Three-Partition Hodge Integrals and the Topological Vertex

Nakatsu, Toshio; Takasaki, Kanehisa

doi:10.1007/s00220-019-03648-5

Cited by 3 publications

(2 citation statements)

References 22 publications

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“…This tau function appears to be a particular case of the generating functions considered in [51]. It is naturally associated to the topological vertex C ∅,∅,ν that enumerates plane partitions with fixed asymptotics ν in one direction [40].…”

Section: Jhep05(2021)216mentioning

confidence: 99%

Intertwining operator and integrable hierarchies from topological strings

Bourgine

2021

J. High Energ. Phys.

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In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

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Section: Jhep05(2021)216mentioning

confidence: 99%

Intertwining operator and integrable hierarchies from topological strings

Bourgine

2021

J. High Energ. Phys.

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“…The special values of the Schur polynomials for these c's play important roles in the work of Orlov and Scherbin as well [1,2,11]. The specialized hypergeometric tau functions are related to our previous work on topological string theory [12,13] and our recent work on the Hurwitz numbers and the Hodge integrals [14,15,16]. We found therein that integrable hierarchies of the Ablowitz-Ladik, Volterra and Gelfand-Dickey types emerge as underlying integrable structures.…”

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confidence: 91%

Integrable structures of specialized hypergeometric tau functions

Takasaki¹

2020

Preprint

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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.

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Cubic Hodge integrals and integrable hierarchies of Volterra type

Takasaki

2021

Proceedings of Symposia in Pure Mathematics

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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 \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 τ \tau . In particular, the discrete series τ = 1 , 2 , … \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.

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Three-Partition Hodge Integrals and the Topological Vertex

Cited by 3 publications

References 22 publications

Intertwining operator and integrable hierarchies from topological strings

Intertwining operator and integrable hierarchies from topological strings

Integrable structures of specialized hypergeometric tau functions

Cubic Hodge integrals and integrable hierarchies of Volterra type

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