Double Ramification Cycles and Quantum Integrable Systems
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.
Keywordsmoduli space of curves cohomological field theories quantum integrable systems double ramification cycle
Mathematics Subject Classification14H10 14H70 81T40 36K10 81R12
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- 3.Buryak, A., Rossi, P.: Recursion relations for double ramification hierarchies. Commun. Math. Phys. (2014). arXiv:1411.6797
- 7.Dubrovin, B.A., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, p. 295. (a new 2005 version of). arXiv:math/0108160v1
- 8.Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory, GAFA 2000 Visions in Mathematics special volume, part II, pp. 560–673 (2000)Google Scholar
- 13.Rossi, P.: Integrable systems and holomorphic curves. In: Proceedings of the Gökova Geometry-topology conference 2009, pp. 34–57. Int. Press, Somerville (2010)Google Scholar
- 14.Zvonkine, D.: Intersection of double loci with boundary strata (2015, unpublished note)Google Scholar