Integrable Systems of Double Ramification Type

  title={Integrable Systems of Double Ramification Type},
  author={Alexandr Buryak and Boris Dubrovin and J'er'emy Gu'er'e and Paolo Rossi},
  journal={International Mathematics Research Notices},
In this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st author, and its quantization. We extend the notion of tau-symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus $1$ quantum correction and, as an application, compute completely the quantization of the $3$- and $4$-KdV hierarchies (the DR hierarchies for Witten’s $3$- and $4$-spin theories). We then focus… 

Figures from this paper

Quadratic double ramification integrals and the noncommutative KdV hierarchy

In this paper we compute the intersection number of two double ramification (DR) cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of

Tautological relations and integrable systems

. We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus g with n marked points. A large part of these relations has a

Quantum KdV hierarchy and quasimodular forms

Dubrovin [10] has shown that the spectrum of the quantization (with respect to the first Poisson structure) of the dispersionless Korteweg–de Vries (KdV) hierarchy is given by shifted symmetric

Bi-Hamiltonian Recursion, Liu–Pandharipande Relations, and Vanishing Terms of the Second Dubrovin–Zhang Bracket

The Dubrovin–Zhang hierarchy is a Hamiltonian infinite-dimensional integrable system associated to a semi-simple cohomological field theory or, alternatively, to a semi-simple Dubrovin–Frobenius

Integrability, Quantization and Moduli Spaces of Curves

This paper has the purpose of presenting in an organic way a new approach to integrable (1+1)-dimensional field systems and their systematic quantization emerging from intersection theory of the

Flat F-Manifolds, F-CohFTs, and Integrable Hierarchies

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat

The quantum Witten–Kontsevich series and one-part double Hurwitz numbers

  • X. Blot
  • Mathematics
    Geometry & Topology
  • 2022
We study the quantum Witten-Kontsevich series introduced by Buryak, Dubrovin, Guere and Rossi in \cite{buryak2016integrable} as the logarithm of a quantum tau function for the quantum KdV hierarchy.

The Bi-Hamiltonian Structures of the DR and DZ Hierarchies in the Approximation up to Genus One

In a recent paper, given an arbitrary homogeneous cohomological field theory ( CohFT), Rossi, Shadrin, and the first author proposed a simple formula for a bracket on the space of local functionals,



Recursion Relations for Double Ramification Hierarchies

In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in Buryak (CommunMath Phys 336(3):1085–1107, 2015) using

Double Ramification Cycles and Quantum Integrable Systems

In this paper, we define a quantization of the Double Ramification Hierarchies of Buryak (Commun Math Phys 336:1085–1107, 2015) and Buryak and Rossi (Commun Math Phys, 2014), using intersection

Tau-Structure for the Double Ramification Hierarchies

In this paper we continue the study of the double ramification hierarchy of Buryak (Commun Math Phys 336(3):1085–1107, 2015). After showing that the DR hierarchy satisfies tau-symmetry we define its

Integrable systems and holomorphic curves

In this paper we attempt a self-contained approach to infinite dimensional Hamiltonian systems appearing from holomorphic curve counting in Gromov-Witten theory. It consists of two parts. The first

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

Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the

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

BCFG Drinfeld–Sokolov hierarchies and FJRW-theory

According to the ADE Witten conjecture, which is proved by Fan, Jarvis and Ruan, the total descendant potential of the FJRW invariants of an ADE singularity is a tau function of the corresponding