Abstract
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.
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References
Buryak A.: Dubrovin–Zhang hierarchy for the Hodge integrals. Commun. Number Theory Phys. 9(2), 239–271 (2015)
Buryak A.: Double ramification cycles and integrable hierarchies. Commun. Math. Phys. 336(3), 1085–1107 (2015)
Buryak, A., Rossi, P.: Recursion relations for double ramification hierarchies. Commun. Math. Phys. (2014). arXiv:1411.6797
Buryak A., Shadrin S., Spitz L., Zvonkine D.: Integrals of psi-classes over double ramification cycles. Am. J. Math. 137(3), 699–737 (2015)
Carlet G., Dubrovin B., Zhang Y.: The extended Toda hierarchy. Mosc. Math. J. 4(2), 313–332 (2004)
Cavalieri R., Marcus S., Wise J.: Polynomial families of tautological classes on \({\mathcal{M}_{g,n}^{rt}}\). J. Pure Appl. Algebra 216(4), 950–981 (2012)
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
Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory, GAFA 2000 Visions in Mathematics special volume, part II, pp. 560–673 (2000)
Fabert, O., Rossi, P.: String, dilaton and divisor equation in symplectic field theory. Int. Math. Res. Not. 19, 4384–4404 (2011)
Ionel E.-N.: Topological recursive relations in \({H^{2g}(\overline{\mathcal{M}}_{g,n})}\). Invent. Math. 148(3), 627–658 (2002)
Pandharipande R., Pixton A., Zvonkine D.: Relations on \({\overline{\mathcal{M}}_{g,n}}\) via 3-spin structures. J. Am. Math. Soc. 28(1), 279–309 (2015)
Rossi P.: Gromov-Witten invariants of target curves via symplectic field theory. J. Geom. Phys. 58(8), 931–941 (2008)
Rossi, P.: Integrable systems and holomorphic curves. In: Proceedings of the Gökova Geometry-topology conference 2009, pp. 34–57. Int. Press, Somerville (2010)
Zvonkine, D.: Intersection of double loci with boundary strata (2015, unpublished note)
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Buryak, A., Rossi, P. Double Ramification Cycles and Quantum Integrable Systems. Lett Math Phys 106, 289–317 (2016). https://doi.org/10.1007/s11005-015-0814-6
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DOI: https://doi.org/10.1007/s11005-015-0814-6