# A modified melting crystal model and the Ablowitz–Ladik hierarchy

@article{Takasaki2013AMM, title={A modified melting crystal model and the Ablowitz–Ladik hierarchy}, author={Kanehisa Takasaki}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2013}, volume={46} }

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…

## 17 Citations

### Orbifold melting crystal models and reductions of Toda hierarchy

- Mathematics
- 2014

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 Z a × Z b ?>…

### Modified melting crystal model and Ablowitz-Ladik hierarchy

- Physics
- 2014

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…

### Rational reductions of the 2D-Toda hierarchy and mirror symmetry

- MathematicsJournal of the European Mathematical Society
- 2017

We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a…

### 4D limit of melting crystal model and its integrable structure

- MathematicsJournal of Geometry and Physics
- 2019

### Toda hierarchies and their applications

- Mathematics
- 2018

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…

### Cubic Hodge integrals and integrable hierarchies of Volterra type

- Mathematics
- 2019

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.…

### Open string amplitudes of closed topological vertex

- Physics
- 2015

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…

### Integrable structures of specialized hypergeometric tau functions By Kanehisa Takasaki

- Mathematics
- 2020

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…

### Integrable structures of specialized hypergeometric tau functions

- Mathematics
- 2020

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…

### Intertwining operator and integrable hierarchies from topological strings

- MathematicsJournal of High Energy Physics
- 2021

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…

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