Papers and Preprints
A larger list of publications, talks, etc. (in Japanese)
is available here.
Eprints can be downloaded from the arxiv.org
and its mirror sistes.

K. Takasaki
Singular Cauchy problems for a class of weakly hyperbolic
differential operators
Proc. Japan Acad. 57A (1981), 393397.
[scanned image]

K. Takasaki
Singular Cauchy problems for a class of weakly hyperbolic
differential operators
Comm. Partial Differential Equations 7 (1982), 11511188.
[scanned image]

K. Ueno and K. Takasaki
Toda lattice hierarchy I and II
Proc. Japan Acad. 59A (1983), 167170, 215218.
[scanned image of I 
scanned image of II]

K. Takasaki
A class of solutions to the selfdual YangMills equations
Proc. Japan Acad. 59A (1983), 308311.
[scanned image]

K. Takasaki
On the structure of solutions to the selfdual YangMills equations
Proc. Japan Acad. 59A (1983), 418421.
[scanned image]

Ueno, K., and K. Takasaki
Toda lattice hierarchy
Comments: 1) The contents of section 2.4 are wrong except that
the tau functions for s = 0,1 can be defined and satisfy
the bilinear equaztions of the 2component BKP herarrchy.
2) The second line of (A.41) should be multiplied by
the inverse of lambda, cf. (A.44).
K. Okamoto (ed.),
{\it Group Representations and Systems of Differential Equations},
Advanced Studies in Pure Math. 4, pp. 195
(Kinokuniya, Tokyo, 1984).
[scanned image]

K. Takasaki
Initial value problem for the Toda lattice hierarchy
K. Okamoto (ed.),
{\it Group Representations and Systems of Differential Equations},
Advanced Studies in Pure Math. 4, pp. 139163
(Kinokuniya, Tokyo, 1984).
[scanned image]

K. Takasaki
A new approach to the selfdual YangMills equations
Commun. Math. Phys. 94 (1984), 3559.
Keywords:
gauge field theory, yangmills/ field equations, soliton/ gauge field
theory, geometrical/ field theory, completely integrable/ algebra,
commutation relations/ mathematical methods/

K. Takasaki
A new approach to the selfdual YangMills equations II
Saitama Math. J. 3 (1985), 1140.
Comments: The last paragraph of section 2.6 is wrong.
Namely, a rational initial data can give rise to
nonrational solution.
[scanned image]

K. Takasaki
Aspects of integrability in selfdual Einstein metrics
and related Equations
Publ. RIMS, Kyoto Univ., 22 (1986), 949990.
Reportno: RIMS669 (October 1989)
Comments: The last paragraph of section 5 of this paper is
wrong. Namely, as opposed to the statement therein,
the time evolution can be defined only on the orbit of
a loop group action, not on the whole Grassmannian
introduced therein. An improved reformulation can be
found in the appendix of "Symmetries of hyperK\"{a}hler
(or Poisson gauge field) hierarchy",
J. Math. Phys. 31 (1990), 18771888.
[scanned image]

K. Takasaki
Issues of multidimensional integrable systems
M. Kashiwara and T. Kawai (eds.),
{\it Algebraic Analysis}, Vol. II, pp. 853866 (Academic Press, 1988).
[scanned image]
Reportno: RIMS588

K. Takasaki
Integrable systems as deformations of ${\cal D}$modules
Proc. Symp. Pure Math. 49 , Part I, pp. 143168
(American Mathematical Society, 1989).
[scanned image]
Comments: talk at the AMS Summer Institute "Theta Functions,"
Bowdoin College, New Brunswick, 1987

K. Takasaki
An infinite number of hidden variables
in hyperK\"{a}hler metrics
J. Math. Phys. 30 (1989), 15151521.
Reportno: RIMS621 (March 1988),
Keywords:
mathematical methods, differential geometry/ spacetime, kaehler/
mathematical methods, twistor/

K. Takasaki
Geometry of universal Grassmann manifold
from algebraic point of view
Reviews in Math. Phys. 1 (1989), 146.
Reportno: RIMS623 (May 1988).
Keywords:
mathematical methods, differential geometry/ mathematical methods, fibre
bundle/ spacetime, grassmann/ group theory, geometrical/ algebra, lie/
commutation relations/ fibre bundle/

K. Takasaki
An infinite number of Hamiltonian flows
arising from hyperK\"{a}hler metric
Y. SaintAubin and L. Vinet (eds.),
{\it the XVIIth International Colloquium on
Group Theoretical Methods in Physics}, SainteAd\`{e}le 1988,
pp. 516519. (World Scientific, Sigapore, 1989).
[scanned image]

K. Takasaki
Symmetries of the super KP hierarchy
Lett. Math. Phys. 17 (1989), 351357.
Reportno: RIMS641 (November 1988).

K. Takasaki
Differential algebras and ${\cal D}$modules
in super Toda lattice hierarchy
Lett. Math. Phys. 19 (1990), 229236.
Reportno: RIMS665 (July 1989)

K. Takasaki
Hierarchy structure in integrable systems of
gauge fields and underlying Lie algebras
Commun. Math. Phys. 127 (1990), 225238.
Reportno: RIMS637 (October 1988)
Keywords:
gauge field theory, yangmills/ duality/ field equations, integrability/
algebra, lie/ riemannhilbert transformation/

K. Takasaki
Symmetries of hyperK\"{a}hler
(or Poisson gauge field) hierarchy
J. Math. Phys. 31 (1990), 18771888.
Reportno: RIMS669 (October 1989)
Keywords:
spacetime, kaehler/ loop space/ hamiltonian formalism/ differential
geometry/

Miyajima, T., Nakayashiki, A. and K. Takasaki
Structure and duality of ${\cal D}$modules related to KP hierarchy
J. Math. Soc. Japan 43 (1991), 751773.
[manuscript]
Reportno: RIMS689 (March 1990)

K. Takasaki
Integrable systems in gauge theory, K\"ahler geometry
and super KP hierarchy  symmetries and algebraic point of view
Proc. International Congress of Mathematicians, Kyoto, 1990,
pp. 12051214 (SpringerVerlag, 1991).
Reportno: RIMS714 (September 1990)
[manuscript]
Keywords:
talk, kyoto 1990/08/ gauge field theory, yangmills/ field theory, kaehler/
differential equations, nonlinear/ differential equations,
kadomtsevpetviashvili/ integrability/ duality/ mathematical methods/

K. Takasaki
Analytic expression of Voros coefficients
and its application to WKB connection problem
M. Kashiwara and T. Miwa (eds.),
{\it Special functions\/},
ICM90 Satellite Conference Proceedings,
pp. 294315
(SpringerVerlag, BerlinNew YorkTokyo, 1991).
[scanned image]
Reportno: RIMS725 (November 1990)
Abstract:
Usually, the
WKB method starts from formal solutions (WKB or LiouvilleGreen
solutions) expanded in powers of the Planck constant, and connects
these solutions by asymptotic matching at turning points.
A resummation prescription of these formal calculations was
proposed by A. Voros, after an idea of Balian and Bloch, and
illustrated for a homogeneous quartic oscillator. Voros argued
that his results should be deeply related with J. Ecalle's theory
of "resurgent functions." Further progress along that line has
been made by F. Pham and his coworkers. I shall report another
approach based upon a classical idea of F. Olver.

K. Takasaki
Hidden symmetries of integrable system in YangMills theory
and K\"ahler geometry
S\'eminaire sur les \'Equations aux D\'eriv\'ees Partielles,
19901991, Expos\'e n$^o$ VIII, pp. 115, 22 Janvier 1991
(Ecole Polytechnique, 1991).
[seminar notes]
Reportno: RIMS743 (March 1991)
Keywords:
talk, paris 1991/01/22/ gauge field theory, yangmills/ field equations,
duality/ field theory, euclidean/ integrability/ differential forms/
einstein equation/ differential geometry, kaehler/
Abstract:
This article is an r\'esum\'e of the present author's work [Ta24]
in recent years on symmetries of the selfduality equations and
related equations. A central issue was to find an explicit form
of {\sl infinitesimal\/} symmetries of the selfdual Einstein
equation that should correspond to the {\sl finite\/} symmetries
of Boyer and Plebanski [BoPl]. Main results are presented in
Section 5. In general, finite symmetries of nonlinear systems
are given by very complicated nonlinear transformation of
dependent (and, sometimes, of independent) variables. This is
already so for soliton equations and the selfdual YangMills
equations whose RiemannHilbert factorization is still relatively
well understood. The case of the selfdual Einstein equations
is far worse; there is no effective way to solve the factorization,
only the existence of a solution being ensured by a general theorem.
Infinitesimal symmetries, on the other hand, should have a more
explicit, and even beautiful expression as experiences in soliton
equation and the selfdual YangMills equation advocate. This
indeed turns out to be the case for the selfdual Einstein equation.

K. Takasaki and T. Takebe
SDiff(2) Toda equation 
hierarchy, tau function and symmetries
Lett. Math. Phys. 23 (1991), 205214.
Reportno: RIMS790 (August, 1991), hepth/9112042
Keywords:
field theory, toda/ lattice field theory/ continuum limit/ dimension, 2/
field equations/ integrability/ differential equations, lax/ transformation,
diffeomorphism/ transformation, symplectic/
Abstract:
A continuum limit of the Toda lattice field theory, called the SDiff(2)
Toda equation, is shown to have a Lax formalism and an infinite hierarchy of
higher flows. The Lax formalism is very similar to the case of the selfdual
vacuum Einstein equation and its hyperK\"ahler version, however now based
upon a symplectic structure and the group SDiff(2) of area preserving
diffeomorphisms on a cylinder $S^1 \times \R$. An analogue of the Toda
lattice tau function is introduced. The existence of hidden SDiff(2)
symmetries are derived from a RiemannHilbert problem in the SDiff(2) group.
Symmetries of the tau function turn out to have commutator anomalies, hence
give a representation of a central extension of the SDiff(2) algebra.

K. Takasaki, and T. Takebe
SDiff(2) KP hierarchy
A. Tsuchiya, T. Eguchi and T. Miwa (eds.),
{\it Infinite Analysis\/}, Adv. Ser. Math. Phys. 16
(World Scientific, Singapore, 1992), part B, pp.889922.
Reportno: RIMS814 (October, 1991, revised version December, 1991),
hepth/9112046
Abstract:
An analogue of the KP hierarchy, the SDiff(2) KP hierarchy, related to
the group of areapreserving diffeomorphisms on a cylinder is proposed. An
improved Lax formalism of the KP hierarchy is shown to give a prototype of
this new hierarchy. Two important potentials, $S$ and $\tau$, are
introduced. The latter is a counterpart of the tau function of the ordinary
KP hierarchy. A RiemannHilbert problem relative to the group of area
diffeomorphisms gives a twistor theoretical description (nonlinear graviton
construction) of general solutions. A special family of solutions related to
topological minimal models are identified in the framework of the Riemann
Hilbert problem. Further, infinitesimal symmetries of the hierarchy are
constructed. At the level of the tau function, these symmetries obey
anomalous commutation relations, hence leads to a central extension of the
algebra of infinitesimal areapreserving diffeomorphisms (or of the
associated Poisson algebra).

K. Takasaki
Areapreserving diffeomorphisms and nonlinear integrable systems
J. Mickelsson and O. Pekonen (eds.),
{\it Topological and geometrical methods in field theory\/},
Turku, Finland, May 26  June 1, 1991
pp. 383397
(World Scientific, Singapore, 1992).
Reportno: KUCP0039/91 (September, 1991), hepth/9112041
Abstract:
Present state of the study of nonlinear "integrable" systems related to
the group of areapreserving diffeomorphisms on various surfaces is
overviewed. Roles of areapreserving diffeomorphisms in 4d selfdual
gravity are reviewed. Recent progress in new members of this family, the
SDiff(2) KP and Toda hierarchies, is reported. The group of areapreserving
diffeomorphisms on a cylinder plays a key role just as the infinite matrix
group GL($\infty$) does in the ordinary KP and Toda lattice hierarchies. The
notion of tau functions is also shown to persist in these hierarchies, and
gives rise to a central extension of the corresponding Lie algebra.

K. Takasaki
Volumepreserving diffeomorphisms in integrable deformations of
selfdual gravity
Phys. Lett. B285 (1992), 187190.
Reportno: KUCP0046/92 (March, 1992), hepth/9203034
Abstract:
A group of volumepreserving diffeomorphisms in 3D turns out to play a key
role in an EinsteinMaxwell theory whose Weyl tensor is selfdual and whose
Maxwell tensor has algebraically general antiselfdual part. This model was
first introduced by Flaherty and recently studied by Park as an integrable
deformation of selfdual gravity. A twisted volume form on the corresponding
twistor space is shown to be the origin of volumepreserving
diffeomorphisms. An immediate consequence is the existence of an infinite
number of symmetries as a generalization of $w_{1+\infty}$ symmetries in
selfdual gravity. A possible relation to Witten's 2D string theory is
pointed out.

K. Takasaki
W algebra, twistor, and nonliear integrable systems
RIMS Kokyuroku 810 (September, 1992).
Comments: Expanded version of talk at RIMS workshop
"Algebraic Analysis and Number Theory," March 2328, 1992.
Reportno: KUCP0049/92 (June, 1992), hepth/9206030
Keywords:
talk/ differential equations, nonlinear/ integrability/ gravitation,
duality/ kadomtsevpetviashvili equation/ differential equations, toda/
differential equations, hierarchy/ algebra, w(n)/ twistor/
Abstract:W algebras arise in the study of various nonlinear
integrable systems such as: selfdual gravity, the KP and Toda
hierarchies, their quasiclassical (or dispersionless) limit, etc.
Twistor theory provides a geometric background for these algebras.
Present state of these topics is overviewed. A few ideas on possible
deformations of selfdual gravity (including quantum deformations)
are presented.

K. Takasaki and T. Takebe
Quasiclassical limit of KP hierarchy,
Wsymmetries and free fermions
V.E. Matveev (ed),
Proceedings of Lobachevsky Semester of
Euler International Institute, 1992, St. Petersburg.
Zapiski Nauchnykh Seminarov POMI 235 (1996), 295  303.
Reportno: KUCP0050/92 (July, 1992), hepth/9207081
Abstract:
This paper deals with the dispersionless KP hierarchy from the point of view
of quasiclassical limit. Its Lax formalism, Winfinity symmetries and
general solutions are shown to be reproduced from their counterparts in the
KP hierarchy in the limit of $\hbar \to 0$. Free fermions and bosonized
vertex operators play a key role in the description of Winfinity symmetries
and general solutions, which is technically very similar to a recent free
fermion formalism of $c=1$ matrix models.

K. Takasaki, and T. Takebe
Quasiclassical limit of Toda hierarchy
and Winfinity symmetries
Lett. Math. Phys. 28 (1993), 165176.
Reportno: KUCP0057/93 (Januray, 1993), hepth/9301070
Abstract:
Previous results on quasiclassical limit of the KP hierarchy and its W
infinity symmetries are extended to the Toda hierarchy. The Planck constant
$\hbar$ now emerges as the spacing unit of difference operators in the Lax
formalism. Basic notions, such as dressing operators, BakerAkhiezer
functions and tau function, are redefined. $W_{1+\infty}$ symmetries of the
Toda hierarchy are realized by suitable rescaling of the DateJimbo
KashiaraMiwa vertex operators. These symmetries are contracted to
$w_{1+\infty}$ symmetries of the dispersionless hierarchy through their
action on the tau function.

K. Takasaki
Quasiclassical limit of BKP hierarchy
and Winfinity symmetries
Lett. Math. Phys. 28 (1993), 177185.
Comments: The contents of this paper concerning
the BKP hierarchy itself turned out to be
completely wrong, though the results on
the dispersionless limit seem to remain intact.
The main source of errors is (14),
which should read M^{*} =  M + L.
This led to a wrong definition, (53), of
the generators A_{ij} of symmetries.
A correct reformulation of the results of
this paper is presented by M.H. Tu,
Lett. Math. Phys. 81 (2007), 91105
(arXiv:nlin.SI/0611053).
Reportno: KUCP0058/93 (January, 1993), hepth/9301090
Abstract:
Previous results on quasiclassical limit of the KP and Toda hierarchies
are now extended to the BKP hierarchy. Basic tools such as the Lax
representation, the BakerAkhiezer function and the tau function are
reformulated so as to fit into the analysis of quasiclassical limit. Two
subalgebras $\WB_{1+\infty}$ and $\wB_{1+\infty}$ of the Winfinity algebras
$W_{1+\infty}$ and $w_{1+\infty}$ are introduced as fundamental Lie algebras
of the BKP hierarchy and its quasiclassical limit, the dispersionless BKP
hierarchy. The quantum Winfinity algebra $\WB_{1+\infty}$ emerges in
symmetries of the BKP hierarchy. In quasiclassical limit, these
$\WB_{1+\infty}$ symmetries are shown to be contracted into $\wB_{1+\infty}$
symmetries of the dispersionless BKP hierarchy.

K. Takasaki
Integrable hierarchy underlying topological
LandauGinzburg models of Dtype
Lett. Math. Phys. 29 (1993), 111121.
Reportno: Kyoto University KUCP0061/93 (March, 1993), hepth/9305053
Abstract:
A universal integrable hierarchy underlying topological LandauGinzburg
models of Dtye is presented. Like the dispersionless Toda hierarchy, the
new hierarchy has two distinct ("positive" and "negative") set of flows.
Special solutions corresponding to topological LandauGinzburg models of D
type are characterized by a RiemannHilbert problem, which can be converted
into a generalized hodograph transformation. This construction gives an
embedding of the finite dimensional small phase space of these models into
the full space of flows of this hierarchy. One of flat coordinates in the
small phase space turns out to be identical to the first "negative" time
variable of the hierarchy, whereas the others belong to the "positive"
flows.

K. Takasaki
Dressing operator approach to Moyal algebraic
deformation of selfdual gravity
Journal of Geometry and Physics 14 (1994), 111120.
Reportno: Kyoto University KUCP0054/92, hepth/9212103
Abstract:
Recently Strachan introduced a Moyal algebraic deformation of selfdual
gravity, replacing a Poisson bracket of the Plebanski equation by a Moyal
bracket. The dressing operator method in soliton theory can be extended to
this Moyal algebraic deformation of selfdual gravity. Dressing operators are
defined as Laurent series with coefficients in the Moyal (or star product)
algebra, and turn out to satisfy a factorization relation similar to the
case of the KP and Toda hierarchies. It is a loop algebra of the Moyal
algebra (i.e., of a $W_\infty$ algebra) and an associated loop group that
characterize this factorization relation. The nonlinear problem is
linearized on this loop group and turns out to be integrable.

K. Takasaki
Nonabelian KP hierarchy with Moyal algebraic coefficients
Journal of Geometry and Physics 14 (1994), 332364.
Reportno: Kyoto University KUCP0062/93, hepth/9305169
Abstract:
A higher dimensional analogue of the KP hierarchy is presented. Fundamental
constituents of the theory are pseudodifferential operators with Moyal
algebraic coefficients. The new hierarchy can be interpreted as large$N$
limit of multicomponent ($\gl(N)$ symmetric) KP hierarchies. Actually, two
different hierarchies are constructed. The first hierarchy consists of
commuting flows and may be thought of as a straightforward extension of the
ordinary and multicomponent KP hierarchies. The second one is a hierarchy
of noncommuting flows, and related to Moyal algebraic deformations of
selfdual gravity. Both hierarchies turn out to possess quasiclassical
limit, replacing Moyal algebraic structures by Poisson algebraic structures.
The language of Winfinity algebras provides a unified point of view to
these results.

K. Takasaki
Dispersionless Toda hierarchy and twodimensional string theory
Commun. Math. Phys. 170 (1995), 101116.
Reportno: Kyoto University KUCP0067/94, hepth/9403190
Abstract:
The dispersionless Toda hierarchy turns out to lie in the heart of a
recently proposed LandauGinzburg formulation of twodimensional string
theory at selfdual compactification radius. The dynamics of massless
tachyons with discrete momenta is shown to be encoded into the structure of
a special solution of this integrable hierarchy. This solution is obtained
by solving a RiemannHilbert problem. Equivalence to the tachyon dynamics is
proven by deriving recursion relations of tachyon correlation functions in
the machinery of the dispersionless Toda hierarchy. Fundamental ingredients
of the LandauGinzburg formulation, such as LandauGinzburg potentials and
tachyon LandauGinzburg fields, are translated into the language of the Lax
formalism. Furthermore, a wedge algebra is pointed out to exist behind the
RiemannHilbert problem, and speculations on its possible role as generators
of "extra" states and fields are presented.

K. Takasaki, and T. Takebe
Integrable hierarchies and dispersionless limit
Reviews in Mathematical Physics 7 (1995), 743808.
Reportno: University of Tokyo UTMS 9435, hepth/9405096
Abstract:
Analogues of the KP and the Toda lattice hierarchy called dispersionless KP
and Toda hierarchy are studied. Dressing operations in the dispersionless
hierarchies are introduced as a canonical transformation, quantization of
which is dressing operators of the ordinary KP and Toda hierarchy. An
alternative construction of general solutions of the ordinary KP and Toda
hierarchy is given as twistor construction which is quatization of the
similar construction of solutions of dispersionless hierarchies. These
results as well as those obtained in previous papers are presented with
proofs and necessary technical details.

K. Takasaki
Symmetries and tau function of higher dimensional
dispersionless integrable hierarchies
J. Math. Phys. 36 (1995), 35743607.
Reportno: Kyoto University KUCP0068, hepth/9407098
Abstract:
A higher dimensional analogue of the dispersionless KP hierarchy is
introduced. In addition to the twodimensional "phase space" variables
$(k,x)$ of the dispersionless KP hierarchy, this hierarchy has extra spatial
dimensions compactified to a two (or any even) dimensional torus.
Integrability of this hierarchy and the existence of an infinite dimensional
of "additional symmetries" are ensured by an underlying twistor
theoretical structure (or a nonlinear RiemannHilbert problem). An analogue
of the tau function, whose logarithm gives the $F$ function ("free energy"
or "prepotential" in the contest of matrix models and topological
conformal field theories), is constructed. The infinite dimensional
symmetries can be extended to this tau (or $F$) function. The extended
symmetries, just like those of the dispersionless KP hierarchy, obey an
anomalous commutation relations.

T. Nakatsu, K. Takasaki and S. Tsujimaru
Quantum and classical aspects of deformed $c=1$ strings
Nucl. Phys. B443 (1995), 155197.
Reportno: INSrep.1087, KUCP0077, hepth/9501038
Abstract:
The quantum and classical aspects of a deformed $c=1$ matrix model proposed
by Jevicki and Yoneya are studied. String equations are formulated in the
framework of Toda lattice hierarchy. The Whittaker functions now play the
role of generalized Airy functions in $c<1$ strings. This matrix model has
two distinct parameters. Identification of the string coupling constant is
thereby not unique, and leads to several different perturbative
interpretations of this model as a string theory. Two such possible
interpretations are examined. In both cases, the classical limit of the
string equations, which turns out to give a formal solution of Polchinski's
scattering equations, shows that the classical scattering amplitudes of
massless tachyons are insensitive to deformations of the parameters in the
matrix model.

K. Takasaki
Integrable hierarchies, dispersionless limit
and string equations
M. Morimoto and T. Kawai (eds.),
{\it Structures of Solutions of Differential Equations\/},
pp. 457481 (World Scientific, Singapore, 1996).
[manuscript]
Abstract:
The notion of string equations was discovered in the end of
the eighties, and has been studied in the language of integrable
hierarchies. String equations in the KP hierarchy are nowadays
relatively well understood. Meanwhile, systematic studies of
string equations in the Toda hierarchy started rather recently.
This article presents the state of art of these issues from
the author's point of view.

T. Nakatsu and K. Takasaki
WhithamToda hierarchy and N = 2 supersymmetric YangMills theory
Mod. Phys. Lett. A11 (2) (1996), 157168.
Reportno: KUCP0083, hepth/9509162
Abstract:
The exact solution of $N=2$ supersymmetric $SU(N)$ YangMills theory is
studied in the framework of the Whitham hierarchies. The solution is
identified with a homogeneous solution of a Whitham hierarchy. This
integrable hierarchy (WhithamToda hierarchy) describes modulation of a
quasiperiodic solution of the (generalized) Toda lattice hierarchy
associated with the hyperelliptic curves over the quantum moduli space. The
relation between the holomorphic prepotential of the low energy effective
action and the $\tau$ function of the (generalized) Toda lattice hierarchy
is also clarified.

K. Takasaki
Toda lattice hierarchy and generalized string equations
Commun. Math. Phys. 181 (1) (1996), 131156.
Reportno: KUCP0079, hepth/9506089.
Abstract:
String equations of the $p$th generalized Kontsevich model and the
compactified $c = 1$ string theory are reexamined in the language of the
Toda lattice hierarchy. As opposed to a hypothesis postulated in the
literature, the generalized Kontsevich model at $p = 1$ does not coincide
with the $c = 1$ string theory at selfdual radius. A broader family of
solutions of the Toda lattice hierarchy including these models are
constructed, and shown to satisfy generalized string equations. The status
of a variety of $c \le 1$ string models is discussed in this new framework.

K. Takasaki and T. Nakatsu
Isomonodromic deformations and supersymmetric YangMills theories
Int. J. Mod. Phys. A 11 (38) (1996), 55055518.
Reportno: KUCP0092, hepth/9603069.
Abstract:
SeibergWitten solutions of fourdimensional supersymmetric gauge theories
possess rich but involved integrable structures. The goal of this paper is
to show that an isomonodromy problem provides a unified framework for
understanding those various features of integrability. The SeibergWitten
solution itself can be interpreted as a WKB limit of this isomonodromy
problem. The origin of underlying Whitham dynamics (adiabatic deformation of
an isomonodromy problem), too, can be similarly explained by a more refined
asymptotic method (multiscale analysis). The case of $N=2$ SU($s$)
supersymmetric YangMills theory without matter is considered in detail for
illustration. The isomonodromy problem in this case is closely related to
the third Painlev\'e equation and its multicomponent analogues. An implicit
relation to $t\tbar$ fusion of topological sigma models is thereby expected.

T. Nakatsu and K. Takasaki
Integrable system and N=2 supersymmetric YangMills theory
H. Itoyama et al. (eds.),
{\it Frontiers in Quantum Field Theory}, 325330
(World Scientific, Singapore, 1996).
Reportno: RITS9601, hepth/9603129
Abstract:
The exact solutions (SeibergWitten type) of $N=2$ supersymmetric YangMills
theory are discussed from the view of WhithamToda hierarchy.

Partha Guha and Kanehisa Takasaki
Dispersionless hierarchies, HamiltonJacobi theory
and twistor correspondences
J. Geom. Phys. 25 (34) (1998), 326340.
Comments: latex, 20pp, no figures.
Reportno:RIMS1124 (Jan, 1997), solvint/9705013.
Abstract:
The dispersionless KP and Toda hierarchies possess an
underlying twistorial structure. A twistorial approach is
partly implemented by the method of RiemannHilbert problem.
This is however still short of clarifying geometric ingredients
of twistor theory, such as twistor lines and twistor surfaces.
A more geometric approach can be developed in a HamiltonJacobi
formalism of Gibbons and Kodama.

K. Takasaki
Spectral curves and Whitham equations in isomonodromic problems
of Schlesinger type
Asian J.Math. 2 (4) (1998), 10491078.
Comments: latex, 40pp, no figures.
Reportno:KUCP0105, solvint/9704004.
Abstract:
The Schlesinger equation is reformulated to include a small
parameter $\epsilon$. In the small$\epsilon$ limit, solutions
of this isomonodromic problem are expected to behave like a
slowly modulated finitegap solution of an isospectral problem.
The modulation is caused by slow deformations of the spectral
curve of the finitegap solution. A modulation equation of this
slow dynamics is derived by a heuristic method. An inverse
period map of SeibergWitten type turns out to give general
solutions of this modulation equation. This construction of
general solution also reveals the existence of deformations
of SeibergWitten type on the same moduli space of spectral
curves. A prepotential is also constructed in the same way as
the prepotential of the SeibergWitten theory.

Kanehisa Takasaki
Dual isomonodromic problems and Whitham equations
Lett. Math. Phys. 43 (2) (1998), 123135.
Comments: 15 pages, latex, no figures
Reportno: KUCP0106, solvint/9705016
Abstract:
The author's recent results on an asymptotic description of the Schlesinger
equation are generalized to the JMMS equation. As in the case of the
Schlesinger equation, the JMMS equation is reformulated to include a small
parameter $\epsilon$. By the method of multiscale analysis, the isomonodromic
problem is approximated by slow modulations of an isospectral problem. A
modulation equation of this slow dynamics is proposed, and shown to possess a
number of properties similar to the Seiberg Witten solutions of low energy
supersymmetric gauge theories.

Kanehisa Takasaki
Gaudin model, KZ Equation, and isomonodromic problem on torus
Lett. Math. Phys. 44 (2) (1998), 143156.
Comments: 15 pages, latex, no figures
Reportno: KUCP0111, hepth/9711058
Abstract:
This paper presents a construction of isomonodromic problems on the torus. The
construction starts from an ${\rm SU}(n)$ version of the XYZ Gaudin model
recently studied by Kuroki and Takebe in the context of a twisted WZW model. In
the classical limit, the quantum Hamiltonians of the generalized Gaudin model
turn into classical Hamiltonians with a natural $r$matrix structure. These
Hamiltonians are used to build a nonautonomous multitime Hamiltonian system,
which is eventually shown to be an isomonodromic problem on the torus. This
isomonodromic problem can also be reproduced from the elliopic KZ equation
of the twisted WZW model. Finally, a geometric interpretation of this
isomonodromicproblem is discussed in the language of a moduli space of
meromorphic connections.

Kanehisa Takasaki
Integrable hierarchies and contact terms in
uplane integrals of topologically twisted
supersymmetric gauge theories
Int. J. Mod. Phys. A 14 (7) (1999), 10011013.
Comments: latex, 17 pages, no figures
Reportno: KUCP0115, hepth/9803217
Comments: I overlooked a factor in identifying
the blowup factor and the tau function. See
"Whitham Deformations and Tau Functions in
N = 2 Supersymmetric Gauge Theories,
Prog. Theor. Phys. Suppl. 135 (1999), 5374."
for a correct identification supplementing
this missing factor.
Abstract:
The $u$plane integrals of topologically twisted $N = 2$ supersymmetric gauge
theories generally contain contact terms of nonlocal topological observables.
This paper proposes an interpretation of these contact terms from the point of
view of integrable hierarchies and their Whitham deformations. This is inspired
by Mari\~no and Moore's remark that that the blowup formula of the $u$plane
integral contains a piece that can be interpreted as a singletime tau function
of an integrable hierarchy. This singletime tau function can be extended to a
multitime version without spoiling the modular invariance of the blowup
formula. The multitime tau function is comprised of a Gaussian factor
$e^{Q(t_1,t_2,\ldots)}$ and a theta function. The time variables $t_n$ play the
role of physical coupling constants of 2observables $I_n(B)$ carried by the
exceptional divisor $B$. The coefficients $q_{mn}$ of the Gaussian part are
identified to be the contact terms of these 2observables. This identification
is further examined in the language of Whitham equations. All relevant
quantities are written in the form of derivatives of the prepotential.

A.J. Bordner, R. Sasaki and K. Takasaki
CalogeroMoser models II: symmetries and foldings
Prog. Thero. Phys. 101 (3) (1999), 487518.
Comments: 35 pages, LaTeX2e with amsfonts, nofigure
Reportno: YITP9860, KUCP0121, hepth/9809068
Abstract:
Universal Lax pairs (the root type and the minimal type) are presented for
CalogeroMoser 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 CalogeroMoser
models known as the `twisted nonsimply laced models'. Among them, a twisted
(BC_n) model is new and it has some novel features. For untwisted nonsimply
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
Whitham deformations of SeibergWitten curves
for classical gauge groups
Int. J. Mod. Phys. A15 (23) (2000), 36353666
Comments: latex, 39pp, no figure
Reportno: KUCP0127, hepth/9901120
Abstract:
Gorsky et al. presented an explicit construction of Whitham
deformations of the SeibergWitten curve for the $SU(N+1)$
$\calN = 2$ SUSY YangMills 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 fourdimensional 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.

Kanehisa Takasaki
Elliptic CalogeroMoser systems and
isomonodromic deformations
J. Math. Phys. 40 (11) (1999), 57875821
Comments: latex2e using amsfonts package
Reportno: KUCP0133, math.QA/9905101
Abstract:
We show that various models of the elliptic CalogeroMoser systems
are accompanied with an isomonodromic system on a torus. The
isomonodromic partner is a nonautonomous 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
CalogeroMoser system turns out to give a Lax representation of
the nonautonomous system as well. This Lax representation
ensures that the nonautonomous 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 CalogeroMoser system. We
use the "root type" Lax pair developed by Bordner et al. to
formulate the associated isomonodromic system on the torus.

Kanehisa Takasaki
Whitham deformations and tau functions in
N = 2 supersymmetric gauge theories
Prog. Theor. Phys. Suppl. 135 (1999), 5374.
Comments: latex2e using amsfonts package
Reportno: KUCP0136, hepth/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
SeibergWitten 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.

S.P. Khastgir, R. Sasaki and K. Takasaki
CalogeroMoser models IV: Limits to Toda theory
Prog. Theor. Phys. 102 (4) (1999), 749776.
Comments: LaTeX2e with amsfonts.sty, 33 pages, no figures
Reportno: YITP9920, KUCP0132, hepth/9907102
Abstract:
CalogeroMoser models and Toda models are wellknown integrable
multiparticle 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 CalogeroMoser 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 CalogeroMoser 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 CalogeroMoser 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.

Takeshi Ikeda and Kanehisa Takasaki
Toroidal Lie algebras and Bogoyavlensky's 2+1dimensional equation
International Mathematics Research Notices 7 (2001), 329369
Comments: LaTeX2e with amsmath and amssymb, 35 pages, no figure
Reportno: 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 grouptheoretic 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
PainleveCalogero correspondence revisited
J. Math. Phys. 42 (3) (2001), 14431473.
Comments: latex2e using amsmath and amssymb packages,
40 pages, no figure
Reportno: KUCP 149, math.QA/0004118
Abstract:
We extend the work of Fuchs, Painlev\'e and Manin on
a Calogerolike expression of the sixth Painlev\'e
equation (the "Painlev\'eCalogero correspondence")
to the other five Painlev\'e equations. The Calogero
side of the sixth Painlev\'e equation is known to be
a nonautonomous 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 "multicomponent" version of the
Painlev\'e equations.

Kanehisa Takasaki
Antiselfdual YangMills equations on noncommutative spacetime
J. Geom. Phys. 37 (4) (2001), 291  306.
Comments: latex2e using packages amsmath and amssymb, 24 pages
Reportno: KUCP153, hepth/0005194
Abstract:
By replacing the ordinary product with the so called $\star$product, one can
construct an analogue of the antiselfdual YangMills (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 zerocurvature representation, an
associated linear system, the RiemannHilbert 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
Hyperelliptic integrable systems on K3 and rational surfaces
Phys. Lett. A283 (2001), 201208.
Comments: latex2e using packages "amsmath and amssymb", 15 pages
Reportno: KUCP0161, 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.

Saburo Kakei, Takeshi Ikeda and Kanehisa Takasaki
Hierarchy of (2+1)dimensional nonlinear Schroedinger equation,
selfdual YangMills equation, and toroidal Lie algebras
Ann. Henri Poincare 3 (2002), 817845.
Comments: 26 pages, Latex2e, uses amsmath, amssymb, amsthm
Reportno: 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
selfdual YangMills equation is also discussed.

R. Sasaki and K. Takasaki
Quantum Inozemtsev model, quasiexact solvability and Nfold
supersymmetry
J.Phys. A34 (2001), 95339554
Comments: LaTeX2e 28 pages, no figures
Reportno: hepth/0109008, YITP0160 and KUCP0191
Abstract
Inozemtsev models are classically integrable multiparticle dynamical systems
related to CalogeroMoser 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 CalogeroMoser models. We show that
quantum Inozemtsev models can be deformed to be a widest class of partly
solvable (or quasiexactly solvable) multiparticle dynamical systems. They
posses Nfold supersymmetry which is equivalent to quasiexact solvability. A
new method for identifying and solving quasiexactly solvable systems, the
method of presuperpotential, is presented.
 Kanehisa Takasaki
Spectral curve and Hamiltonian structure of isomonodromic
SU(2) CalogeroGaudin system
J. Math. Phys. 44 (9) (2003), 39793999.
Comments: The manuscript published in
J. Math. Phys. vol. 44 turned out to contain serious errors.
A corrected version is stored in the eprint arXiv
as nlin.SI/0111019 ver.6.
Reportno: 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
$\rmSU(2)$ CalogeroGaudin 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 $\rmSU(2)$ CalogeroGaudin system
itself. The same machinery turns out to work for the isomonodromic
system on a trous, though the separability is lost and the
nonautonomous 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.

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), 120
Comments: 25 pages, AMSLaTeX, no figure
Reportno: 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 actionangle variables and the separated
variables (in Sklyanin's sense) are related via a canonical transformation,
the generating function of which is the AbelJacobi type integral of
the SeibergWitten differential over the spectral curve.

Kanehisa Takasaki
Spectral curve, Darboux coordinates and
Hamiltonian structure of periodic dressing chains
Commun. Math. Phys. 241 (1) (2003), 111142
Comments: latex2e, 41 pages, no figure
Reportno: nlin.SI/0206049, v3
Abstract
A chain of onedimensional Schr\"odinger operators is called
a "dressing chain" if they are connected by successive
Darboux transformations. Particularly interesting are
periodic dressing chains; they include finiteband 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 odddimensional
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
Integrable systems whose spectral curve is
the graph of a function
Comments: CRM Proceedings and Lecture Notes
vol. 37, pp. 211222 (AMS, Province, 2004).
Contbution to the proceedings of
the conference "Superintegrability in classical and
quantum systems" (Montreal, September 1622, 2002),
latex2e, 15 pages, no figure
Reportno: 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
Tyurin parameters and elliptic analogue of
nonlinear Schr\"odinger hierarchy
J. Math. Sci. Univ. Tokyo 11 (2004), 91131
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
Reportno: 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 zerocurvature
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 RiemannHilbert pair with degeneration points.
The RiemannHilbert 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.

Kanehisa Takasaki
LandauLifshitz hierarchy and infinite dimensional
Grassmann variety
Lett. Math. Phys. 67 (2) (2004), 141152
Comments: latex2e (usepackage:amssyb), 15 pages, no figure;
(v2) minor changes; (v3) typos corrected
Reportno: lin.SI/0312002
Abstract
The LandauLifshitz equation is an example of soliton equations
with a zerocurvature representation defined on an elliptic curve.
This equation can be embedded into an integrable hierarchy of evolution
equations called the LandauLifshitz 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 LandauLifshitz hierarchy. The LandauLifshitz 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
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. 169197
(Birkhauser, Basel/Switzerland, 2005)
Reportno: 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 zerocurvature
representation.

Kanehisa Takasaki
$q$analogue of modified KP hierarchy and its quasiclassical limit
Lett. Math. Phys. 72 (3) (2005), 165181.
Reportno: 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 quasiclassical 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.

Takashi Maeda, Toshio Nakatsu, Kanehisa Takasaki, Takeshi Tamakoshi
Fivedimensional supersymmetric YangMills theories and
random plane partitions
JHEP 03 (2005), 056.
Reportno: OUHET 512, hepth/0412327
Abstract
Fivedimensional $\mathcal{N}=1$ supersymmetric YangMills
theories are investigated from the viewpoint of random
plane partitions. It is shown that random plane partitions
are factorizable as qdeformed random partitions so that
they admit the interpretations as fivedimensional
YangMills and as topological string amplitudes.
In particular, they lead to the exact partition functions
of fivedimensional $\mathcal{N}=1$ supersymmetric
YangMills with the ChernSimons 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.

Takashi Maeda, Toshio Nakatsu, Kanehisa Takasaki, Takeshi Tamakoshi
Free fermion and SeibergWitten differential
in random plane partitions
Nucl. Phys. B715 (2005), 275303
Reportno: OUHET 513, hepth/0412329
Abstract
A model of random plane partitions which describes
fivedimensional $\mathcal{N}=1$ supersymmetric SU(N)
YangMills is studied. We compute the wave functions
of fermions in this statistical model and investigate
their thermodynamic limits or the semiclassical 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 semiclassical wave functions
are obtained in a universal form. We further show that
by taking the fourdimensional limit the semiclassical
wave functions turn to live on the SeibergWitten curve
and that the classical action becomes precisely the integral
of the SeibergWitten differential. When the fermions
are located away from the main diagonal, the semiclassical
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.

Kanehisa Takasaki
Tyurin parameters of commuting pairs and
infinite dimensional Grassmannian manifold
arXiv:nlin.SI/0505005
M. Noumi and K. Takasaki (ed.), "Elliptic Integrable Systems",
Rokko Lectures in Mathematics vol. 18, pp. 289304
(Kobe University, 2005).
Comments: contribution to proceedings of RIMS workshop
"Elliptic Integrable Systems" (RIMS, 2004)
Abstract
Commuting pairs of ordinary differential operators are
classified by a set of algebrogeometric 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.

Ryu Sasaki and Kanehisa Takasaki
Explicit solutions of the classical Calogero & Sutherland
systems for any root system
arXiv: hepth/0510035
J. Math. Phys. 47 (1) (2006), 012701
Comments: 18 pages, LaTeX, no figure
Reportno: YITP0560
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 wellknown 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, Takashi Takebe
Loewner equations and dispersionless hierarchies
arXiv:nlin.SI/0512008
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)
Abstract
Reduction of a dispersionless type integrable system
(dcmKP hierarchy) to the radial Loewner equation is presented.

Kanehisa Takasaki and Takashi Takebe
Radial Loewner equation and dispersionless cmKP hierarchy
arXiv:nlin.SI/0601063
Comments: 18 pages, Latex2e (article, amsmath, amssymb, amsthm)
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
Dispersionless Hirota equations of twocomponent BKP hierarchy
arXiv:nlin.SI/0604003
SIGMA Vol. 2 (2006), Paper 057, 22 pages
Abstract
The BKP hierarchy has a twocomponent analogue (the 2BKP hierarchy).
Dispersionless limit of this multicomponent 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 HamiltonJacobi 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
Universal Whitham hierarchy, dispersionless Hirota equations
and multicomponent KP hierarchy
Physica D235, no. 12 (2007), 109125
arXiv:nlin.SI/0608068
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 multicomponent
KP hierarchy. To this end, the multicomponent
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 multicomponent version of the so called
differential Fay identity are derived from
the Hirota equations of the $\tau$function of
this "charged" multicomponent KP hierarchy.
These multicomponent differential Fay identities
have a welldefined dispersionless limit
(the dispersionless Hirota equations).
The dispersionless Hirota equations turn out
to be equivalent to the HamiltonJacobi 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
scalarvalued wave functions of the multicomponent
KP hierarchy.

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.

Toshio Nakatsu and Kanehisa Takasaki
Melting crystal, quantum torus and Toda hierarchy
arXiv:0710.5339 [hepth]
Commun. Math. Phys. 285 (2009), 445468
Comments: 30 pages, 4 figures
Abstract
Searching for the integrable structures of supersymmetric
gauge theories and topological strings, we study melting crystal,
which is known as random plane partition, from the viewpoint of
integrable systems. We show that a series of partition functions
of melting crystals gives rise to a tau function of the onedimensional
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 subalgebra 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 fivedimensional
$\mathcal{N}=1$ 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
Differential Fay identities and auxiliary linear problem of
integrable hierarchies
Advanced Studies in Pure Mathematics vol. 61 (Mathematical Society of Japan, 2011), pp. 387441.
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, Yui Noma and Kanehisa Takasaki
Integrable structure of $5d$ $\mathcal{N}=1$ supersymmetric YangMills
and melting crystal
arXiv:0806.3675 [hepth]
Int. J. Mod. Phys. A23 (2008), 23322342.
Comments: The solution of the ReimannHilbert problem presented here is wrong.
A correct solution, along with a correct curve, can be obtained by solving
a RiemannHilbert 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 710, 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.

Toshio Nakatsu, Yui Noma and Kanehisa Takasaki
Extended $5d$ SeibergWitten theory and melting crystal
arXiv:0807.0746 [hepth]
Nucl. Phys. B808 (2009), 411440
Comments: The solution of the ReimannHilbert problem presented here is wrong.
A correct solution, along with a correct curve, can be obtained by solving
a RiemannHilbert problem for the primitive function of the Phi potential.
Abstract
We study an extension of the SeibergWitten theory of $5d$ $\mathcal{N}=1$
supersymmetric YangMills 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 fifthdimensional circle and give rise to
physical observables of topologicaltwisted $5d$ $\mathcal{N}=1$ supersymmetric
YangMills 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 1Toda
hierarchy, where the coupling constants of the loop operators are interpreted
as time variables of 1Toda hierarchy. The thermodynamic limit of the partition
function of this model is studied. We solve a RiemannHilbert 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 SeibergWitten differential.

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. 201223.
arXiv:0807.4970 [mathph]
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 711, 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 YangMills theories, topological strings
and the 2D Toda hierarchy are briefly discussed.

Kanehisa Takasaki and Takashi Takebe
Loewner equations, Hirota equations and
reductions of universal Whitham hierarchy
J. Phys. A: Math. Theor. 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 onevariable
reduction, dispersionless Hirota equations turn out to be
a powerful tool for understanding the mechanism of reduction.
All relevant equations describing the reduction (Loewnertype
equations and diagonal hydrodynamic equations) can be thereby
derived and justified in a unified manner. The case of
multivariable 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.

Kanehisa Takasaki
Integrable structure of melting crystal model with two qparameters
arXiv:0903.2607 [mathph]
J. Geometry and Physics 59 (2009), 12441257
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
Auxiliary linear problem, difference Fay identities and
dispersionless limit of PfaffToda 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 Faytype 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 PfaffToda 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 Faylike 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 quasiclassical
deformations of this elliptic spectral curve.

Kanehisa Takasaki and Takashi Takebe
hbarexpansion of KP hierarchy: Recursive construction of solutions
arXiv:0912.4867 [mathph]
Comments: 28 pages
Abstract
The \hbardependent 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 RiemannHilbert problem for the pair (L,M) of Lax and OrlovSchulman
operators is presented. The RiemannHilbert problem is converted to a set of
recursion relations for the coefficients X_n of an \hbarexpansion 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 \hbarexpansion 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 \hbarexpansion of the form \log\tau = \hbar^{2}F_0 + \hbar^{1}F_1 + F_2 +
... .

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
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 2731 July 2009
(World Sci. Publ., Hackensack, NJ), pp. 373391.
arXiv:1003.307 [mathph]
Comments: latex2e, using wsprocs9x6 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 6vertex 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, Takashi Takebe and Lee Peng Teo
Nondegenerate solutions of universal Whitham hierarchy
J. Phys. A: Math. Theor. 43 (2010), 325205
arXiv:1003.5767 [mathph]
Comments: latex2e, using amsmath, amssym and amsthm packages,
32 pages, no figure
Abstract
The notion of nondegenerate 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
RiemannHilbert problem (generalized string equations) with respect to
twodimensional canonical transformations, and may be thought of as a
kind of general solutions of the hierarchy. The RiemannHilbert problem
contains $M$ arbitrary functions $H_a(z_0,z_a)$, $a = 1,\ldots,M$, which
play the role of generating functions of twodimensional canonical
transformations. The solution of the RiemannHilbert 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
Generalized string equations for double Hurwitz numbers
Journal of Geometry and Physics 62 (2012), 11351156
arXiv:1012.5554 [mathph]
Comments: latex2e using amsmath,amssymb,amsthm, 41 pages, no figure
MSCclass: 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 OrlovSchulman 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
OrlovSchulman operators are contained therein in an exponentiated form. These
equations are derived from a set of intertwining relations for fermiom
bilinears in a twodimensional free fermion system. The intertwiner is
constructed from a fermionic counterpart of the cutandjoin 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
Toda tau functions with quantum torus symmetries
Acta Polytechnica 51, No.1 (2011), 7476.
arXiv:1101.4083 [mathph]
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 and Takashi Takebe
An hbar dependent formulation of the KadomtsevPetviashvili hierarchy
Theoretical and Mathematical Physics 171 (2) (2012), 683690.
arXiv:1105.0794v1 [mathph]
Comments: 12 pages, contribution to the Proceedings of
the "International Workshop on Classical and Quantum Integrable Systems 2011"
(January 2427, 2011 Protvino, Russia)
Abstract
This is a summary of a recursive construction of solutions of the hbardependent KP hierarchy. We give recursion relations for the coefficients X_n of an hbarexpansion 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 and Toshio Nakatsu
Thermodynamic limit of random partitions and dispersionless Toda hierarchy
J. Phys. A: Math. Theor. 45 (2012), 025403 (38pp)
arXiv:1110.0657 [mathph]
Comments: 55 pages, no figure
MSCclass: 35Q58, 81T13, 82B20
doi:10.1088/17518113/45/2/025403
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 scalarvalued RiemannHilbert problem. This
RiemannHilbert problem is solved with the aid of a complex curve, which may be
thought of as the SeibergWitten curve of the deformed U(1) gauge theory. This
solution of the RiemannHilbert 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 SeibergWitten differential are also considered.

Kanehisa Takasaki and Takashi Takebe
An hbarexpansion of the Toda hierarchy: a recursive construction of solutions
Analysis and Mathematical Physics 2 (2012), 171214.
arXiv:1112.0601v1 [mathph]
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 \hbardependent Toda hierarchy is presented. The construction is based on a RiemannHilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and OrlovSchulman operators. This RiemannHilbert 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 \hbarexpansions 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 \hbarexpansion of the logarithm of the tau function.

A. Yu. Orlov, T. Shiota, K. Takasaki
Pfaffian structures and certain solutions to BKP hierarchies I. Sums over partitions
arXiv:1201.4518v1 [mathph]
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 smalllarge BKP tau functions (Kac and van de Leur). Tau functions under consideration are equal to certain sums over partitions and to certain multiintegrals 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
Old and new reductions of dispersionless Toda hierarchy
SIGMA 8 (2012), 102, 22 pages
arXiv:1206.1151 [mathph]
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 finitevariable reductions of the dispersionless Toda hierarchy are considered in the geometric perspectives. The reductions are formulated in terms of "LandauGinzburg 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 LandauGinzburg potentials are shown to satisfy (a radial version of) the L\"onwer equations with respect to the critical values of the LandauGinzburg potentials. Integrability conditions of these L\"owner equations are (a radial version of) the GibbonsTsarev equations. These equations are used to formulate hodograph solutions of the reduced hierarchy. Geometric aspects of the GibbonsTsarev 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 LandauGinzburg potentials are introduced, and flat coordinates are constructed explicitly.

Kanehisa Takasaki
Integrable structure of modified melting crystal model
arXiv:1208.4497 [mathph]
Comments: 10 pages, no figure, poster presentation at conference "Integrability
in Gauge and String Theory" (Zurich, August 2024, 2012)
MSCclass: 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 AblowitzLadik hierarchy.

Kanehisa Takasaki
Remarks on partition functions of topological string theory on
generalized conifolds
RIMS Kokyuroku No. 1913 (2014), 182201
arXiv:1301.4548 [mathph]
Comments: 20 pages, 3 figures, contribution to the proceedings of the RIMS
campstyle 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
MSCclass: 05E05, 37K10, 81T30
Abstract
The notion of topological vertex and the construction of topological string
partition functions on local toric CalabiYau 3folds 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
BakerAkhiezer 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
Modified melting crystal model and AblowitzLadik hierarchy
J. Phys. A: Math. Theor. 46 (2013), 245202 (23 pages)
arXiv:1302.6129 [mathph]
Comments: 33 pages, no figure
MSC classes: 17B65, 35Q55, 81T30, 82B20
doi:10.1088/17518113/46/24/245202
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 AblowitzLadik (equivalently, relativistic Toda) hierarchy.
As a byproduct, the shift symmetries are shown to be related
to matrixvalued quantum dilogarithmic functions.

Kanehisa Takasaki
Generalized AblowitzLadik hierarchy in topological string theory
J. Phys. A: Math. Theor. 47 (2014), 165201 (20 pages)
doi:10.1088/17518113/47/16/165201
arXiv:1312.7184 [mathph]
Comments: 24pages, 1 figre
MSCclass: 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 AblowitzLadik hierarchy embedded in the 2D Toda
hierarchy. The generalized AblowitzLadik hierarchy is thus identified as a
fundamental integrable structure of topological string theory on the
generalized conifolds.

Kanehisa Takasaki
Modified melting crystal model and AblowitzLadik hierarchy
J. Phys.: Conf. Ser. 482 (2014), 012041
[open access]
doi:10.1088/17426596/482/1/012041
arXiv:1312.7276 [mathph]
Comments: 10 pages, 4 figures, contribution to proceedings of the conference
"Physics and Mathematic of Nonlinear Phenomena", Gallipoli, Italy, June 2328, 2013
MSCclass: 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 AblowitzLadik
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
Orbifold melting crystal models and reductions of Toda hierarchy
J. Phys. A: Math. Theor. 48 (2015), 215201 (34 pages)
doi:10.1088/17518113/48/21/215201
arXiv:1410.5060 [mathph]
Comments: 41 pages, no figure
MSCclass: 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 bigraded Toda hierarchy
of bidegree $(a,b)$. That of the orbifold version of the modified
melting crystal model is the rational reduction of bidegree $(a,b)$.
This result seems to be in accord with recent work of Brini et al.
on a mirror description of the genuszero GromovWitten theory
on a $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifold of the resolved conifold.

Kanehisa Takasaki and Toshio Nakatsu
Open string amplitudes of closed topological vertex
J. Phys. A: Math. Theor. 49 (2016), 025201 (28pp)
doi:10.1088/17518113/49/2/025201
arXiv:1507.07053
Comments: latex2e, package amsmath,amssymb,amsthm,graphicx, 10 figures
MSCclass: 17B81, 33E20, 81T30
Abstract
The closed topological vertex is the simplest ``offstrip'' case
of noncompact toric CalabiYau 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 ``onstrip'' subdiagram of the treelike web diagram.
If nontrivial 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 and Toshio Nakatsu
$q$difference KacSchwarz operators in topological string theory
SIGMA 13 (2017), 009, 28 pages
doi:10.3842/SIGMA.2017.009
arXiv:1609.00882 [mathph]
Comments: Contribution to the Special Issue on Combinatorics of Moduli Spaces:
Integrability, Cohomology, Quantisation, and Beyond
MSCclass: 37K10, 39A13, 81T30
Key words: topological vertex; mirror symmetry; quantum curve;
qdifference equation; KP hierarchy; KacSchwarz operator
Abstract
The perspective of KacSchwarz 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 multivariate
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
KacSchwarz 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
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 137C (2019), 184203
DOI: 10.1016/j.geomphys.2018.12.012
arXiv:1704.02750 [mathph]
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
MSCclass: 14N35, 37K10, 39A13
Abstract
This paper addresses the problems of quantum spectral curves and 4D limit for
the melting crystal model of 5D SUSY $U(1)$ YangMills 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
singlevariate 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 GromovWitten 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 allgenus GromovWitten invariants of $\mathbb{CP}^1$. Faytype
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
Toda hierarchies and their applications
J. Phys. A: Math. Theor. 51 (2018) 203001 (35pp)
doi: 10.1088/17518121/aabc14
arXiv:1801.09924 [mathph]
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 AblowitzLadik (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
Hurwitz numbers and integrable hierarchy of Volterra type
J. Phys. A: Math. Theor. 51 (2018), 43LT01 (9 pages)
doi: 10.1088/17518121/aae10b
arXiv:1807.00085
Comments: latex2e, amsmath,amssymb,amsthm, 12 pages, no figure
MSCclass: 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 differencedifferential 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 BogoyavlenskyItoh (aka hungry
LotkaVolterra) hierarchies. Emergence of this underlying integrable structure
is further explained in the language of generalized string equations for the
Lax and OrlovSchulman operators of the 2D Toda hierarchy. This leads to
logarithmic string equations, which are confirmed with the help of a
factorization problem of operators.

Toshio Nakatsu and Kanehisa Takasaki
Threepartition Hodge integrals and the topological vertex
Communications in Mathematical Physics, 376(1) (2020), 201234
https://doi.org/10.1007/s00220019036485
arXiv:1812.11726
Comments: 44 pages, 2 figures
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 twodimensional 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 GelfandDickey)
hierarchies.

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. 481502
arXiv:1909.13095
Comments: latex2e, amsmath,amssymb,amsthm, 29pp, no figure
Abstract
A tau function of the 2D Toda hierarchy can be obtained from a generating
function of the twopartition 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 tausymmetric integrable Hamiltonian PDEs.

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 911, 2019)
MSCclass: 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 GelfandDickey hierarchy, the BogoyavlenskyItohNarita
lattice and the AblowitzLadik hierarchy. These reductions contain previously
unknown integrable hierarchies as well.

Kanehisa Takasaki
Dressing operators in equivariant GromovWitten theory of $\mathbb{CP}^1$
J. Phys. A: Math. Theor. 54 (2021), 35LT02
doi: 10.1088/17518121/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
MSCclass: 14N35, 37K10
Abstract
Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs
the equivariant GromovWitten 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
GromovWitten theory of $\mathbb{CP}^1$. Moreover, the nonequivariant limit of
these operators turns out to capture the integrable structure of the
nonequivariant GromovWitten theory correctly.