• Corpus ID: 231709494

Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality

@inproceedings{Dubrovin2021GeometryAA,
  title={Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality},
  author={Boris Dubrovin and Di Yang and Don Zagier},
  year={2021}
}
For each of the simple Lie algebras g = Al, Dl or E6, we show that the all-genera one-point FJRW invariants of g-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of g-type evaluated at a special point. For the A4 (5-spin) case we also find two other… 
4 Citations

Figures from this paper

On tau-functions for the KdV hierarchy

For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic

On the large genus asymptotics of psi-class intersection numbers

Based on an explicit formula of the generating series for the n-point psi-class intersection numbers (cf. Bertola et. al. [4]), we give a novel proof of a conjecture of Delecroix et. al. [10]

Punctures and p-Spin Curves from Matrix Models II

We report here an extension of a previous work in which we have shown that matrix models provide a tool to compute the intersection numbers of p-spin curves. We discuss further an extension to

Punctures and p-Spin Curves from Matrix Models III. Dl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_l$$\end{docume

  • S. Hikami
  • Materials Science
    Journal of Statistical Physics
  • 2022
The intersection numbers for p spin curves of the moduli space M¯g,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}

References

SHOWING 1-10 OF 127 REFERENCES

Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of higher spin curves to Gelfand–Dickey hierarchies. That is, we show that intersection numbers on the moduli

Simple singularities and integrable hierarchies

The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In [12], it is proved that the total descendent potential corresponding to K.

The structure of 2D semi-simple field theories

I classify the cohomological 2D field theories based on a semi-simple complex Frobenius algebra A. They are controlled by a linear combination of κ-classes and by an extension datum to the

A_{n-1} singularities and nKdV hierarchies

According to a conjecture of E. Witten proved by M. Kontsevich, a certain generating function for intersection indices on the Deligne -- Mumford moduli spaces of Riemann surfaces coincides with a

Arithmetic and Topology of Differential Equations

The talk will describe arithmetic properties of solutions of linear differential equations, especially Picard-Fuchs equations, and their connections with modular forms, mirror symmetry, and

Computing topological invariants with one and two-matrix models

A generalization of the Kontsevich Airy-model allows one to compute the intersection numbers of the moduli space of p-spin curves. These models are deduced from averages of characteristic polynomials

THE N-POINT FUNCTIONS FOR INTERSECTION NUMBERS ON MODULI SPACES OF CURVES

Abstract. Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the n-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the

Open Saito Theory for A and D Singularities

A well-known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper, we present a

A polynomial bracket for the Dubrovin--Zhang hierarchies

We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the

Semisimple Frobenius structures at higher genus

In the context of equivariant Gromov-Witten theory of tori actions with isolated fixed points, we compute genus g > 1 Gromov-Witten potentials and their generalizations with gravitational
...