# Integrable Systems of Double Ramification Type

@article{Buryak2016IntegrableSO, 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}, year={2016} }

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…

## 22 Citations

### Quadratic double ramification integrals and the noncommutative KdV hierarchy

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### Tautological relations and integrable systems

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. 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…

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### Quantum KdV hierarchy and quasimodular forms

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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

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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

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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

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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

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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

- MathematicsFunctional Analysis and Its Applications
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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,…

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