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On the strong DR/DZ equivalence conjecture

Xavier Blot, Danilo Lewanski, Sergey Shadrin

TL;DR

This work proves the strong DR/DZ equivalence by showing that, for a semi-simple CohFT, the Dubrovin–Zhang hierarchy is Miura-equivalent to Buryak’s double ramification hierarchy after a normal Miura transformation driven by a specific differential polynomial. Central to the argument is the A=B tautological relation, reformulated via the master relation in the Gorenstein quotient and expressed through leveled, degree-labeled trees with $B$- and $A$-type classes. The authors establish three key degree-vanishing statements for intersection numbers involving these classes, reducing the problem to combinatorial and tautological-geometry arguments, and then relate the master relation to the DR/DZ equivalence. As a corollary, the results extend to partial and F-CohFTs, remove the semi-simplicity assumption in many cases, and imply the existence and polynomiality of the DZ hierarchy in broader settings, thereby unifying two fundamental integrable systems in enumerative geometry with a concrete, computable linking map.

Abstract

We establish the Miura equivalence of two integrable systems associated to a semi-simple cohomological field theory: the double ramification hierarchy of Buryak and the Dubrovin-Zhang hierarchy. This equivalence was conjectured by Buryak and further refined by Buryak, Dubrovin, Guéré, and Rossi.

On the strong DR/DZ equivalence conjecture

TL;DR

This work proves the strong DR/DZ equivalence by showing that, for a semi-simple CohFT, the Dubrovin–Zhang hierarchy is Miura-equivalent to Buryak’s double ramification hierarchy after a normal Miura transformation driven by a specific differential polynomial. Central to the argument is the A=B tautological relation, reformulated via the master relation in the Gorenstein quotient and expressed through leveled, degree-labeled trees with - and -type classes. The authors establish three key degree-vanishing statements for intersection numbers involving these classes, reducing the problem to combinatorial and tautological-geometry arguments, and then relate the master relation to the DR/DZ equivalence. As a corollary, the results extend to partial and F-CohFTs, remove the semi-simplicity assumption in many cases, and imply the existence and polynomiality of the DZ hierarchy in broader settings, thereby unifying two fundamental integrable systems in enumerative geometry with a concrete, computable linking map.

Abstract

We establish the Miura equivalence of two integrable systems associated to a semi-simple cohomological field theory: the double ramification hierarchy of Buryak and the Dubrovin-Zhang hierarchy. This equivalence was conjectured by Buryak and further refined by Buryak, Dubrovin, Guéré, and Rossi.
Paper Structure (34 sections, 17 theorems, 55 equations)

This paper contains 34 sections, 17 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Cohomological field theories
  3. Dubrovin-Zhang hierarchy
  4. Buryak's double ramification hierarchy
  5. Normal Miura equivalence
  6. Generalizations and approach to the proof
  7. Structure of the paper
  8. Acknowledgments
  9. A further development
  10. The A=B relation in the Gorenstein quotient
  11. Basic notation for trees
  12. Trees and strata
  13. Leveled stable rooted trees
  14. Definition of the B class
  15. The conjectural A=B relation revisited
  16. ...and 19 more sections

Key Result

Theorem 1.2

Conjecture conj:main holds.

Theorems & Definitions (43)

  • Conjecture 1.1: Strong DR/DZ equivalence conjecture
  • Theorem 1.2
  • Remark 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • Conjecture 2.6: generalized $A=B$ relations
  • Lemma 2.7
  • proof
  • ...and 33 more