Double ramification cycles and quantum integrable systems

International audienceIn 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