Letters in Mathematical Physics

, Volume 106, Issue 3, pp 289–317 | Cite as

Double Ramification Cycles and Quantum Integrable Systems

  • Alexandr Buryak
  • Paolo RossiEmail author


In this paper, we define a quantization of the Double Ramification Hierarchies of Buryak (Commun Math Phys 336:1085–1107, 2015) and Buryak and Rossi (Commun Math Phys, 2014), using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with the first descendant of the unit of the cohomological field theory only. We study various examples which provide, in very explicit form, new (1+1)-dimensional integrable quantum field theories whose classical limits are well-known integrable hierarchies such as KdV, Intermediate Long Wave, extended Toda, etc. Finally, we prove polynomiality in the ramification multiplicities of the integral of any tautological class over the double ramification cycle.


moduli space of curves cohomological field theories quantum integrable systems double ramification cycle 

Mathematics Subject Classification

14H10 14H70 81T40 36K10 81R12 


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsETH ZurichZurichSwitzerland
  2. 2.IMB, UMR 5584 CNRS, Université de Bourgogne 9Dijon CedexFrance

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