# Cubic Hodge integrals and integrable hierarchies of Volterra type

@article{Takasaki2019CubicHI, title={Cubic Hodge integrals and integrable hierarchies of Volterra type}, author={K. Takasaki}, journal={arXiv: Mathematical Physics}, year={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. 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… Expand

#### 3 Citations

KP integrability of triple Hodge integrals, I. From Givental group to hierarchy symmetries

- Mathematics, Physics
- 2020

In this paper we investigate a relation between the Givental group of rank one and Heisenberg-Virasoro symmetry group of the KP hierarchy. We prove, that only a two-parameter family of the Givental… Expand

Integrable structures of specialized hypergeometric tau functions By Kanehisa Takasaki

- 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… Expand

Integrable structures of specialized hypergeometric tau functions

- Mathematics, Physics
- 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… Expand

#### References

SHOWING 1-10 OF 44 REFERENCES

Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs

- Mathematics, Physics
- 2014

For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of… Expand

Fractional Volterra hierarchy

- Mathematics, Physics
- 2017

The generating function of cubic Hodge integrals satisfying the local Calabi–Yau condition is conjectured to be a tau function of a new integrable system which can be regarded as a fractional… Expand

Hurwitz numbers and integrable hierarchy of Volterra type

- Physics, Mathematics
- 2018

A generating function of the single Hurwitz numbers of the Riemann sphere is a tau function of the lattice KP hierarchy. The associated Lax operator L turns out to be expressed as , where is a… Expand

Three-Partition Hodge Integrals and the Topological Vertex

- Mathematics, Physics
- 2018

We give a new proof of the equivalence between the cubic Hodge integrals and the topological vertex in topological string theory. A central role is played by the notion of generalized shift… Expand

The extended bigraded Toda hierarchy

- Physics, Mathematics
- 2006

We generalize the Toda lattice hierarchy by considering N + M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that… Expand

Toda hierarchies and their applications

- Mathematics, Physics
- 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… Expand

Hodge Integrals and Integrable Hierarchies

- Mathematics
- 2003

We show that the generating series of some Hodge integrals involving one or two partitions are τ-functions of the KP hierarchy or the 2-Toda hierarchy, respectively. We also reformulate the results… Expand

On an Explicitly Soluble System of Nonlinear Differential Equations Related to Certain Toda Lattices

- Mathematics
- 1975

Publisher Summary This chapter discusses an explicitly soluble system of nonlinear differential equations related to certain Toda lattices. Toda lattice was solved by applying a discrete version of… Expand

Soliton Equations and Hamiltonian Systems

- Mathematics
- 2003

Integrable Systems Generated by Linear Differential nth Order Operators Hamiltonian Structures Hamiltonian Structure of the GD Hierarchies Modified KdV and GD. The Kupershmidt-Wilson Theorem The KP… Expand

Generalized Ablowitz–Ladik hierarchy in topological string theory

- Physics, Mathematics
- 2014

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… Expand