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 on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometric origin, we use it to characterize and construct a new family of quantum and classical integrable systems that we call of DR type, as they satisfy all of the main properties of the DR hierarchy. In the 2nd part, we obtain new insight towards the Miura equivalence conjecture between the DR and Dubrovin-Zhang (DZ) hierarchies, via a geometric interpretation of the correlators forming the DR tau-function. We then show that the candidate Miura transformation between the DR and DZ hierarchies (which we uniquely identified in our previous paper) indeed turns the DZ Poisson structure into the standard form. Eventually, we focus on integrable hierarchies associated with rank-1 cohomological field theories and their deformations, and we prove the DR/DZ equivalence conjecture up to genus 5 in this context.

Integrable systems of double ramification type / Buryak, A.; Dubrovin, B.; Guéré, J.; Rossi, Paolo. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - 2020:24(2020). [10.1093/imrn/rnz029]

Integrable systems of double ramification type

Dubrovin B.;
2020-01-01

Abstract

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 on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometric origin, we use it to characterize and construct a new family of quantum and classical integrable systems that we call of DR type, as they satisfy all of the main properties of the DR hierarchy. In the 2nd part, we obtain new insight towards the Miura equivalence conjecture between the DR and Dubrovin-Zhang (DZ) hierarchies, via a geometric interpretation of the correlators forming the DR tau-function. We then show that the candidate Miura transformation between the DR and DZ hierarchies (which we uniquely identified in our previous paper) indeed turns the DZ Poisson structure into the standard form. Eventually, we focus on integrable hierarchies associated with rank-1 cohomological field theories and their deformations, and we prove the DR/DZ equivalence conjecture up to genus 5 in this context.
2020
2020
24
https://arxiv.org/abs/1609.04059
Buryak, A.; Dubrovin, B.; Guéré, J.; Rossi, Paolo
File in questo prodotto:
File Dimensione Formato  
BDGR-2019.pdf

non disponibili

Tipologia: Versione Editoriale (PDF)
Licenza: Non specificato
Dimensione 1.03 MB
Formato Adobe PDF
1.03 MB Adobe PDF   Visualizza/Apri   Richiedi una copia
Buryak_Dubrovin_Guere_Rossi_2.pdf

Open Access dal 20/02/2020

Tipologia: Documento in Post-print
Licenza: Non specificato
Dimensione 578.58 kB
Formato Adobe PDF
578.58 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/87478
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 16
  • ???jsp.display-item.citation.isi??? 12
social impact