arXiv subjects

Kanehisa Takasaki

Publications and source records attributed to Kanehisa Takasaki.

At least 19 recordsLinked to original sources

arXiv2026

Direct linearization, Cauchy matrix and Sato Grassmannian

Fu and Nijhoff's direct linearization scheme for the KP hierarchy and its reductions employs a system of evolution equations of an infinite matrix $U$ with quadratic nonlinearity. Part of the matrix elements of $U$ can be identified with affine coordinates $w_{ij}$ of the top cell of the Sato Grassmannian. The evolution equations of these matrix elements are identical to the evolution equations of $w_{ij}$ representing the KP hierarchy in geometric terms. This geometric interpretation can be extended to other matrix elements of $U$ by introducing negative flows to the evolution equations of $U$. The extended system turns out to be substantially equivalent to the two-component KP hierarchy. The Cauchy matrix approach to the KP hierarchy can be explained in this geometric perspective. Multi-component generalizations of Fu and Nijhoff's nonlinear system are related to the AKNS and ASDYM hierarchies. Multi-component Sato Grassmannians show up therein as the relevant geometric structure.

nlin.SI↗

arXiv2023

Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy

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.

nlin.SI↗

arXiv2022

Extended lattice Gelfand-Dickey hierarchy

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.

nlin.SI↗

arXiv2021

Dressing operators in equivariant Gromov-Witten theory of $\mathbb{CP}^1$

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.

math-ph↗

arXiv2021

Integrable structures of specialized hypergeometric tau functions

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.

math-ph↗

arXiv2020

Cubic Hodge integrals and integrable hierarchies of Volterra type

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 $τ$ 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 $τ$. In particular, the discrete series $τ= 1,2,\ldots$ 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.

math-ph↗

arXiv2018

Three-partition Hodge integrals and the topological vertex

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.

math-ph↗

arXiv2018

4D limit of melting crystal model and its integrable structure

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.

math-ph↗

arXiv2018

Hurwitz numbers and integrable hierarchy of Volterra type

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.

math-ph↗

arXiv2018

Toda hierarchies and their applications

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.

math-ph↗

arXiv2017

$q$-Difference Kac-Schwarz Operators in Topological String Theory

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 $\{Φ_j(x)\}_{j=0}^\infty$ of $W$. $q$-difference analogues $A$, $B$ of Kac-Schwarz operators are defined with the aid of $G$. $Φ_j(x)$'s satisfy the linear equations $AΦ_j(x) = q^jΦ_j(x)$, $BΦ_j(x) = Φ_{j+1}(x)$. The lowest equation $AΦ_0(x) = Φ_0(x)$ reproduces the quantum mirror curve in the authors' previous work.

math-ph↗

arXiv2015

Open string amplitudes of closed topological vertex

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.

math-ph↗

arXiv2015

Remarks on partition functions of topological string theory on generalized conifolds

The method of topological vertex for topological string theory on toric Calabi-Yau 3-folds is reviewed. Implications of an explicit formula of partition functions in the "on-strip" case, typically 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 Sułkowski.

math-ph↗

arXiv2015

Orbifold melting crystal models and reductions of Toda hierarchy

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.

math-ph↗

arXiv2014

Generalized Ablowitz-Ladik hierarchy in topological string theory

This paper addresses the issue of integrable structures 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 becomes 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.

math-ph↗

arXiv2013

Modified melting crystal model and Ablowitz-Ladik hierarchy

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.

math-ph↗

arXiv2013

Modified melting crystal model and Ablowitz-Ladik hierarchy

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.

math-ph↗

arXiv2012

Old and New Reductions of Dispersionless Toda Hierarchy

This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of "Landau-Ginzburg potentials" that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric polynomial. The other is a transcendental function, the logarithm of which resembles the waterbag models of the dispersionless KP hierarchy. They both satisfy a radial version of the Löwner equations. Consistency of these Löwner equations yields 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). Flat coordinates of the underlying Egorov metrics are presented.

math-ph↗