Communications in Mathematical Physics

, Volume 342, Issue 2, pp 533–568 | Cite as

Recursion Relations for Double Ramification Hierarchies

  • Alexandr BuryakEmail author
  • Paolo Rossi


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 intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the Hamiltonians, the one associated to the first descendant of the unit of a cohomological field theory. Moreover, we introduce analogues of the topological recursion relations and the divisor equation both for the Hamiltonian densities and for the string solution of the double ramification hierarchy. This machinery is very efficient and we apply it to various computations for the trivial and Hodge cohomological field theories, and for the r -spin Witten’s classes. Moreover, we prove the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line (extended Toda hierarchy).


Recursion Relation Hamiltonian Operator String Solution Local Functional Hamiltonian Density 
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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.IMB, UMR 5584 CNRS, Université de BourgogneDijon CedexFrance

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