Communications in Mathematical Physics

, Volume 363, Issue 1, pp 191–260 | Cite as

Tau-Structure for the Double Ramification Hierarchies

  • Alexandr Buryak
  • Boris Dubrovin
  • Jérémy Guéré
  • Paolo RossiEmail author


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 partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton, and divisor equations plus some important degree constraints). We then formulate a stronger version of the conjecture from Buryak (2015): for any semisimple cohomological field theory, the Dubrovin–Zhang and double ramification hierarchies are related by a normal [i.e. preserving the tau-structure (Dubrovin et al. in Adv Math 293:382–435, 2016)] Miura transformation which we completely identify in terms of the partition function of the CohFT. In fact, using only the partition functions, the conjecture can be formulated even in the non-semisimple case (where the Dubrovin–Zhang hierarchy is not defined). We then prove this conjecture for various CohFTs (trivial CohFT, Hodge class, Gromov–Witten theory of \({\mathbb{CP}^1}\), 3-, 4- and 5-spin classes) and in genus 1 for any semisimple CohFT. Finally we prove that the higher genus part of the DR hierarchy is basically trivial for the Gromov–Witten theory of smooth varieties with non-positive first Chern class and their analogue in Fan–Jarvis–Ruan–Witten quantum singularity theory (Fan et al. in Ann Math 178(1):1–106, 2013).


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We would like to thank Andrea Brini, Guido Carlet, Rahul Pandharipande, Sergey Shadrin, and Dimitri Zvonkine for useful discussions. A. B. was supported by Grant ERC-2012-AdG-320368-MCSK in the group of R. Pandharipande at ETH Zurich, Grant RFFI-16-01-00409 and the Marie Curie Fellowship (Project ID 797635). B. D. was partially supported by PRIN 2010-11 Grant “Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions” of Italian Ministry of Universities and Researches. J. G. was supported by the Einstein foundation. P. R. was partially supported by a Chaire CNRS/Enseignement superieur 2012-2017 Grant. Part of the work was completed during the visits of B. D. and P. R to the Forschungsinstitut für Mathematik at ETH Zürich in 2014 and 2015.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsUK
  2. 2.SISSATriesteItaly
  3. 3.Univ. Grenoble Alpes, CNRS, Institut FourierGrenobleFrance
  4. 4.IMB, UMR5584 CNRSUniversité de Bourgogne Franche-ComtéDijonFrance

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