Orbifold melting crystal models and reductions of Toda hierarchy

Takasaki, Kanehisa

doi:10.1088/1751-8113/48/21/215201

Cited by 6 publications

(13 citation statements)

References 47 publications

(108 reference statements)

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“…Note here that the sum with respect to α 3 resembles the partition function of the modified melting model [11,12]: The main part of the Boltzmann weight therein takes the product form s t α 3 (q −ρ )s α 3 (q −ρ ), and this weight is deformed by external potentials depending on α 3 . To calculate this sum, we use the machinery of 2D charged free fermions.…”

Section: Reformulation Of Amplitudementioning

confidence: 99%

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Open string amplitudes of closed topological vertex

Takasaki

Nakatsu

2015

J. Phys. A: Math. Theor.

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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.

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Section: Reformulation Of Amplitudementioning

confidence: 99%

“…The setup of the fermionic Fock space and operators is the same as used for the melting crystal models [9,10,11,12]. Let ψ n , ψ * n , n ∈ Z, denote the Fourier modes of the 2D charged free fermion fields ψ(z), ψ * (z).…”

Section: Fermionic Fock Space and Operatorsmentioning

confidence: 99%

Open string amplitudes of closed topological vertex

Takasaki

Nakatsu

2015

J. Phys. A: Math. Theor.

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“…We proved, with the aid of symmetries of a quantum torus algebra, that Z(t, s) is essentially a tau function of the 1D Toda hierarchy [37]. This result has been extended to some other types of melting crystal models [32,33].…”

Section: Introductionmentioning

confidence: 75%

4D limit of melting crystal model and its integrable structure

Takasaki

2019

Journal of Geometry and Physics

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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 R 4 ×S 1 . The partition function Z(t) deformed by an infinite number of external potentials is a tau function of the KP hierarchy with respect to the coupling constants t = (t 1 , t 2 , . . .). A single-variate specialization Z(x) of Z(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 R 4 × S 1 tends to 0, it turns into a difference equation for a 4D counterpart Z 4D (X) of Z(x). This difference equation reproduces the quantum spectral curve of Gromov-Witten theory of CP 1 . Z 4D (X) is obtained from Z(x) by letting R → 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(t) with an R-dependent transformation t = t(T , R) of t to T = (T 1 , T 2 , . . .). This yields a 4D counterpart Z 4D (T ) of Z(t). Z 4D (T ) agrees with a generating function of all-genus Gromov-Witten invariants of CP 1 . Fay-type bilinear equations for Z 4D (T ) can be derived from similar equations satisfied by Z(t). The bilinear equations imply that Z 4D (T ), too, is a tau function of the KP hierarchy. These results are further extended to deformations Z(t, s) and Z 4D (T , s) by a discrete variable s ∈ Z, which are shown to be tau functions of the 1D Toda hierarchy.

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“…To prove the algebraic relation of the Lax operators, we use a method developed in our previous work on the melting crystal model and a family of topological string theory [17,18,19]. This method is based on a factorization problem that characterizes the dressing operators behind the Lax operators.…”

Section: Introductionmentioning

confidence: 99%

Cubic Hodge integrals and integrable hierarchies of Volterra type

Takasaki

2019

Preprint

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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, . . . 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.

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Orbifold melting crystal models and reductions of Toda hierarchy

Cited by 6 publications

References 47 publications

Open string amplitudes of closed topological vertex

Open string amplitudes of closed topological vertex

4D limit of melting crystal model and its integrable structure

Cubic Hodge integrals and integrable hierarchies of Volterra type

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