices have several extra poles. Tyurin parameters consist of the coordinates of those poles and some additional parameters that describe the structure of $A_n(z)$'s. Two distinct solutions of the auxiliary linear equations are constructed, and shown to form a Riemann-Hilbert pair with degeneration points. The Riemann-Hilbert pair is used to define a mapping to an infinite dimensional Grassmann variety. The elliptic analogues of the nonlinear Schr\"odinger hierarchy are thereby mapped to a simple dynamical system on a special subset of the Grassmann variety.

Soliton(in Japanese)
To appear in "Encyclopedic Dictionary of Mathematics" Ver. 4 (Iwanami Shoten, Tokyo) [pdf] A revised version of "Soliton" in the previous edition. Fully revised to cover such topics as 'Hamiltonian structure', 'bilinearization method', 'KP hierarchy' etc.

Kanehisa Takasaki and Takashi Takebe
Integrable system on the space of pairs of meromorphic functions (in Japanese)
Abstract of talk at Mathematical Society Annual Conference 2003 (University of Tokyo, March 23--26, 2003) [pdf]
Abstract The space of rational functions with the Atiyah-Hitchin symplectic strucutre is known to have the structure of an integrable system. We extend this result to the space of paris of meromorphic functions on a complex algebraic curve of arbitrary genus.

Kanehisa Takasaki
Landau-Lifshitz equation in perspectives of Sato theory (in Japanese)
Abstract of talk at Mathematical Society Annual Conference 2002 (University of Tokyo, March 23--26, 2003) [pdf]
Abstract The Landau-Lifshitz equation is known to have a Lax representation with matrices depending on a spectral parameter on the torus. We show how to formulate this equation in the framework of M. Sato's approach to soliton equations.

Kanehisa Takasaki
Complex WKB method viewed from Liouville plane (in Japanese)
Contribution to the proceedings of the workshop "In search of ideal mathematical analysis", Kyushu University, December 12 -- 14, 2002 [pdf]
Abstract An alternative approach to the exact WKB method is proposed. This approach is based on the so called Liouville transformation'' that converts the problem on the physical coordinate space to a scattering problem on another space.

Kanehisa Takasaki
Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains
Commun. Math. Phys. 241 (1) (2003), 111--142
Comments: latex2e, 41 pages no figure; Revised version (version 3), considerably shortened (56pages --> 41 pages), but some results are improved.
Report-no: nlin.SI/0206049
Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains
Report-no: nlin.SI/0206049
Abstract A chain of one-dimensional Schr\"odinger operators is called a dressing chain'' if they are connected by successive Darboux transformations. Particularly interesting are periodic dressing chains; they include finite-band operators and Painlev\'e equations as a special case. We investigate the Hamiltonian structure of these nonlinear lattices using V.~Adler's $2 \times 2$ Lax pair. The Lax equation and the auxiliary linear problem of this Lax pair contain a shift, rather than a derivative, in the spectral parameter. Despite this unusual feature, we can construct a transition matrix around the periodic chain, an associated spectral curve'' and a set of Darboux coordinates (spectral Darboux coordinates''). The dressing chain is thereby converted to a Hamiltonian system in these Darboux coordinates. Moreover, the Hamiltonian formalism is accompanied by an odd-dimensional Poisson structure. This induces a quadratic Poisson algebra of the matrix elements of the transition matrix. As a byproduct, we show that this Poisson structure is equivalent to another Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.

Kanehisa Takasaki
Similarity of periodic Darboux chains and periodic Toda lattices (in Japanese)
Contribution to the proceedings of the RIAM workshop "Recent topis of Nonlinear Waves and Nonlinear Dynamical Systems" (November 6--8, 2002) [pdf]
Abstract The peridic dressing chains turn out to resemble the periodic Toda chains in a variety of aspects, such as Lax representations, spectral curves, Hamiltonian structures, Poisson strucures, etc. This article presents a part of the similarity.

Kanehisa Takasaki
Integrable systems whose spectral curve is the graph of a function
Comments: CRM Proceedings and Lecture Notes vol. 37, pp. 211--222 (AMS, Province, 2004). Contribution to the proceedings of the conference "Superintegrability in classical and quantum systems" (Montreal, September 16--22, 2002), latex2e, 15 pages, no figure
Report-no: nlin.SI/0211021
Abstract For some integrable systems, such as the open Toda molecule, the spectral curve of the Lax representation becomes the graph $C = \{(\lambda,z) \mid z = A(\lambda)\}$ of a ffunction $A(\lambda)$. Those integrable systems provide an interesting toy model'' of separation of variables. Examples of this type of integrable systems are presented along with generalizations for which $A(\lambda)$ lives on a cylinder, a torus or a Riemann surface of higher genus.

Kanehisa Takasaki
Spectral curve and Hamiltonian structure of string equations (in Japanese)
TAlk at RIMS workshop "Deformations and asymptotic analysis of differential equations" (June 3--7, 2002) [pdf]
Abstract The commutator equation [Q,P] = 1 for differential operators in one variable is called the string equation. In this paper, we present a result on the Hamiltonian structure of this equation in the case where Q is of second order and P of odd (2g + 1) order. The string eqution in this case has a Lax representation of 2 x 2 matrices. We use this Lax representation to define the spectral curve (which is a hyperelliptic curve of genus g) and 2g canonical (Darboux) variables, and to rewrite the string equation to a Hamiltonian system.

kanehisa Takasaki and Takashi Takebe
Integrable system on a space of rational functions and its variants --- a perspective from separation of variables, algebraic surfaces and Seiberg-Witten theory (in Japanese)
Talk at Annual Conference of Mathematical Society of Japan (March 28 -- 31, 2002, Meiji University, Tokyo) [pdf]
Abstract Atiyah and Hithin introduced a symplectic structure on a space of rational functions, which they used for a twistorial description of the moduli space of SU(2) Yang-Mills monopoles. The symmetric polynomials of the poles of the rational function are Poisson-commutative, so that one obtains an integrable system on the space of rational functions. This integrable system can be identified with the one that Moser derived as an avatar of the finite nonperiodic Toda lattice in his classical work. In this talk, we note that this (almost trivial) integrable is actually a simple model of "separation of variables, algebraic surfaces and Seiberg-Witten theory", and present a few extensions based on that point of view.

Kanehisa Takasaki and Takashi Takebe
An integrable system on the moduli space of rational functions and its variants
Journal of Geometry and Physics 47 (1) (2003), 1--20
Comments: 25 pages, AMS-LaTeX, no figure
Report-no: nlin.SI/0202042
Abstract We study several integrable Hamiltonian systems on the moduli spaces of meromorphic functions on Riemann surfaces (the Riemann sphere, a cylinder and a torus). The action-angle variables and the separated variables (in Sklyanin's sense) are related via a canonical transformation, the generating function of which is the Abel-Jacobi type integral of the Seiberg-Witten differential over the spectral curve.

Kanehisa Takasaki
Spectral curve and Hamiltonian structure of dressing chain (in Japanese)
TAlk at RIMS workshop "Bilinearization Method for Integrable Systems and Related Topics" (July 2--4, 2001) [pdf]
Abstract The notion of "Dressing chain", introduced by Shabat et al., means a sequence of one-dimensional Schrodinger operators linked by a combination of the Darboux transformation and shift by a constant. Originating in the studies on spectral properties of Schrodinger operators, dressing chains are also known to be closely related to finite band potentials and Painleve equations. In this article, we first review the general features of dressing chains (in particular, the existence of two distinct Lax representations), then turn to the structre of the transfer matrix of a periodic chain and its spectral curve, and present a few results on the Hamiltonian structure.

Kanehisa Takasaki
Spectral curve and Hamiltonian structure of isomonodromic SU(2) Calogero-Gaudin system
J. Math. Phys. 44 (9) (2003), 3979--3999
Comments: The manuscript published in J. Math. Phys. vol. 44 turned out to contain serious errors. A corrected version is stored in the e-print arXiv as nlin.SI/0111019 ver.6.
Report-no: nlin.SI/0111019
Abstract This paper presents an approach to the Hamiltonian structure of isomonodromic systems of matrix ODE's on a torus from their spectral curve. An isomonodromic analogue of the so called $\rmSU(2)$ Calogero-Gaudin system is used for a case study of this approach. A clue of this approach is a mapping from the Lax equation to a dynamical system of a finite number of points on the spectral curve. The coordinates of these moving points give a new set of canonical variables, which have been used in the literature for separation of variables of many integrable systems including the usual $\rmSU(2)$ Calogero-Gaudin system itself. The same machinery turns out to work for the isomonodromic system on a trous, though the separability is lost and the non-autonomous nature of the system causes technical complications. Strong evidence is shown which suggests that this isomonodromic system is equivalent to a previously known isomonodromic system of second order scalar ODE's on a torus.

R. Sasaki and K. Takasaki
Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
J.Phys. A34 (2001), 9533--9554
Comments: LaTeX2e 28 pages, no figures
Report-no: hep-th/0109008, YITP-01-60 and KUCP-0191
Abstract Inozemtsev models are classically integrable multi-particle dynamical systems related to Calogero-Moser models. Because of the additional q^6 (rational models) or sin^2(2q) (trigonometric models) potentials, their quantum versions are not exactly solvable in contrast with Calogero-Moser models. We show that quantum Inozemtsev models can be deformed to be a widest class of partly solvable (or quasi-exactly solvable) multi-particle dynamical systems. They posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A new method for identifying and solving quasi-exactly solvable systems, the method of pre-superpotential, is presented.

Saburo Kakei, Takeshi Ikeda and Kanehisa Takasaki
Hierarchy of (2+1)-dimensional nonlinear Schroedinger equation, self-dual Yang-Mills equation, and toroidal Lie algebras
Ann. Henri Poincare 3 (2002), 817--845.
Comments: 26 pages, Latex2e, uses amsmath, amssymb, amsthm
Report-no: nlin.SI/0107065
Abstract The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed.

Kanehisa Takasaki
Integrable systems with rational spectral curves (in Japanese)
Talk at Annual Conference of Mathematical Society of Japan (Keio University, March 26 - 29, 2001) [pdf]
Absract Several examples of integrable systems with rational spectral curves are presented.

Kanehisa Takasaki
Painleve-Calogero correspondence
Talk at RIMS workshop "Analysis of Painleve equations" (October 23 - 27, 2000) [pdf]
Abstract The so called Painlev\'e-Calogero correspondence'' relates the sixth Painlev\'e equation with an integrable system of the Calogero type. This relation was recently generaized to the other Painlev\'e equations and a multi-component'' analogue. This paper reviews these results.

Kanehisa Takasaki
Separation of variables revived (in Japanese)
Lecture at the organized session "Applied analysis of dynamical systems -- point of view of geometry and integrable systems --", the annual conference of the Japan SIAM (Tokyo Institute of Technology, October 6 - 8, 2000) [pdf]
Abstract The separation of variables'' in the 19c has been revived in the light of modern integrable systems. I illustrate the basic ideas of the moderninzed separation of variable with a simple integrable system.

Kanehisa Takasaki
Hyperelliptic Integrable Systems on K3 and Rational Surfaces
Phys. Lett. A283 (2001), 201--208.
Comments: latex2e using packages "amsmath and amssymb", 15 pages
Report-no: KUCP-0161, math.AG/0007073
Abstract: We show several examples of integrable systems related to special K3 and rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double covering of the projective plane, a rational elliptic surface, etc.). The construction, based on Beauvilles's general idea, is considerably simplified by the fact that all examples are described by hyperelliptic curves and Jacobians. This also enables to compare these integrable systems with more classical integrable systems, such as the Neumann system and the periodic Toda chain, which are also associated with rational surfaces. A delicate difference between the cases of K3 and of rational surfaces is pointed out therein.

Kanehisa Takasaki
Anti-self-dual Yang-Mills equations on noncommutative spacetime
J. Geom. Phys. 37 (2001), 291 - 306.
Comments: latex2e using packages amsmath and amssymb, 24 pages
Report-no: KUCP-153, hep-th/0005194
Abstract: By replacing the ordinary product with the so called $\star$-product, one can construct an analogue of the anti-self-dual Yang-Mills (ASDYM) equations on the noncommutative $\bbR^4$. Many properties of the ordinary ASDYM equations turn out to be inherited by the $\star$-product ASDYM equation. In particular, the twistorial interpretation of the ordinary ASDYM equations can be extended to the noncommutative $\bbR^4$, from which one can also derive the fundamental strutures for integrability such as a zero-curvature representation, an associated linear system, the Riemann-Hilbert problem, etc. These properties are further preserved under dimensional reduction to the principal chiral field model and Hitchin's Higgs pair equations. However, some structures relying on finite dimensional linear algebra break down in the $\star$-product analogues.

Kanehisa Takasaki
Painleve-Calogero Correspondence Revisited
J. Math. Phys. 42 (3) (2001), 1443-1473.
Comments: latex2e using amsmath and amssymb packages, 40 pages, no figure
Report-no: KUCP 149, math.QA/0004118
Abstract: We extend the work of Fuchs, Painlev\'e and Manin on a Calogero-like expression of the sixth Painlev\'e equation (the Painlev\'e-Calogero correspondence'') to the other five Painlev\'e equations. The Calogero side of the sixth Painlev\'e equation is known to be a non-autonomous version of the (rank one) elliptic model of Inozemtsev's extended Calogero systems. The fifth and fourth Painlev\'e equations correspond to the hyperbolic and rational models in Inozemtsev's classification. Those corresponding to the third, second and first are not included therein. We further extend the correspondence to the higher rank models, and obtain a multi-component'' version of the Painlev\'e equations.

Kanehisa Takasaki and Takeshi Ikeda
Toroidal Lie algebras and Bogoyavlensky's 2+1-dimensional equation
International Mathematics Research Notices 7 (2001), 329--369
Comments: LaTeX2e with amsmath and amssymb, 35 pages, no figures
Report-no: nlin.SI/0004015
Abstract:We introduce an extension of the $\ell$-reduced KP hierarchy, which we call the $\ell$-Bogoyavlensky hierarchy. Bogoyavlensky's $2+1$-dimensional extension of the KdV equation is the lowest equation of the hierarchy in case of $\ell=2$. We present a group-theoretic characterization of this hierarchy on the basis of the $2$-toroidal Lie algebra ${\fraksl}_\ell^\tor$. This reproduces essentially the same Hirota bilinear equations as those recently introduced by Billig and Iohara et al. We can further derive these Hirota bilinear equation from a Lax formalism of the hierarchy.This Lax formalism also enables us to construct a family of special solutions that generalize the breaking soliton solutions of Bogoyavlensky. These solutions contain the $N$-soliton solutions, which are usually constructed by use of vertex operators.

Kanehisa Takasaki
Hyperelliptic integrable systems on elliptic K3 and rational surfaces
Talk at Anual Conference of Mathematical Society of Japan 2000 (Waseda University, March 27 - 30, 2000) [pdf]
Abstract After the work of Mukai and Beauville, it has come to be widely recognized that the moduli spaces of a certain class of sheaves on a K3 surface have the structure of algebraically integrable Hamiltonian systems (AIHS). The aim of this talk is to present a special example of those AIHS's on elliptic K3 (and rational) surfaces, along with an explicit description of action-angle variables''.

Kanehisa Takasaki
Isomonodromic deformation on tori (in Japanese)
Plenary talk at session "Infinite Analysis", Anual Conference of Mathematical Society Japan 2000 (Waseda University, March 27 - 30, 2000) [pdf]
Abstract This is a review of recent studies on isomonodromic deformations of ODEs on tori and their relation to integrable systems, conformal field theories and Painleve equations.

Kanehisa Takasaki
Painleve equations from the point of view of Calogero-Moser systems (in Japanese)
Based on lectures at RIMS workshop "Painleve systems, hypergeometric systems, and asymptotic analysis" (June 7 -10, 1999) and Kobe University workshop "Global analysis of Painleve equations" (October 26- 29, 1999) [pdf]
Abstract Manin proposed an alternative expression of Painleve's sixth equation. This expression takes the form of a non-autonomous version of the elliptic Inozemtsev system, which is a kind of elliptic Calogero-Moser systems. Recently, it was discovered that similar non-autonomous systems can be obtained from various elliptic Calogero-Moser systems, that those non-autonomous systems, like their autonomous partners, possess a Lax representation, and that this Lax representation leads to a characterization of those non-autonomous systems as isomonodromic deformations on a torus. I will show an overview of these results and their backgrounds. A few related topics are also presented, such as the degeneration to the other Painleve equations, a possible generalization to the Garnier systems, etc.

Kanehisa Takasaki
Modulation of nonlinear waves and Whitham equations (in Japanese)
Contribution to the proceedings of RIMS workshop "Applications of Renormalization Groups to Mathematical Sciences" (July 21 - 23, 1999) [pdf]
Abstract The so called Whitham averaging method is a generalization of the Bogoliubov-Krilov-Mitropol'ski method to nonlinear waves. Originally developed as a general technique for dispersive nonlinear waves, this method has turned out to be linked with a variety of mathematical structures, such as the geometry of hyperelliptic Riemann surfaces.

The Whitham method extracts the slow spacetime dynamics of parameters of the carrier wave by averaging the fast oscillation. This kind of methods, called the "adiabatic approximation", is widely used in physics. A typical example is the Born-Oppenheimer approximation in quantum mechanics, whose mathematical treatment is almost the same as the Whitham averaging method. The central idea of these methods is somehow very similar to the notion of renormalization groups.

Anticipating some link with the renormalization groups, we now illustrate the Whitham averaging method for the KdV equation.

Kanehisa Takasaki
Elliptic Calogero-Moser systems and isomonodromic deformations
Talk at Autumn Meeting of Mathemtical Society of Japan (Hiroshima University, September 1999) [pdf]
Abstract We consider a non-autonomous analogue of various elliptic Calogero-Moser systems. A particularly interesting case is the Inozemtsev system, which contains Manin's elliptic ($\mu$-equation'') form of the sixth Painlev\'e equation as a special case. We show that the non-autonomous analogues, like their autonomous counterparts, have a Lax pair. This Lax pair enables us to interpret the non-autonomous systems as isomonodromic deformations on a torus.

S.P. Khastgir, R. Sasaki and Kanehisa Takasaki
Calogero-Moser Models IV: Limits to Toda theory
Prog. Theor. Phys. 102 (4) (1999), 749-776.
Comments: LaTeX2e with amsfonts.sty, 33 pages, no figures
Report-no: YITP-99-20, KUCP-0132, hep-th/9907102
Abstract: Calogero-Moser models and Toda models are well-known integrable multi-particle dynamical systems based on root systems associated with Lie algebras. The relation between these two types of integrable models is investigated at the levels of the Hamiltonians and the Lax pairs. The Lax pairs of Calogero-Moser models are specified by the representations of the reflection groups, which are not the same as those of the corresponding Lie algebras. The latter specify the Lax pairs of Toda models. The Hamiltonians of the elliptic Calogero-Moser models tend to those of Toda models as one of the periods of the elliptic function goes to infinity, provided the dynamical variables are properly shifted and the coupling constants are scaled. On the other hand most of Calogero-Moser Lax pairs, for example, the root type Lax pairs, do not a have consistent Toda model limit. The minimal type Lax pairs, which corresponds to the minimal representations of the Lie algebras, tend to the Lax pairs of the corresponding Toda models.

Whitham Deformations and Tau Functions in N = 2 Supersymmetric Gauge Theories
Prog. Theor. Phys. Suppl. 135 (1999), 53-74.
Report-no: KUCP-0136, hep-th/9905224
Abstract: We review new aspects of integrable systems discovered recently in $N=2$ supersymmetric gauge theories and their topologically twisted versions. The main topics are (i) an explicit construction of Whitham deformations of the Seiberg-Witten curves for classical gauge groups, (ii) its application to contact terms in the $u$-plane integral of topologically twisted theories, and (iii) a connection between the tau functions and the blowup formula in topologically twisted theories.

Elliptic Calogero-Moser Systems and Isomonodromic Deformations
J. Math. Phys. 40 (11) (1999), 5787-5821
Report-no: KUCP-0133, math.QA/9905101
Abstract: We show that various models of the elliptic Calogero-Moser systems are accompanied with an isomonodromic system on a torus. The isomonodromic partner is a non-autonomous Hamiltonian system defined by the same Hamiltonian. The role of the time variable is played by the modulus of the base torus. A suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic Calogero-Moser system turns out to give a Lax representation of the non-autonomous system as well. This Lax representation ensures that the non-autonomous system describes isomonodromic deformations of a linear ordinary differential equation on the torus on which the spectral parameter of the Lax pair is defined. A particularly interesting example is the extended twisted $BC_\ell$ model'' recently introduced along with some other models by Bordner and Sasaki, who remarked that this system is equivalent to Inozemtsev's generalized elliptic Calogero-Moser system. We use the root type'' Lax pair developed by Bordner et al. to formulate the associated isomonodromic system on the torus.

Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups
Int. J. Mod. Phys. A15 (23) (2000), 3635-3666
Report-no: KUCP-0127, hep-th/9901120
Abstract: Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the $SU(N+1)$ $\calN = 2$ SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the $A^{(2)}_{2N}$ affine Toda system. Our construction, too, uses fractional powers of the superpotential $W(x)$ that characterizes the curve. We also consider the $u$-plane integral of topologically twisted theories on four-dimensional manifolds $X$ with $b_2^{+}(X) = 1$ in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind.

A.J. Bordner, R. Sasaki and K. Takasaki
Calogero-Moser Models II: Symmetries and Foldings
Prog. Thero. Phys. 101 (3) (1999), 487-518.
Comments: 35 pages, LaTeX2e with amsfonts, no-figure
Report-no: YITP-98-60, KUCP-0121, hep-th/9809068
Abstract: Universal Lax pairs (the root type and the minimal type) are presented for Calogero-Moser models based on simply laced root systems including (E_8). They are with and without spectral parameter and they work for all of the four choices of potentials: the rational, trigonometric, hyperbolic and elliptic. For the elliptic potential, the discrete symmetries of the simply laced models, originating from the automorphism of the extended Dynkin diagrams are combined with the periodicity of the potential to derive a class of Calogero-Moser models known as the twisted non-simply laced models'. Among them, a twisted (BC_n) model is new and it has some novel features. For untwisted non-simply laced models, two kinds of root type Lax pairs (based on long roots and short roots) are derived which contain independent coupling constants for the long and short roots. The (BC_n) model contains three independent couplings, for the long, middle and short roots. The (G_2) model based on long roots exhibits a new feature which deserves further study.

Kanehisa Takasaki
Construction of isomonodromic problems on torus
Talk at Autumn Meeting of Mathemtical Society of Japan (Osaka University, September 1998) [pdf]

This talk deals with the problem of constructuting isomonodromic deformations of a first order matrix system $\frac{dY}{dz} = L(z) Y$ on the torus. The matrix linear system is assumed to have regular singularities at $N$ points $z = t_1,\cdots,t_N$. The problem is to construct isomonodromic deformations that leave a set of monodromy data invariant while the position of the poles and the modulus $\tau$ being varied. I shall present two types of such isomonodromic systems. Both have an isospectral partner (the elliptic Calogero-Gaudin system / the ordinary elliptic Gaudin system), and are also related to a conformal field theory (the ordinary WZW model / its `twisted'' version by Kuroki and Takebe) on a torus.