Papers and Preprints

A larger list of publications, talks, etc. (in Japanese) is available here.

E-prints can be downloaded from the arxiv.org and its mirror sistes.

1. K. Takasaki
   Singular Cauchy problems for a class of weakly hyperbolic differential operators
   Proc. Japan Acad. 57A (1981), 393-397. [scanned image]
   

2. K. Takasaki
   Singular Cauchy problems for a class of weakly hyperbolic differential operators
   Comm. Partial Differential Equations 7 (1982), 1151-1188. [scanned image]
   

3. K. Ueno and K. Takasaki
   Toda lattice hierarchy I and II
   Proc. Japan Acad. 59A (1983), 167-170, 215-218. [scanned image of I | scanned image of II]
   

4. K. Takasaki
   A class of solutions to the self-dual Yang-Mills equations
   Proc. Japan Acad. 59A (1983), 308-311. [scanned image]
   

5. K. Takasaki
   On the structure of solutions to the self-dual Yang-Mills equations
   Proc. Japan Acad. 59A (1983), 418-421. [scanned image]
   

6. 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 2-component 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. 1-95 (Kinokuniya, Tokyo, 1984). [scanned image]
   

7. 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. 139-163 (Kinokuniya, Tokyo, 1984). [scanned image]
   

8. K. Takasaki
   A new approach to the self-dual Yang-Mills equations
   Commun. Math. Phys. 94 (1984), 35-59.
   Keywords: gauge field theory, yang-mills/ field equations, soliton/ gauge field theory, geometrical/ field theory, completely integrable/ algebra, commutation relations/ mathematical methods/
   

9. K. Takasaki
   A new approach to the self-dual Yang-Mills equations II
   Saitama Math. J. 3 (1985), 11-40.
   Comments: The last paragraph of section 2.6 is wrong. Namely, a rational initial data can give rise to non-rational solution. [scanned image]
   

10. K. Takasaki
    Aspects of integrability in self-dual Einstein metrics and related Equations
    Publ. RIMS, Kyoto Univ., 22 (1986), 949-990.
    Report-no: RIMS-669 (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 hyper-K\"{a}hler (or Poisson gauge field) hierarchy", J. Math. Phys. 31 (1990), 1877-1888. [scanned image]
    

11. K. Takasaki
    Issues of multi-dimensional integrable systems
    M. Kashiwara and T. Kawai (eds.), {\it Algebraic Analysis}, Vol. II, pp. 853-866 (Academic Press, 1988). [scanned image]
    Report-no: RIMS-588
    

12. K. Takasaki
    Integrable systems as deformations of ${\cal D}$-modules
    Proc. Symp. Pure Math. 49 , Part I, pp. 143-168 (American Mathematical Society, 1989). [scanned image]
    Comments: talk at the AMS Summer Institute "Theta Functions,"
    Bowdoin College, New Brunswick, 1987
    

13. K. Takasaki
    An infinite number of hidden variables in hyper-K\"{a}hler metrics
    J. Math. Phys. 30 (1989), 1515-1521.
    Report-no: RIMS-621 (March 1988),
    Keywords: mathematical methods, differential geometry/ space-time, kaehler/ mathematical methods, twistor/
    

14. K. Takasaki
    Geometry of universal Grassmann manifold from algebraic point of view
    Reviews in Math. Phys. 1 (1989), 1-46.
    Report-no: RIMS-623 (May 1988).
    Keywords: mathematical methods, differential geometry/ mathematical methods, fibre bundle/ space-time, grassmann/ group theory, geometrical/ algebra, lie/ commutation relations/ fibre bundle/
    

15. K. Takasaki
    An infinite number of Hamiltonian flows arising from hyper-K\"{a}hler metric
    Y. Saint-Aubin and L. Vinet (eds.), {\it the XVIIth International Colloquium on Group Theoretical Methods in Physics}, Sainte-Ad\`{e}le 1988, pp. 516-519. (World Scientific, Sigapore, 1989). [scanned image]
    

16. K. Takasaki
    Symmetries of the super KP hierarchy
    Lett. Math. Phys. 17 (1989), 351-357.
    Report-no: RIMS-641 (November 1988).

17. K. Takasaki
    Differential algebras and ${\cal D}$-modules in super Toda lattice hierarchy
    Lett. Math. Phys. 19 (1990), 229-236.
    Report-no: RIMS-665 (July 1989)
    

18. K. Takasaki
    Hierarchy structure in integrable systems of gauge fields and underlying Lie algebras
    Commun. Math. Phys. 127 (1990), 225-238.
    Report-no: RIMS-637 (October 1988)
    Keywords: gauge field theory, yang-mills/ duality/ field equations, integrability/ algebra, lie/ riemann-hilbert transformation/
    

19. K. Takasaki
    Symmetries of hyper-K\"{a}hler (or Poisson gauge field) hierarchy
    J. Math. Phys. 31 (1990), 1877-1888.
    Report-no: RIMS-669 (October 1989)
    Keywords: space-time, kaehler/ loop space/ hamiltonian formalism/ differential geometry/
    

20. Miyajima, T., Nakayashiki, A. and K. Takasaki
    Structure and duality of ${\cal D}$-modules related to KP hierarchy
    J. Math. Soc. Japan 43 (1991), 751-773. [manuscript]
    Report-no: RIMS-689 (March 1990)
    

21. 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. 1205-1214 (Springer-Verlag, 1991).
    Report-no: RIMS-714 (September 1990) [manuscript]
    Keywords: talk, kyoto 1990/08/ gauge field theory, yang-mills/ field theory, kaehler/ differential equations, nonlinear/ differential equations, kadomtsev-petviashvili/ integrability/ duality/ mathematical methods/
    

22. K. Takasaki
    Analytic expression of Voros coefficients and its application to WKB connection problem
    M. Kashiwara and T. Miwa (eds.), {\it Special functions\/}, ICM-90 Satellite Conference Proceedings, pp. 294-315 (Springer-Verlag, Berlin-New York-Tokyo, 1991). [scanned image]
    Report-no: RIMS-725 (November 1990)
    Abstract: Usually, the WKB method starts from formal solutions (WKB or Liouville-Green solutions) expanded in powers of the Planck constant, and connects these solutions by asymptotic matching at turning points. A resummation prescription of these formal calculations was proposed by A. Voros, after an idea of Balian and Bloch, and illustrated for a homogeneous quartic oscillator. Voros argued that his results should be deeply related with J. Ecalle's theory of "resurgent functions." Further progress along that line has been made by F. Pham and his coworkers. I shall report another approach based upon a classical idea of F. Olver.
    

23. K. Takasaki
    Hidden symmetries of integrable system in Yang-Mills theory and K\"ahler geometry
    S\'eminaire sur les \'Equations aux D\'eriv\'ees Partielles, 1990-1991, Expos\'e n$^o$ VIII, pp. 1-15, 22 Janvier 1991 (Ecole Polytechnique, 1991). [seminar notes]
    Report-no: RIMS-743 (March 1991)
    Keywords: talk, paris 1991/01/22/ gauge field theory, yang-mills/ 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 [Ta2-4] in recent years on symmetries of the self-duality equations and related equations. A central issue was to find an explicit form of {\sl infinitesimal\/} symmetries of the self-dual Einstein equation that should correspond to the {\sl finite\/} symmetries of Boyer and Plebanski [Bo-Pl]. 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 self-dual Yang-Mills equations whose Riemann-Hilbert factorization is still relatively well understood. The case of the self-dual 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 self-dual Yang-Mills equation advocate. This indeed turns out to be the case for the self-dual Einstein equation.
    

24. K. Takasaki and T. Takebe
    SDiff(2) Toda equation --- hierarchy, tau function and symmetries
    Lett. Math. Phys. 23 (1991), 205-214.
    Report-no: RIMS-790 (August, 1991), hep-th/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 self-dual vacuum Einstein equation and its hyper-K\"ahler version, however now based upon a symplectic structure and the group SDiff(2) of area preserving diffeomorphisms on a cylinder $S^1 \times \R$. An analogue of the Toda lattice tau function is introduced. The existence of hidden SDiff(2) symmetries are derived from a Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function turn out to have commutator anomalies, hence give a representation of a central extension of the SDiff(2) algebra.
    

25. 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.889-922.
    Report-no: RIMS-814 (October, 1991, revised version December, 1991), hep-th/9112046
    Abstract: An analogue of the KP hierarchy, the SDiff(2) KP hierarchy, related to the group of area-preserving 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 Riemann-Hilbert 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 area-preserving diffeomorphisms (or of the associated Poisson algebra).
    

26. K. Takasaki
    Area-preserving 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. 383-397 (World Scientific, Singapore, 1992).
    Report-no: KUCP-0039/91 (September, 1991), hep-th/9112041
    Abstract: Present state of the study of nonlinear "integrable" systems related to the group of area-preserving diffeomorphisms on various surfaces is overviewed. Roles of area-preserving diffeomorphisms in 4-d self-dual gravity are reviewed. Recent progress in new members of this family, the SDiff(2) KP and Toda hierarchies, is reported. The group of area-preserving diffeomorphisms on a cylinder plays a key role just as the infinite matrix group GL($\infty$) does in the ordinary KP and Toda lattice hierarchies. The notion of tau functions is also shown to persist in these hierarchies, and gives rise to a central extension of the corresponding Lie algebra.
    

27. K. Takasaki
    Volume-preserving diffeomorphisms in integrable deformations of selfdual gravity
    Phys. Lett. B285 (1992), 187-190.
    Report-no: KUCP-0046/92 (March, 1992), hep-th/9203034
    Abstract: A group of volume-preserving diffeomorphisms in 3D turns out to play a key role in an Einstein-Maxwell theory whose Weyl tensor is selfdual and whose Maxwell tensor has algebraically general anti-selfdual part. This model was first introduced by Flaherty and recently studied by Park as an integrable deformation of selfdual gravity. A twisted volume form on the corresponding twistor space is shown to be the origin of volume-preserving diffeomorphisms. An immediate consequence is the existence of an infinite number of symmetries as a generalization of $w_{1+\infty}$ symmetries in selfdual gravity. A possible relation to Witten's 2D string theory is pointed out.
    

28. 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 23-28, 1992.
    Report-no: KUCP-0049/92 (June, 1992), hep-th/9206030
    Keywords: talk/ differential equations, nonlinear/ integrability/ gravitation, duality/ kadomtsev-petviashvili 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: self-dual gravity, the KP and Toda hierarchies, their quasi-classical (or dispersionless) limit, etc. Twistor theory provides a geometric background for these algebras. Present state of these topics is overviewed. A few ideas on possible deformations of self-dual gravity (including quantum deformations) are presented.
    

29. K. Takasaki and T. Takebe
    Quasi-classical limit of KP hierarchy, W-symmetries 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.
    Report-no: KUCP-0050/92 (July, 1992), hep-th/9207081
    Abstract: This paper deals with the dispersionless KP hierarchy from the point of view of quasi-classical limit. Its Lax formalism, W-infinity symmetries and general solutions are shown to be reproduced from their counterparts in the KP hierarchy in the limit of $\hbar \to 0$. Free fermions and bosonized vertex operators play a key role in the description of W-infinity symmetries and general solutions, which is technically very similar to a recent free fermion formalism of $c=1$ matrix models.
    

30. K. Takasaki, and T. Takebe
    Quasi-classical limit of Toda hierarchy and W-infinity symmetries
    Lett. Math. Phys. 28 (1993), 165-176.
    Report-no: KUCP-0057/93 (Januray, 1993), hep-th/9301070
    Abstract: Previous results on quasi-classical limit of the KP hierarchy and its W- infinity symmetries are extended to the Toda hierarchy. The Planck constant $\hbar$ now emerges as the spacing unit of difference operators in the Lax formalism. Basic notions, such as dressing operators, Baker-Akhiezer functions and tau function, are redefined. $W_{1+\infty}$ symmetries of the Toda hierarchy are realized by suitable rescaling of the Date-Jimbo- Kashiara-Miwa vertex operators. These symmetries are contracted to $w_{1+\infty}$ symmetries of the dispersionless hierarchy through their action on the tau function.
    

31. K. Takasaki
    Quasi-classical limit of BKP hierarchy and W-infinity symmetries
    Lett. Math. Phys. 28 (1993), 177-185.
    Report-no: KUCP-0058/93 (January, 1993), hep-th/9301090
    Abstract: Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebras $\WB_{1+\infty}$ and $\wB_{1+\infty}$ of the W-infinity algebras $W_{1+\infty}$ and $w_{1+\infty}$ are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantum W-infinity algebra $\WB_{1+\infty}$ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, these $\WB_{1+\infty}$ symmetries are shown to be contracted into $\wB_{1+\infty}$ symmetries of the dispersionless BKP hierarchy.
    

32. K. Takasaki
    Integrable hierarchy underlying topological Landau-Ginzburg models of D-type
    Lett. Math. Phys. 29 (1993), 111-121.
    Report-no: Kyoto University KUCP-0061/93 (March, 1993), hep-th/9305053
    Abstract: A universal integrable hierarchy underlying topological Landau-Ginzburg models of D-tye is presented. Like the dispersionless Toda hierarchy, the new hierarchy has two distinct ("positive" and "negative") set of flows. Special solutions corresponding to topological Landau-Ginzburg models of D- type are characterized by a Riemann-Hilbert problem, which can be converted into a generalized hodograph transformation. This construction gives an embedding of the finite dimensional small phase space of these models into the full space of flows of this hierarchy. One of flat coordinates in the 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.
    

33. K. Takasaki
    Dressing operator approach to Moyal algebraic deformation of selfdual gravity
    Journal of Geometry and Physics 14 (1994), 111-120.
    Report-no: Kyoto University KUCP-0054/92, hep-th/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.
    

34. K. Takasaki
    Nonabelian KP hierarchy with Moyal algebraic coefficients
    Journal of Geometry and Physics 14 (1994), 332-364.
    Report-no: Kyoto University KUCP-0062/93, hep-th/9305169
    Abstract: A higher dimensional analogue of the KP hierarchy is presented. Fundamental constituents of the theory are pseudo-differential operators with Moyal algebraic coefficients. The new hierarchy can be interpreted as large-$N$ limit of multi-component ($\gl(N)$ symmetric) KP hierarchies. Actually, two different hierarchies are constructed. The first hierarchy consists of commuting flows and may be thought of as a straightforward extension of the ordinary and multi-component KP hierarchies. The second one is a hierarchy of noncommuting flows, and related to Moyal algebraic deformations of selfdual gravity. Both hierarchies turn out to possess quasi-classical limit, replacing Moyal algebraic structures by Poisson algebraic structures. The language of W-infinity algebras provides a unified point of view to these results.
    

35. K. Takasaki
    Dispersionless Toda hierarchy and two-dimensional string theory
    Commun. Math. Phys. 170 (1995), 101-116.
    Report-no: Kyoto University KUCP-0067/94, hep-th/9403190
    Abstract: The dispersionless Toda hierarchy turns out to lie in the heart of a recently proposed Landau-Ginzburg formulation of two-dimensional string theory at self-dual compactification radius. The dynamics of massless tachyons with discrete momenta is shown to be encoded into the structure of a special solution of this integrable hierarchy. This solution is obtained by solving a Riemann-Hilbert problem. Equivalence to the tachyon dynamics is proven by deriving recursion relations of tachyon correlation functions in the machinery of the dispersionless Toda hierarchy. Fundamental ingredients of the Landau-Ginzburg formulation, such as Landau-Ginzburg potentials and tachyon Landau-Ginzburg fields, are translated into the language of the Lax formalism. Furthermore, a wedge algebra is pointed out to exist behind the Riemann-Hilbert problem, and speculations on its possible role as generators of "extra" states and fields are presented.
    

36. K. Takasaki, and T. Takebe
    Integrable hierarchies and dispersionless limit
    Reviews in Mathematical Physics 7 (1995), 743-808.
    Report-no: University of Tokyo UTMS 94-35, hep-th/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.
    

37. K. Takasaki
    Symmetries and tau function of higher dimensional dispersionless integrable hierarchies
    J. Math. Phys. 36 (1995), 3574-3607.
    Report-no: Kyoto University KUCP-0068, hep-th/9407098
    Abstract: A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional "phase space" variables $(k,x)$ of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions compactified to a two (or any even) dimensional torus. Integrability of this hierarchy and the existence of an infinite dimensional of "additional symmetries" are ensured by an underlying twistor theoretical structure (or a nonlinear Riemann-Hilbert problem). An analogue of the tau function, whose logarithm gives the $F$ function ("free energy" or "prepotential" in the contest of matrix models and topological conformal field theories), is constructed. The infinite dimensional symmetries can be extended to this tau (or $F$) function. The extended symmetries, just like those of the dispersionless KP hierarchy, obey an anomalous commutation relations.
    

38. T. Nakatsu, K. Takasaki and S. Tsujimaru
    Quantum and classical aspects of deformed $c=1$ strings
    Nucl. Phys. B443 (1995), 155-197.
    Report-no: INS-rep.-1087, KUCP-0077, hep-th/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.
    

39. K. Takasaki
    Integrable hierarchies, dispersionless limit and string equations
    M. Morimoto and T. Kawai (eds.), {\it Structures of Solutions of Differential Equations\/}, pp. 457-481 (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.
    

40. T. Nakatsu and K. Takasaki
    Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory
    Mod. Phys. Lett. A11 (2) (1996), 157-168.
    Report-no: KUCP-0083, hep-th/9509162
    Abstract: The exact solution of $N=2$ supersymmetric $SU(N)$ Yang-Mills theory is studied in the framework of the Whitham hierarchies. The solution is identified with a homogeneous solution of a Whitham hierarchy. This integrable hierarchy (Whitham-Toda hierarchy) describes modulation of a quasi-periodic solution of the (generalized) Toda lattice hierarchy associated with the hyperelliptic curves over the quantum moduli space. The relation between the holomorphic pre-potential of the low energy effective action and the $\tau$ function of the (generalized) Toda lattice hierarchy is also clarified.
    

41. K. Takasaki
    Toda lattice hierarchy and generalized string equations
    Commun. Math. Phys. 181 (1) (1996), 131-156.
    Report-no: KUCP-0079, hep-th/9506089.
    Abstract: String equations of the $p$-th generalized Kontsevich model and the compactified $c = 1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c \le 1$ string models is discussed in this new framework.
    

42. K. Takasaki and T. Nakatsu
    Isomonodromic deformations and supersymmetric Yang-Mills theories
    Int. J. Mod. Phys. A 11 (38) (1996), 5505-5518.
    Report-no: KUCP-0092, hep-th/9603069.
    Abstract: Seiberg-Witten solutions of four-dimensional supersymmetric gauge theories possess rich but involved integrable structures. The goal of this paper is to show that an isomonodromy problem provides a unified framework for understanding those various features of integrability. The Seiberg-Witten solution itself can be interpreted as a WKB limit of this isomonodromy problem. The origin of underlying Whitham dynamics (adiabatic deformation of an isomonodromy problem), too, can be similarly explained by a more refined asymptotic method (multiscale analysis). The case of $N=2$ SU($s$) supersymmetric Yang-Mills theory without matter is considered in detail for illustration. The isomonodromy problem in this case is closely related to the third Painlev\'e equation and its multicomponent analogues. An implicit relation to $t\tbar$ fusion of topological sigma models is thereby expected.
    

43. T. Nakatsu and K. Takasaki
    Integrable system and N=2 supersymmetric Yang-Mills theory
    H. Itoyama et al. (eds.),
    {\it Frontiers in Quantum Field Theory}, 325-330 (World Scientific, Singapore, 1996).
    Report-no: RITS-96-01, hep-th/9603129
    Abstract: The exact solutions (Seiberg-Witten type) of $N=2$ supersymmetric Yang-Mills theory are discussed from the view of Whitham-Toda hierarchy.
    

44. Partha Guha and Kanehisa Takasaki
    Dispersionless hierarchies, Hamilton-Jacobi theory and twistor correspondences
    J. Geom. Phys. 25 (3-4) (1998), 326-340.
    Comments: latex, 20pp, no figures.
    Report-no:RIMS-1124 (Jan, 1997), solv-int/9705013.
    Abstract: The dispersionless KP and Toda hierarchies possess an underlying twistorial structure. A twistorial approach is partly implemented by the method of Riemann-Hilbert problem. This is however still short of clarifying geometric ingredients of twistor theory, such as twistor lines and twistor surfaces. A more geometric approach can be developed in a Hamilton-Jacobi formalism of Gibbons and Kodama.
    

45. K. Takasaki
    Spectral curves and Whitham equations in isomonodromic problems of Schlesinger type
    Asian J.Math. 2 (4) (1998), 1049-1078.
    Comments: latex, 40pp, no figures.
    Report-no:KUCP-0105, solv-int/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 finite-gap solution of an isospectral problem. The modulation is caused by slow deformations of the spectral curve of the finite-gap solution. A modulation equation of this slow dynamics is derived by a heuristic method. An inverse period map of Seiberg-Witten type turns out to give general solutions of this modulation equation. This construction of general solution also reveals the existence of deformations of Seiberg-Witten type on the same moduli space of spectral curves. A prepotential is also constructed in the same way as the prepotential of the Seiberg-Witten theory.
    

46. Kanehisa Takasaki
    Dual isomonodromic problems and Whitham equations
    Lett. Math. Phys. 43 (2) (1998), 123-135.
    Comments: 15 pages, latex, no figures
    Report-no: KUCP-0106, solv-int/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.
    

47. Kanehisa Takasaki
    Gaudin model, KZ Equation, and isomonodromic problem on torus
    Lett. Math. Phys. 44 (2) (1998), 143-156.
    Comments: 15 pages, latex, no figures
    Report-no: KUCP-0111, hep-th/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 non-autonomous multi-time Hamiltonian system, which is eventually shown to be an isomonodromic problem on the torus. This isomonodromic problem can also be reproduced from 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.
    

48. Kanehisa Takasaki
    Integrable hierarchies and contact terms in u-plane integrals of topologically twisted supersymmetric gauge theories
    Int. J. Mod. Phys. A 14 (7) (1999), 1001-1013.
    Comments: latex, 17 pages, no figures
    Report-no: KUCP-0115, hep-th/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), 53-74." 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 single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multi-time version without spoiling the modular invariance of the blowup formula. The multi-time tau function is comprised of a Gaussian factor $e^{Q(t_1,t_2,\ldots)}$ and a theta function. The time variables $t_n$ play the role of physical coupling constants of 2-observables $I_n(B)$ carried by the exceptional divisor $B$. The coefficients $q_{mn}$ of the Gaussian part are identified to be the contact terms of these 2-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential.
    

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

50. Kanehisa Takasaki
    Whitham deformations of Seiberg-Witten curves for classical gauge groups
    Int. J. Mod. Phys. A15 (23) (2000), 3635-3666
    Comments: latex, 39pp, no figure
    Report-no: KUCP-0127, hep-th/9901120
    Abstract: Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the $SU(N+1)$ $\calN = 2$ SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the $A^{(2)}_{2N}$ affine Toda system. Our construction, too, uses fractional powers of the superpotential $W(x)$ that characterizes the curve. We also consider the $u$-plane integral of topologically twisted theories on four-dimensional manifolds $X$ with $b_2^{+}(X) = 1$ in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind.
    

51. Kanehisa Takasaki
    Elliptic Calogero-Moser systems and isomonodromic deformations
    J. Math. Phys. 40 (11) (1999), 5787-5821
    Comments: latex2e using amsfonts package
    Report-no: KUCP-0133, math.QA/9905101
    Abstract: We show that various models of the elliptic Calogero-Moser systems are accompanied with an isomonodromic system on a torus. The isomonodromic partner is a non-autonomous Hamiltonian system defined by the same Hamiltonian. The role of the time variable is played by the modulus of the base torus. A suitably chosen Lax pair (with an elliptic spectral parameter) of the elliptic Calogero-Moser system turns out to give a Lax representation of the non-autonomous system as well. This Lax representation ensures that the non-autonomous system describes isomonodromic deformations of a linear ordinary differential equation on the torus on which the spectral parameter of the Lax pair is defined. A particularly interesting example is the "extended twisted $BC_\ell$ model" recently introduced along with some other models by Bordner and Sasaki, who remarked that this system is equivalent to Inozemtsev's generalized elliptic Calogero-Moser system. We use the "root type" Lax pair developed by Bordner et al. to formulate the associated isomonodromic system on the torus.
    

52. Kanehisa Takasaki
    Whitham deformations and tau functions in N = 2 supersymmetric gauge theories
    Prog. Theor. Phys. Suppl. 135 (1999), 53-74.
    Comments: latex2e using amsfonts package
    Report-no: KUCP-0136, hep-th/9905224
    Abstract: We review new aspects of integrable systems discovered recently in $N=2$ supersymmetric gauge theories and their topologically twisted versions. The main topics are (i) an explicit construction of Whitham deformations of the Seiberg-Witten curves for classical gauge groups, (ii) its application to contact terms in the $u$-plane integral of topologically twisted theories, and (iii) a connection between the tau functions and the blowup formula in topologically twisted theories.
    

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

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

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

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

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

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

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

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

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

62. Kanehisa Takasaki
    Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains
    Commun. Math. Phys. 241 (1) (2003), 111--142
    Comments: latex2e, 41 pages, no figure
    Report-no: nlin.SI/0206049, v3
    Abstract A chain of one-dimensional Schr\"odinger operators is called a "dressing chain" if they are connected by successive Darboux transformations. Particularly interesting are periodic dressing chains; they include finite-band operators and Painlev\'e equations as a special case. We investigate the Hamiltonian structure of these nonlinear lattices using V.~Adler's $2 \times 2$ Lax pair. The Lax equation and the auxiliary linear problem of this Lax pair contain a shift, rather than a derivative, in the spectral parameter. Despite this unusual feature, we can construct a transition matrix around the periodic chain, an associated "spectral curve" and a set of Darboux coordinates ("spectral Darboux coordinates"). The dressing chain is thereby converted to a Hamiltonian system in these Darboux coordinates. Moreover, the Hamiltonian formalism is accompanied by an odd-dimensional Poisson structure. This induces a quadratic Poisson algebra of the matrix elements of the transition matrix. As a byproduct, we show that this Poisson structure is equivalent to another Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada.
    

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

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

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

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

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

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

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

70. 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. 289--304 (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 algebro-geometric data called "algebraic spectral data". These data consist of an algebraic curve ("spectral curve") $\Gamma$ with a marked point $\gamma_\infty$, a holomorphic vector bundle $E$ on $\Gamma$ and some additional data related to the local structure of $\Gamma$ and $E$ in a neighborhood of $\gamma_\infty$. If the rank $r$ of $E$ is greater than $1$, one can use the so called "Tyurin parameters" in place of $E$ itself. The Tyurin parameters specify the pole structure of a basis of joint eigenfunctions of the commuting pair. These data can be translated to the language of an infinite dimensional Grassmann manifold. This leads to a dynamical system of the standard exponential flows on the Grassmann manifold, in which the role of Tyurin parameters and some other parameters is made clear.
    

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

72. 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.
    

73. 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.
    

74. Kanehisa Takasaki
    Dispersionless Hirota equations of two-component BKP hierarchy
    arXiv:nlin.SI/0604003
    SIGMA Vol. 2 (2006), Paper 057, 22 pages
    Abstract The BKP hierarchy has a two-component analogue (the 2-BKP hierarchy). Dispersionless limit of this multi-component hierarchy is considered on the level of the $\tau$-function. The so called dispersionless Hirota equations are obtained from the Hirota equations of the $\tau$-function. These dispersionless Hirota equations turn out to be equivalent to a system of Hamilton-Jacobi equations. Other relevant equations, in particular, dispersionless Lax equations, can be derived from these fundamental equations. For comparison, another approach based on auxiliary linear equations is also presented.
    

75. Kanehisa Takasaki and Takashi Takebe
    Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy
    Physica D235, no. 1-2 (2007), 109-125
    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 multi-component KP hierarchy. To this end, the multi-component KP hierarchy is (re)formulated to depend on several discrete variables called "charges". These discrete variables play the role of lattice coordinates in underlying Toda field equations. A multi-component version of the so called differential Fay identity are derived from the Hirota equations of the $\tau$-function of this "charged" multi-component KP hierarchy. These multi-component differential Fay identities have a well-defined dispersionless limit (the dispersionless Hirota equations). The dispersionless Hirota equations turn out to be equivalent to the Hamilton-Jacobi equations for the $S$-functions of the universal Whitham hierarchy. The differential Fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar-valued wave functions of the multi-component KP hierarchy.
    

76. 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.
    

77. Toshio Nakatsu and Kanehisa Takasaki
    Melting crystal, quantum torus and Toda hierarchy
    arXiv:0710.5339 [hep-th]
    Commun. Math. Phys. 285 (2009), 445--468
    Comments: 30 pages, 4 figures
    Abstract Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories and $A$-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.
    

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

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

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

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

82. 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 one-variable reduction, dispersionless Hirota equations turn out to be a powerful tool for understanding the mechanism of reduction. All relevant equations describing the reduction (Loewner-type equations and diagonal hydrodynamic equations) can be thereby derived and justified in a unified manner. The case of multi-variable reductions is not so straightforward. Nevertheless, the reduction procedure can be formulated in a general form, and justified with the aid of dispersionless Hirota equations. As an application, previous results of Guil, Ma\~{n}as and Mart\'{\i}nez Alonso are reconfirmed in this formulation.
    

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

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

85. Kanehisa Takasaki and Takashi Takebe
    hbar-expansion of KP hierarchy: Recursive construction of solutions
    arXiv:0912.4867 [math-ph]
    Comments: 28 pages
    Abstract The \hbar-dependent KP hierarchy is a formulation of the KP hierarchy that depends on the Planck constant \hbar and reduces to the dispersionless KP hierarchy as \hbar -> 0. A recursive construction of its solutions on the basis of a Riemann-Hilbert problem for the pair (L,M) of Lax and Orlov-Schulman operators is presented. The Riemann-Hilbert problem is converted to a set of recursion relations for the coefficients X_n of an \hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 +... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). Given the lowest order term X_0, one can solve the recursion relations to obtain the higher order terms. The wave function \Psi associated with W turns out to have the WKB form \Psi = \exp(S/\hbar), and the coefficients S_n of the \hbar-expansion S = S_0 + \hbar S_1 + \hbar^2 S_2 +..., too, are determined by a set of recursion relations. This WKB form is used to show that the associated tau function has an \hbar-expansion of the form \log\tau = \hbar^{-2}F_0 + \hbar^{-1}F_1 + F_2 + ... .
    

86. 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.
    

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

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

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

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

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

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

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

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

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

96. Kanehisa Takasaki
    Integrable structure of modified melting crystal model
    arXiv:1208.4497 [math-ph]
    Comments: 10 pages, no figure, poster presentation at conference "Integrability in Gauge and String Theory" (Zurich, August 20-24, 2012)
    MSC-class: 17B65, 35Q58, 82B20
    Abstract Our previous work on a hidden integrable structure of the melting crystal model (the U(1) Nekrasov function) is extended to a modified crystal model. As in the previous case, "shift symmetries" of a quantum torus algebra plays a central role. With the aid of these algebraic relations, the partition function of the modified model is shown to be a tau function of the 2D Toda hierarchy. We conjecture that this tau function belongs to a class of solutions (the so called Toeplitz reduction) related to the Ablowitz-Ladik hierarchy.
    

97. Kanehisa Takasaki
    Remarks on partition functions of topological string theory on generalized conifolds
    RIMS Kokyuroku No. 1913 (2014), 182--201
    arXiv:1301.4548 [math-ph]
    Comments: 20 pages, 3 figures, contribution to the proceedings of the RIMS camp-style seminar "Algebraic combinatorics related to Young diagrams and statistical physics", August, 2012, International Institute for Advanced Studies, Kyoto, organized by M. Ishikawa, S. Okada and H. Tagawa
    MSC-class: 05E05, 37K10, 81T30
    Abstract The notion of topological vertex and the construction of topological string partition functions on local toric Calabi-Yau 3-folds are reviewed. Implications of an explicit formula of partition functions for the generalized conifolds are considered. Generating functions of part of the partition functions are shown to be tau functions of the KP hierarchy. The associated Baker-Akhiezer functions play the role of wave functions, and satisfy $q$-difference equations. These $q$-difference equations represent the quantum mirror curves conjectured by Gukov and Su{\l}kowski.

98. Kanehisa Takasaki
    Modified melting crystal model and Ablowitz-Ladik hierarchy
    J. Phys. A: Math. Theor. 46 (2013), 245202 (23 pages)
    arXiv:1302.6129 [math-ph]
    Comments: 33 pages, no figure
    MSC classes: 17B65, 35Q55, 81T30, 82B20
    doi:10.1088/1751-8113/46/24/245202
    Abstract This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of "shift symmetries". They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can be characterized by a factorization problem of $\ZZ\times\ZZ$ matrices as well. The associated Lax operators turn out to be quotients of first order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz-Ladik (equivalently, relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.

99. Kanehisa Takasaki
    Generalized Ablowitz-Ladik hierarchy in topological string theory
    J. Phys. A: Math. Theor. 47 (2014), 165201 (20 pages)
    doi:10.1088/1751-8113/47/16/165201
    arXiv:1312.7184 [math-ph]
    Comments: 24pages, 1 figre
    MSC-class: 17B80, 35Q55, 81T30
    Abstract This paper addresses the issue of integrable structure in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. The generating function of these amplitudes with respect to the product of two independent Schur functions become a tau function of the 2D Toda hierarchy. The associated Lax operators turn out to have a particular factorized form. This factorized form of the Lax operators characterizes a generalization of the Ablowitz-Ladik hierarchy embedded in the 2D Toda hierarchy. The generalized Ablowitz-Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.

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

101. Kanehisa Takasaki
     Orbifold melting crystal models and reductions of Toda hierarchy
     J. Phys. A: Math. Theor. 48 (2015), 215201 (34 pages)
     doi:10.1088/1751-8113/48/21/215201
     arXiv:1410.5060 [math-ph]
     Comments: 41 pages, no figure
     MSC-class: 17B65, 35Q55, 81T30, 82B20
     Abstract Orbifold generalizations of the ordinary and modified melting crystal models are introduced. They are labelled by a pair $a,b$ of positive integers, and geometrically related to $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifolds of local $\mathbf{CP}^1$ geometry of the $\mathcal{O}(0)\oplus\mathcal{O}(-2)$ and $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ types. The partition functions have a fermionic expression in terms of charged free fermions. With the aid of shift symmetries in a fermionic realization of the quantum torus algebra, one can convert these partition functions to tau functions of the 2D Toda hierarchy. The powers $L^a,\bar{L}^{-b}$ of the associated Lax operators turn out to take a special factorized form that defines a reduction of the 2D Toda hierarchy. The reduced integrable hierarchy for the orbifold version of the ordinary melting crystal model is the bi-graded Toda hierarchy of bi-degree $(a,b)$. That of the orbifold version of the modified melting crystal model is the rational reduction of bi-degree $(a,b)$. This result seems to be in accord with recent work of Brini et al. on a mirror description of the genus-zero Gromov-Witten theory on a $\mathbf{Z}_a\times\mathbf{Z}_b$ orbifold of the resolved conifold.

102. Kanehisa Takasaki and Toshio Nakatsu
     Open string amplitudes of closed topological vertex
     J. Phys. A: Math. Theor. 49 (2016), 025201 (28pp)
     doi:10.1088/1751-8113/49/2/025201
     arXiv:1507.07053
     Comments: latex2e, package amsmath,amssymb,amsthm,graphicx, 10 figures
     MSC-class: 17B81, 33E20, 81T30
     Abstract The closed topological vertex is the simplest ``off-strip'' case of non-compact toric Calabi-Yau threefolds with acyclic web diagrams. By the diagrammatic method of topological vertex, open string amplitudes of topological string theory therein can be obtained by gluing a single topological vertex to an ``on-strip'' subdiagram of the tree-like web diagram. If non-trivial partitions are assigned to just two parallel external lines of the web diagram, the amplitudes can be calculated with the aid of techniques borrowed from the melting crystal models. These amplitudes are thereby expressed as matrix elements, modified by simple prefactors, of an operator product on the Fock space of 2D charged free fermions. This fermionic expression can be used to derive $q$-difference equations for generating functions of special subsets of the amplitudes. These $q$-difference equations may be interpreted as the defining equation of a quantum mirror curve.

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

104. Kanehisa Takasaki
     4D limit of melting crystal model and its integrable structure
     (previous title: Quantum curve and 4D limit of melting crystal model)
     Journal of Geometry and Physics 137 (2019), 184--203
     DOI: 10.1016/j.geomphys.2018.12.012
     arXiv:1704.02750 [math-ph]
     Comments: latex2e using packages amsmath,amssymb,amsthm, 35 pages, no figure; (v2) the title is changed, and an appendix on the relevance to the Toda hierarchy is added; (v3) texts in Introduction and Sect. 4.2 are modified, a few typos are corrected, final version for publication
     MSC-class: 14N35, 37K10, 39A13
     Abstract This paper addresses the problems of quantum spectral curves and 4D limit for the melting crystal model of 5D SUSY $U(1)$ Yang-Mills theory on $\mathbb{R}^4\times S^1$. The partition function $Z(\mathbf{t})$ deformed by an infinite number of external potentials is a tau function of the KP hierarchy with respect to the coupling constants $\mathbf{t} = (t_1,t_2,\ldots)$. A single-variate specialization $Z(x)$ of $Z(\mathbf{t})$ satisfies a $q$-difference equation representing the quantum spectral curve of the melting crystal model. In the limit as the radius $R$ of $S^1$ in $\mathbb{R}^4\times S^1$ tends to $0$, it turns into a difference equation for a 4D counterpart $Z_{\mathrm{4D}}(X)$ of $Z(x)$. This difference equation reproduces the quantum spectral curve of Gromov-Witten theory of $\mathbb{CP}^1$. $Z_{\mathrm{4D}}(X)$ is obtained from $Z(x)$ by letting $R \to 0$ under an $R$-dependent transformation $x = x(X,R)$ of $x$ to $X$. A similar prescription of 4D limit can be formulated for $Z(\mathbf{t})$ with an $R$-dependent transformation $\mathbf{t} = \mathbf{t}(\mathbf{T},R)$ of $\mathbf{t}$ to $\mathbf{T} = (T_1,T_2,\ldots)$. This yields a 4D counterpart $Z_{\mathrm{4D}}(\mathbf{T})$ of $Z(\mathbf{t})$. $Z_{\mathrm{4D}}(\mathbf{T})$ agrees with a generating function of all-genus Gromov-Witten invariants of $\mathbb{CP}^1$. Fay-type bilinear equations for $Z_{\mathrm{4D}}(\mathbf{T})$ can be derived from similar equations satisfied by $Z(\mathbf{t})$. The bilinear equations imply that $Z_{\mathrm{4D}}(\mathbf{T})$, too, is a tau function of the KP hierarchy. These results are further extended to deformations $Z(\mathbf{t},s)$ and $Z_{\mathrm{4D}}(\mathbf{T},s)$ by a discrete variable $s \in \mathbb{Z}$, which are shown to be tau functions of the 1D Toda hierarchy.

105. Kanehisa Takasaki
     Toda hierarchies and their applications
     J. Phys. A: Math. Theor. 51 (2018) 203001 (35pp)
     doi: 10.1088/1751-8121/aabc14
     arXiv:1801.09924 [math-ph]
     Comments: 46 pages, no figure, contribution to JPhysA Special Issue "Fifty years of the Toda lattice"
     MSC classes: 17B65, 37K10, 82B20
     Abstract The 2D Toda hierarchy occupies a central position in the family of integrable hierarchies of the Toda type. The 1D Toda hierarchy and the Ablowitz-Ladik (aka relativistic Toda) hierarchy can be derived from the 2D Toda hierarchy as reductions. These integrable hierarchies have been applied to various problems of mathematics and mathematical physics since 1990s. A recent example is a series of studies on models of statistical mechanics called the melting crystal model. This research has revealed that the aforementioned two reductions of the 2D Toda hierarchy underlie two different melting crystal models. Technical clues are a fermionic realization of the quantum torus algebra, special algebraic relations therein called shift symmetries, and a matrix factorization problem. The two melting crystal models thus exhibit remarkable similarity with the Hermitian and unitary matrix models for which the two reductions of the 2D Toda hierarchy play the role of fundamental integrable structures.

106. Kanehisa Takasaki
     Hurwitz numbers and integrable hierarchy of Volterra type
     J. Phys. A: Math. Theor. 51 (2018), 43LT01 (9 pages)
     doi: 10.1088/1751-8121/aae10b
     arXiv:1807.00085
     Comments: latex2e, amsmath,amssymb,amsthm, 12 pages, no figure
     MSC-class: 14N10, 37K10
     Abstract A generating function of the single Hurwitz numbers of the Riemann sphere $\mathbb{CP}^1$ is a tau function of the lattice KP hierarchy. The associated Lax operator $L$ turns out to be expressed as $L = e^{\mathfrak{L}}$, where $\mathfrak{L}$ is a difference-differential operator of the form $\mathfrak{L} = \partial_s - ve^{-\partial_s}$. $\mathfrak{L}$ satisfies a set of Lax equations that form a continuum version of the Bogoyavlensky-Itoh (aka hungry Lotka-Volterra) hierarchies. Emergence of this underlying integrable structure is further explained in the language of generalized string equations for the Lax and Orlov-Schulman operators of the 2D Toda hierarchy. This leads to logarithmic string equations, which are confirmed with the help of a factorization problem of operators.

107. Toshio Nakatsu and Kanehisa Takasaki
     Three-partition Hodge integrals and the topological vertex
     Communications in Mathematical Physics, 376(1) (2020), 201-234
     doi: 10.1007/s00220-019-03648-5
     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 two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters therein is shown to be a tau function of the generalized KdV (aka Gelfand-Dickey) hierarchies.

108. Kanehisa Takasaki
     Cubic Hodge integrals and integrable hierarchies of Volterra type
     Proceedings of Symposia in Pure Mathematics, vol. 103.1, Amer. Math. Soc., Providence, RI, 2021, pp. 481-502
     arXiv:1909.13095
     Comments: latex2e, amsmath,amssymb,amsthm, 29pp, no figure
     Abstract A tau function of the 2D Toda hierarchy can be obtained from a generating function of the two-partition cubic Hodge integrals. The associated Lax operators turn out to satisfy an algebraic relation. This algebraic relation can be used to identify a reduced system of the 2D Toda hierarchy that emerges when the parameter $\tau$ of the cubic Hodge integrals takes a special value. Integrable hierarchies of the Volterra type are shown to be such reduced systems. They can be derived for positive rational values of $\tau$. In particular, the discrete series $\tau = 1,2,\ldots$ correspond to the Volterra lattice and its hungry generalizations. This provides a new explanation to the integrable structures of the cubic Hodge integrals observed by Dubrovin et al. in the perspectives of tau-symmetric integrable Hamiltonian PDEs.

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

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

111. Kanehisa Takasaki
     Extended lattice Gelfand-Dickey hierarchy
     J. Phys. A: Math. Theor. 55 (2022), 305203 (14pp)
     doi:10.1088/1751-8121/ac7ca2
     arXiv:2203.06621
     MSC classes: 14N35, 37K10
     Abstract The lattice Gelfand-Dickey hierarchy is a lattice version of the Gelfand-Dickey hierarchy. A special case is the lattice KdV hierarchy. Inspired by recent work of Buryak and Rossi, we propose an extension of the lattice Gelfand-Dickey hierarchy. The extended system has an infinite number of logarithmic flows alongside the usual flows. We present the Lax, Sato and Hirota equations and a factorization problem of difference operators that captures the whole set of solutions. The construction of this system resembles the extended 1D and bigraded Toda hierarchy, but exhibits several novel features as well.

112. 高崎金久
     Matrix-tree theoremの起源
     第31回数学史シンポジウム(2021.10.16〜17)報告集
     津田塾大学数学・計算機科学研究所報43(2021), 61-76
     要旨 Matrix-tree theorem（行列と木の定理）はグラフの全域木の個数が ある行列の余因子として表せることを主張する．さらに，グラフの辺に 重みを付けて，全域木の重みの総和を行列式として表す一般化もある． この定理は数え上げ問題に対する線形代数的技法の中でも古くから 知られているもので，組合せ論の枠内にとどまらない内容をもつ． この定理の原型が登場する19 世紀の文献を紹介し，この定理の歴史的経緯や， それに関して流布している誤解などを紹介する．

113. Kanehisa Takasaki
     Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy
     J. Phys. A: Math. Theor. 56 (2023) 165201 (25pp)
     doi: 10.1088/1751-8121/acc495
     arXiv:2211.11353
     Comments: latex2e using amsmath,amssymb,amsthm, 30 pages, no figure
     MSC-class: 14N35, 37K10
     Abstract The intermediate long wave (ILW) hierarchy and its generalization, labelled by a positive integer $N$, can be formulated as reductions of the lattice KP hierarchy. The integrability of the lattice KP hierarchy is inherited by these reduced systems. In particular, all solutions can be captured by a factorization problem of difference operators. A special solution among them is obtained from Okounkov and Pandharipande's dressing operators for the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. This indicates a hidden link with the equivariant Toda hierarchy. The generalized ILW hierarchy is also related to the lattice Gelfand-Dickey (GD) hierarchy and its extension by logarithmic flows. The logarithmic flows can be derived from the generalized ILW hierarchy by a scaling limit as a parameter of the system tends to $0$. This explains an origin of the logarithmic flows. A similar scaling limit of the equivariant Toda hierarchy yields the extended 1D/bigraded Toda hierarchy.